Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2013 Article ID 193697 11 pageshttpdxdoiorg1011552013193697

Research ArticleQR-Submanifolds of (119901 minus 1) QR-Dimension in a QuaternionicProjective Space QP(119899+119901)4 under Some Curvature Conditions

Hyang Sook Kim1 and Jin Suk Pak2

1 Department of Applied Mathematics Institute of Basic Science Inje University Gimhae 621-749 Republic of Korea2 Kyungpook National University Daegu 702-701 Republic of Korea

Correspondence should be addressed to Hyang Sook Kim mathkiminjeackr

Received 4 March 2013 Accepted 9 May 2013

Academic Editor Luc Vrancken

Copyright copy 2013 H S Kim and J S Pak This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The purpose of this paper is to study n-dimensional QR-submanifolds of (119901 minus 1) QR-dimension in a quaternionic projective spaceQP(119899+119901)4 and especially to determine such submanifolds under some curvature conditions

1 Introduction

Let 119872 be a connected real 119899-dimensional submanifold ofreal codimension 119901 of a quaternionic Kahler manifold 119872

with quaternionic Kahler structure 119865 119866119867 If there existsan 119903-dimensional normal distribution ] of the normal bundle119879119872perp such that

119865]119909 sub ]119909 119866]119909 sub ]119909 119867]119909 sub ]119909

119865]perp

119909 sub 119879119909119872 119866]perp

119909 sub 119879119909119872 119867]perp

119909 sub 119879119909119872(1)

at each point 119909 in 119872 then 119872 is called a QR-submanifoldof 119903 QR-dimension where ]perp denotes the complementaryorthogonal distribution to ] in 119879119872perp (cf [1ndash3]) Real hyper-surfaces which are typical examples of QR-submanifold with119903 = 0 have been investigated by many authors (cf [2ndash9]) inconnection with the shape operator and the induced almostcontact 3-structure (for definition see [10ndash13]) In their paper[2 3] Kwon and Pak had studied QR-submanifolds of (119901 minus1) QR-dimension isometrically immersed in a quaternionicprojective space QP(119899+119901)4 and proved the following theoremas a quaternionic analogy to theorems given in [14 15] whichare natural extensions of theorems proved in [6] to the case

of QR-submanifolds with (119901 minus 1) QR-dimension and alsoextensions of theorems in [16]

Theorem K-P Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension isometrically immersed in a quater-nionic projective space119876119875(119899+119901)4 and let the normal vector field1198731 be parallel with respect to the normal connection If theshape operator 1198601 corresponding to1198731 satisfies

1198601120601 = 1206011198601 1198601120595 = 1205951198601 1198601120579 = 1205791198601 (2)

then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)On the other hand when 119872 is a real hypersurface

of QP(119899+119901)4 if 120587minus1(119872) is (1) an Einstein space or (2) alocally symmetric space then 120587minus1(119872) has a parallel secondfundamental form (cf [4 6 7 9]) Projecting the quantitieson 120587minus1(119872) onto 119872 in QP(119899+119901)4 we can consider QR-submanifolds of (119901 minus 1) QR-dimension with the conditionscorresponding to (1) or (2) In this paper we will study suchQR-submanifolds isometrically immersed in QP(119899+119901)4 andobtainTheorem 3 and other results stated in the last Section 5as quaternionic analogies to theorems given in [16 17] and asthe extensions of theorems given in [18] by using TheoremK-P

2 International Journal of Mathematics and Mathematical Sciences

2 Preliminaries

Let 119872 be a real (119899 + 119901)-dimensional quaternionic KahlermanifoldThen by definition there is a 3-dimensional vectorbundle 119881 consisting of tensor fields of type (1 1) over 119872satisfying the following conditions (a) (b) and (c)

(a) In any coordinate neighborhood U there is a localbasis 119865 119866119867 of 119881 such that

1198652= minus119868 119866

2= minus119868 119867

2= minus119868

119865119866 = minus119866119865 = 119867 119866119867 = minus119867119866 = 119865

119867119865 = minus119865119867 = 119866

(3)

(b) There is a Riemannianmetric119892which isHermitewithrespect to all of 119865 119866 and119867

(c) For the Riemannian connection nabla with respect to 119892

(

nabla119865

nabla119866

nabla119867

) = (

0 119903 minus119902

minus119903 0 119901

119902 minus119901 0

)(

119865

119866

119867

) (4)

where 119901 119902 and 119903 are local 1-forms defined inU Sucha local basis 119865 119866119867 is called a canonical local basisof the bundle 119881 inU (cf [10 19 20])

For canonical local bases 119865 119866119867 and 1015840119865 1015840119866 1015840119867 of 119881in coordinate neighborhoodsU and

1015840U it follows that inUcap

1015840U

(

101584011986510158401198661015840119867

) = (119904119909119910)(

119865

119866

119867

) (119909 119910 = 1 2 3) (5)

where 119904119909119910 are local differentiable functionswith (119904119909119910) isin SO(3)as a consequence of (3) As is well known (cf [19]) everyquaternionic Kahler manifold is orientable

Now let119872 be an 119899-dimensional QR-submanifold of (119901 minus1) QR-dimension isometrically immersed in 119872 Then bydefinition there is a unit normal vector field 119873 such that]perp119909 = Span119873 at each point 119909 in119872 We set

119880 = minus119865119873 119881 = minus119866119873 119882 = minus119867119873 (6)

Denoting by D119909 the maximal quaternionic invariant sub-space 119879119909119872 cap 119865119879119909119872 cap 119866119879119909119872 cap 119867119879119909119872 of 119879119909119872 we haveDperp119909 = Span119880 119881119882 where Dperp119909 means the complementaryorthogonal subspace toD119909 in 119879119909119872 (cf [1ndash3]) Thus we have

119879119909119872 = D119909 oplus Span 119880 119881119882 119909 isin 119872 (7)

which together with (3) and (6) implies

119865119879119909119872 119866119879119909119872 119867119879119909119872 sub 119879119909119872oplus Span 119873 (8)

Therefore for any tangent vector field 119883 and for a localorthonormal basis 119873120572120572=1119901 (1198731 = 119873) of normal vectorsto119872 we have

119865119883 = 120601119883 + 119906 (119883)119873

119866119883 = 120595119883 + V (119883)119873

119867119883 = 120579119883 + 119908 (119883)119873

(9)

119865119873120572 = minus119880120572 + 1198751119873120572

119866119873120572 = minus119881120572 + 1198752119873120572

119867119873120572 = minus119882120572 + 1198753119873120572

(10)

(120572 = 1 119901) Then it is easily seen that 120601 120595 120579 and1198751 1198752 1198753 are skew-symmetric endomorphisms acting on119879119909119872 and119879119909119872

perp respectivelyMoreover the Hermitian prop-erty of 119865 119866119867 implies

119892 (119883 120601119880120572) = minus119906 (119883) 119892 (1198731 1198751119873120572)

119892 (119883 120595119881120572) = minusV (119883) 119892 (1198731 1198752119873120572) 120572 = 1 119901

119892 (119883 120579119882120572) = minus119908 (119883) 119892 (1198731 1198753119873120572)

(11)

119892 (119880120572 119880120573) = 120575120572120573 minus 119892 (1198751119873120572 1198751119873120573)

119892 (119881120572 119881120573) = 120575120572120573 minus 119892 (1198752119873120572 1198752119873120573) 120572 120573 = 1 119901

119892 (119882120572119882120573) = 120575120572120573 minus 119892 (1198753119873120572 1198753119873120573)

(12)

Also from the hermitian properties 119892(119865119883119873120572) =

minus119892(119883 119865119873120572) 119892(119866119883119873120572) = minus119892(119883 119866119873120572) and 119892(119867119883119873120572) =

minus119892(119883119867119873120572) it follows that

119892 (119883119880120572) = 119906 (119883) 1205751120572 119892 (119883 119881120572) = V (119883) 1205751120572

119892 (119883119882120572) = 119908 (119883) 1205751120572(13)

and hence

119892 (1198801 119883) = 119906 (119883) 119892 (1198811 119883) = V (119883)

119892 (1198821 119883) = 119908 (119883) 119880120572 = 0

119881120572 = 0 119882120572 = 0 120572 = 2 119901

(14)

On the other hand comparing (6) and (10) with 120572 = 1 wehave 1198801 = 119880 1198811 = 119881 and1198821 = 119882 which together with (6)and (14) implies

119892 (119880119883) = 119906 (119883) 119892 (119881119883) = V (119883)

119892 (119882119883) = 119908 (119883) 119906 (119880) = 1 V (119881) = 1 119908 (119882) = 1

(15)

In the sequel we will use the notations 119880 119881 and119882 insteadof 1198801 1198811 and1198821

Next applying119865 to the first equation of (9) and using (10)(14) and (15) we have

1206012119883 = minus119883 + 119906 (119883)119880 119906 (119883) 1198751119873 = minus119906 (120601119883)119873 (16)

International Journal of Mathematics and Mathematical Sciences 3

Similarly we have

1206012119883 = minus119883 + 119906 (119883)119880 120595

2119883 = minus119883 + V (119883)119881

1205792119883 = minus119883 + 119908 (119883)119882

(17)

119906 (119883) 1198751119873 = minus119906 (120601119883)119873 V (119883) 1198752119873 = minusV (120595119883)119873

119908 (119883) 1198753119873 = minus119908 (120579119883)119873

(18)

from which taking account of the skew symmetry of 1198751 1198752and 1198753 and using (11) we also have

119906 (120601119883) = 0 V (120595119883) = 0 119908 (120579119883) = 0

120601119880 = 0 120595119881 = 0 120579119882 = 0 1198751119873 = 0

1198752119873 = 0 1198753119873 = 0

(19)

So (10) can be rewritten in the form

119865119873 = minus119880 119866119873 = minus119881 119867119873 = minus119882

119865119873120572 = 1198751119873120572 119866119873120572 = 1198752119873120572 119867119873120572 = 1198753119873120572

(20)

(120572 = 2 119901) Applying 119866 and119867 to the first equation of (9)and using (3) (9) and (20) we have

120579119883 + 119908 (119883)119873 = minus120595 (120601119883) minus V (120601119883)119873 + 119906 (119883)119881

120595119883 + V (119883)119873 = 120579 (120601119883) + 119908 (120601119883)119873 minus 119906 (119883)119882(21)

and consequently

120595 (120601119883) = minus120579119883 + 119906 (119883)119881 V (120601119883) = minus119908 (119883)

120579 (120601119883) = 120595119883 + 119906 (119883)119882 119908 (120601119883) = V (119883) (22)

Similarly the other equations of (9) yield

120601 (120595119883) = 120579119883 + V (119883)119880 119906 (120595119883) = 119908 (119883)

120579 (120595119883) = minus120601119883 + V (119883)119882 119908 (120595119883) = minus119906 (119883)

120601 (120579119883) = minus120595119883 + 119908 (119883)119880 119906 (120579119883) = minusV (119883)

120595 (120579119883) = 120601119883 + 119908 (119883)119881 V (120579119883) = 119906 (119883)

(23)

From the first three equations of (20) we also have

120595119880 = minus119882 V (119880) = 0 120579119880 = 119881

119908 (119880) = 0 120601119881 = 119882 119906 (119881) = 0

120579119881 = minus119880 119908 (119881) = 0 120601119882 = minus119881

119906 (119882) = 0 120595119882 = 119880 V (119882) = 0

(24)

Equations (14)ndash(17) (19) and (22)ndash(24) tell us that 119872admits the so-called almost contact 3-structure and conse-quently 119899 = 4119898 + 3 for some integer119898 (cf [12])

Now let nabla be the Levi-Civita connection on 119872 and letnablaperp be the normal connection induced from nabla in the normal

bundle of119872Then Gauss andWeingarten formulae are givenby

nabla119883119884 = nabla119883119884 + ℎ (119883 119884) (25)

nabla119883119873120572 = minus119860120572119883 + nablaperp

119883119873120572 120572 = 1 119901 (26)

for 119883 119884 tangent to 119872 Here ℎ denotes the second fun-damental form and 119860120572 the shape operator correspondingto 119873120572 They are related by ℎ(119883 119884) = sum

119901

120572=1 119892(119860120572119883119884)119873120572Furthermore put

nablaperp

119883119873120572 =

119901

sum120573=1

119904120572120573 (119883)119873120573 (27)

where (119904120572120573) is the skew-symmetric matrix of connectionforms of nablaperp

Differentiating the first equation of (9) covariantly andusing (4) (9) (10) (14) (25) and (26) we have

(nabla119884120601)119883 = 119903 (119884) 120595119883 minus 119902 (119884) 120579119883 + 119906 (119883)1198601119884

minus 119892 (1198601119884119883)119880

(nabla119884119906)119883 = 119903 (119884) V (119883) minus 119902 (119884)119908 (119883) + 119892 (1206011198601119884119883)

(28)

From the other equations of (9) we also have

(nabla119884120595)119883 = minus 119903 (119884) 120601119883 + 119901 (119884) 120579119883 + V (119883)1198601119884

minus 119892 (1198601119884119883)119881

(nabla119884V)119883 = minus119903 (119884) 119906 (119883) + 119901 (119884)119908 (119883) + 119892 (1205951198601119884119883)

(nabla119884120579)119883 = 119902 (119884) 120601119883 minus 119901 (119884)120595119883 + 119908 (119883)1198601119884

minus 119892 (1198601119884119883)119882

(nabla119884119908)119883 = 119902 (119884) 119906 (119883) minus 119901 (119884) V (119883) + 119892 (1205791198601119884119883)

(29)

Next differentiating the first equation of (20) covariantlyand comparing the tangential and normal parts we have

nabla119884119880 = 119903 (119884)119881 minus 119902 (119884)119882 + 1206011198601119884

119892 (119860120572119880119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198751120573120572 120572 = 2 119901(30)

From the other equations of (20) we have similarly

nabla119884119881 = minus119903 (119884)119880 + 119901 (119884)119882 + 1205951198601119884

119892 (119860120572119881119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198752120573120572 120572 = 2 119901

nabla119884119882 = 119902 (119884)119880 minus 119901 (119884)119881 + 1205791198601119884

119892 (119860120572119882119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198753120573120572 120572 = 2 119901

(31)

4 International Journal of Mathematics and Mathematical Sciences

Finally the equation of Gauss is given as follows (cf [21])

119892 (119877 (119883 119884)119885119882)

= 119892 (119877 (119883 119884)119885119882)

+sum120572

119892 (119860120572119883119885) 119892 (119860120572119884119882)

minus119892 (119860120572119884119885) 119892 (119860120572119883119882)

(32)

for 119883119884 and 119885 tangent to 119872 where 119877 and 119877 denote theRiemannian curvature tensor of119872 and119872 respectively

In the rest of this paper we assume that the distinguishednormal vector field 1198731 = 119873 is parallel with respect to thenormal connection nablaperp Then it follows from (27) that 1199041120573 = 0and consequently (30)-(31) imply

119860120572119880 = 0 119860120572119881 = 0 119860120572119882 = 0 120572 = 2 119901

(33)

On the other hand since the curvature tensor 119877 ofQP(119899+119901)4 is of the form

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883119885)119884

+ 119892 (119865119884 119885) 119865119883 minus 119892 (119865119883119885)119865119884

minus 2119892 (119865119883 119884) 119865119885 + 119892 (119866119884119885)119866119883

minus 119892 (119866119883119885)119866119884 minus 2119892 (119866119883 119884)119866119885

+ 119892 (119867119884119885)119867119883 minus 119892 (119867119883119885)119867119884

minus 2119892 (119867119883119884)119867119885

(34)

for119883119884 and 119885 tangent to QP(119899+119901)4 (32) reduces to

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883 119885) 119884

+ 119892 (120601119884 119885) 120601119883 minus 119892 (120601119883119885) 120601119884

minus 2119892 (120601119883 119884) 120601119885 + 119892 (120595119884 119885)120595119883

minus 119892 (120595119883119885)120595119884 minus 2119892 (120595119883 119884)120595119885

+ 119892 (120579119884 119885) 120579119883 minus 119892 (120579119883 119885) 120579119884

minus 2119892 (120579119883 119884) 120579119885

+sum120572

119892 (119860120572119884119885)119860120572119883 minus 119892 (119860120572119883119885)119860120572119884

(35)

3 Fibrations and Immersions

Fromnowon 119899-dimensionalQR-submanifolds of (119901minus1) QR-dimension isometrically immersed in QP(119899+119901)4 only will beconsidered Moreover we will use the assumption and thenotations as in Section 2

Let 119878119899+119901+3(119886) be the hypersphere of radius 119886 (gt0) in119876(119899+119901+4)4 the quaternionic space of quaternionic dimension

(119899 + 119901 + 4)4 which is identified with the Euclidean (119899 +

119901 + 4)-spaceR119899+119901+4 The unit sphere 119878119899+119901+3(1) will be brieflydenoted by 119878119899+119901+3 Let 119878119899+119901+3 rarr QP(119899+119901)4 be thenatural projection of 119878119899+119901+3 onto QP(119899+119901)4 defined by theHopf fibration 1198783 rarr 119878

119899+119901+3rarr QP(119899+119901)4 As is well known

(cf [10 11 20]) 119878119899+119901+3 admits a Sasakian 3-structure whereby120585 120578 and 120577 are mutually orthogonal unit Killing vector fieldsThus it follows that

nabla120585120585 = 0 nabla120578120578 = 0 nabla

120577120577 = 0

nabla120577120578 = minusnabla120578120577 = 120585 nabla

120585120577 = minusnabla

120577120585 = 120578

nabla120578120585 = minusnabla120585120578 = 120577

(36)

where nabla denotes the Riemannian connection with respect tothe canonical metric 119892 on 119878119899+119901+3 (cf [6 9ndash13]) Moreovereach fibre minus1(119909) of 119909 in QP(119899+119901)4 is a maximal integralsubmanifold of the distribution spanned by 120585 120578 and 120577 Thusthe base space QP(119899+119901)4 admits the induced quaternionicKahler structure of constant 119876-sectional curvature 4 (cf[10 11]) We have especially a fibration 120587 120587

minus1(119872) rarr

119872 which is compatible with the Hopf fibration Moreprecisely speaking 120587 120587

minus1(119872) rarr 119872 is a fibration withtotally geodesic fibers such that the following diagram iscommutative

120587minus1(M) S

n+p+3

M QP(n+p)4i

i998400

120587 (37)

where 1198941015840 120587minus1(119872) rarr 119878119899+119901+3 and 119894 119872 rarr QP(119899+119901)4 areisometric immersions

Now let 120585 120578 and 120577 be the unit vector fields tangent to thefibers of 120587minus1(119872) such that 1198941015840lowast120585 = 120585 1198941015840lowast120578 = 120578 and 1198941015840lowast120577 = 120577(In what follows we will again delete the 1198941015840 and 1198941015840lowast in ournotation) Furthermore we denote by 119883lowast the horizontal liftof a vector field 119883 tangent to 119872 Then the horizontal lifts119873lowast120572 (120572 = 1 119901) of the normal vectors 119873120572 to 119872 form an

orthonormal basis of normal vectors to 120587minus1(119872) in 119878119899+119901+3 Let1198601015840120572 and 119904

1015840120572120573 be the corresponding shape operators and normal

connection forms respectively Then as shown in [3 9 22]the fundamental equations for the submersion 120587 are given by

1015840nabla119883lowast119884

lowast= (nabla119883119884)

lowast+ 1198921015840((120601119883)

lowast 119884lowast) 120585 + 119892

1015840((120595119883)

lowast 119884lowast) 120578

+ 1198921015840((120579119883)

lowast 119884lowast) 120577

(38)

International Journal of Mathematics and Mathematical Sciences 5

[119883lowast 119884lowast] = [119883 119884]

lowast+ 21198921015840((120601119883)

lowast 119884lowast) 120585

+ 21198921015840((120595119883)

lowast 119884lowast) 120578 + 2119892

1015840((120579119883)

lowast 119884lowast) 120577

(39)

1015840nabla119883lowast120585 =

1015840nabla120585119883lowast= minus(120601119883)

lowast

1015840nabla119883lowast120578 =

1015840nabla120578119883lowast= minus(120595119883)

lowast

1015840nabla119883lowast120577 =

1015840nabla120577119883lowast= minus(120579119883)

lowast

(40)

[119883lowast 120585] = 0 [119883

lowast 120578] = 0 [119883

lowast 120577] = 0 (41)

where 1198921015840 denotes the Riemannian metric of 120587minus1(119872) inducedfrom 119892 in 119878119899+119901+3 and 1015840nabla the Levi-Civita connection withrespect to1198921015840The same equations are valid for the submersion by replacing 120601 120595 and 120579 (resp 120585 120578 and 120577) with 119865119866 and119867(resp 120585 120578 and 120577) respectivelyWe denote by 1015840nabla

perp the normalconnection of 120587minus1(119872) induced from nabla Since the diagram iscommutative nabla119883lowast119873

lowast120572 implies

1015840nablaperp

119883lowast119873lowast

120572 minus 1198601015840

120572119883lowast= (nabla119883119873120572)

lowast+ 119892 ((119865119883)

lowast 119873lowast

120572 ) 120585

+ 119892 ((119866119883)lowast 119873lowast

120572 ) 120578 + 119892 ((119867119883)lowast 119873lowast

120572 ) 120577

= minus (119860120572119883)lowast+ 119892(119880120572 119883)

lowast120585 + 119892(119881120572 119883)

lowast120578

+ 119892(119882120572 119883)lowast120577 + (nabla

perp

119883119873120572)lowast

(42)

because of (10) (26) and (38) from which comparing thetangential part we have

1198601015840

120572119883lowast= (119860120572119883)

lowastminus 119892(119880120572 119883)

lowast120585

minus 119892(119881120572 119883)lowast120578 minus 119892(119882120572 119883)

lowast120577

(43)

Next calculating nabla120585119873lowast120572 and using (10) (26) and (40) we have

1015840nablaperp

120585119873lowast

120572 minus 1198601015840

120572120585 = minus(119865119873120572)lowast= 119880lowast

120572 minus (1198751119873120572)lowast (44)

which yields

1198601015840

120572120585 = minus119880lowast

120572(45)

and similarly

1198601015840

120572120585 = minus119880lowast

120572 1198601015840

120572120578 = minus119881lowast

120572 1198601015840

120572120577 = minus119882lowast

120572 (46)

Hence (43) and (46) with 120572 = 1 imply

1198601015840

1119883lowast= (1198601119883)

lowastminus 119892(119880119883)

lowast120585 minus 119892(119881119883)

lowast120578 minus 119892(119882119883)

lowast120577

1198601015840

1120585 = minus119880lowast 119860

1015840

1120578 = minus119881lowast 119860

1015840

1120577 = minus119882lowast

(47)

4 Co-Gauss Equations for the Submersion120587120587minus1(119872) rarr 119872

In this section we derive the co-Gauss and co-Codazziequations of the submersion 120587 120587minus1(119872) rarr 119872 for later use

Differentiating (38) with119884 = 119880 covariantly along120587minus1(119872)

and using (24) (38) and (39) we have

1015840nabla119884lowast1015840nabla119883lowast119880

lowast

= (nabla119884nabla119883119880)lowast+ V (119883) 120579119884 minus 119908 (119883)120595119884

lowast

+ 119892(120601119884 nabla119883119880)lowast120585

+ 119892 (120595119884 nabla119883119880) + 119892 (nabla119884119883119882) + 119892 (119883 nabla119884119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (nabla119884119883119881) minus 119892 (119883 nabla119884119881)lowast120577

(48)

Similarly (38) with 119884 = 119881 and (38) with 119884 = 119882 give

1015840nabla119884lowast1015840nabla119883lowast119881

lowast

= (nabla119884nabla119883119881)lowast+ 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892 (120601119884 nabla119883119881) minus 119892 (nabla119884119883119882) minus 119892 (119883 nabla119884119882)lowast120585

+ 119892(120595119884 nabla119883119881)lowast120578

+ 119892 (120579119884 nabla119883119881) + 119892 (nabla119884119883119880) + 119892 (119883 nabla119884119880)lowast120577

(49)1015840nabla119884lowast1015840nabla119883lowast119882

lowast

= (nabla119884nabla119883119882)lowast

minus V (119883) 120601119884 minus 119906 (119883)120595119884lowast

+ 119892 (120601119884 nabla119883119882) + 119892 (nabla119884119883119881) + 119892 (119883 nabla119884119881)lowast120585

+ 119892 (120595119884 nabla119883119882) minus 119892 (nabla119884119883119880) minus 119892 (119883 nabla119884119880)lowast120578

+ 119892(120579119884 nabla119883119882)lowast120577

(50)

respectively On the other hand it follows from (19) (24)(38) and (39) that

1015840nabla[119884lowast 119883lowast]119880

lowast= (nabla[119884119883]119880)

lowast+ 2119892(120595119884119883)

lowast119882lowast

minus 2119892(120579119884119883)lowast119881lowast+ 119892([119884119883]119882)

lowast120578

minus 119892([119884119883] 119881)lowast120577

(51)

6 International Journal of Mathematics and Mathematical Sciences

1015840nabla[119884lowast119883lowast]119881

lowast= (nabla[119884119883]119881)

lowastminus 2119892(120601119884119883)

lowast119882lowast

+ 2119892(120579119884119883)lowast119880lowastminus 119892([119884119883] 119882)

lowast120585

+ 119892([119884119883] 119880)lowast120577

(52)

1015840nabla[119884lowast119883lowast]119882

lowast= (nabla[119884119883]119882)

lowast+ 2119892(120601119884119883)

lowast119881lowast

minus 2119892(120595119884119883)lowast119880lowast+ 119892([119884119883] 119881)

lowast120585

minus 119892([119884119883] 119880)lowast120578

(53)

By means of (48) and (51) we have

1015840119877 (119884lowast 119883lowast) 119880lowast= 119877 (119884119883)119880

lowast

+ 119908 (119884) 120595119883 minus 119908 (119883)120595119884 minus V (119884) 120579119883

+ V (119883) 120579119884 + 2119892 (120579119884119883)119881

minus 2119892 (120595119884119883)119882lowast

+ 119892 (120601119884 nabla119883119880) minus 119892 (120601119883 nabla119884119880)lowast120585

+ 119892 (120595119884 nabla119883119880) minus 119892 (120595119883 nabla119884119880)

+119892 (119883 nabla119884119882) minus 119892 (119884 nabla119883119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (120579119883 nabla119884119880)

minus119892(119883 nabla119884119881) + 119892(119884 nabla119883119881)lowast120577

(54)

where 1015840119877 denotes the curvature tensor of 120587minus1(119872) withrespect to the connection 1015840nabla Using (30) (31) (33) and (35)we can easily see that

1015840119877 (119884lowast 119883lowast) 119880lowast= 119906 (119883)119884 minus 119906 (119884)119883

+ 119906 (1198601119883)1198601119884 minus 119906 (1198601119884)1198601119883lowast

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119906 (119883) 119906 (1198601119884) minus 119906 (119884) 119906 (1198601119883)lowast120585

+ 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V (119883) 119906 (1198601119884) minus V (119884) 119906 (1198601119883)lowast120578

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908(119883)119906(1198601119884) minus 119908(119884)119906(1198601119883)lowast120577

(55)

By the same method we can easily verify that (49) (50) (52)and (53) yield

1015840119877 (119884lowast 119883lowast) 119881lowast= V (119883) 119884 minus V (119884)119883 + V (1198601119883)1198601119884

minus V (1198601119884)1198601119883lowast

+ 119902 (119883) 119906 (119884) minus 119902 (119884) 119906 (119883)

minus 119906 (119884) V (1198601119883) + 119906 (119883) V (1198601119884)lowast120585

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ V (119883) V (1198601119884) minus V (119884) V (1198601119883)lowast120578

+ 119902 (119883)119908 (119884) minus 119902 (119884)119908 (119883)

minus 119908 (119884) V (1198601119883) + 119908 (119883) V (1198601119884)lowast120577

1015840119877 (119884lowast 119883lowast)119882lowast= 119908 (119883)119884 minus 119908 (119884)119883 + 119908 (1198601119883)1198601119884

minus 119908 (1198601119884)1198601119883lowast

+ 119903 (119883) 119906 (119884) minus 119903 (119884) 119906 (119883)

minus 119906 (119884)119908 (1198601119883) + 119906 (119883)119908 (1198601119884)lowast120585

+ 119903 (119883) V (119884) minus 119903 (119884) V (119883)

minus V (119884)119908 (1198601119883) + V (119883)119908 (1198601119884)lowast120578

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ 119908(119883)119908(1198601119884) minus 119908(119884)119908(1198601119883)lowast120577

(56)

Differentiating (38)with119883 = 119880 covariantly along120587minus1(119872)

and using (24) we have

1015840nabla119884lowast1015840nabla119880lowast119883

lowast

= (nabla119884nabla119880119883)lowast+ 119908 (119883)120595119884 minus V (119883) 120579119884

lowast

+ 119892(120601119884 nabla119880119883)lowast120585

+ 119892 (120595119884 nabla119880119883) minus 119892 (nabla119884119882119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (120579119884 nabla119880119883) + 119892 (nabla119884119881119883) + 119892 (119881 nabla119884119883)lowast120577

(57)

International Journal of Mathematics and Mathematical Sciences 7

Similarly (38) with119883 = 119881 and (38) with119883 = 119882 respectivelygive

1015840nabla119884lowast1015840nabla119881lowast119883

lowast

= (nabla119884nabla119881119883)lowastminus 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892(120595119884 nabla119881119883)lowast120578

+ 119892(120601119884 nabla119881119883) + 119892(nabla119884119882119883) + 119892(119882 nabla119884119883)lowast120585

+ 119892(120579119884 nabla119881119883) minus 119892(nabla119884119880119883) minus 119892(119880 nabla119884119883)lowast120577

(58)

1015840nabla119884lowast1015840nabla119882lowast119883

lowast

= (nabla119884nabla119882119883)lowast+ V (119883) 120601119884 minus 119906 (119883)120595119884

lowast

+ 119892(120579119884 nabla119882119883)lowast120577

+ 119892 (120601119884 nabla119882119883) minus 119892 (nabla119884119881119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (120595119884 nabla119882119883) + 119892 (nabla119884119880119883) + 119892 (119880 nabla119884119883)lowast120578

(59)

Differentiating (38) also covariantly in the direction of119880lowast andusing (24) we have

1015840nabla119880lowast1015840nabla119884lowast119883

lowast

= (nabla119880nabla119884119883)lowast

+ 119892(120595119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119881lowast

+ 119892 (nabla119880 (120601119884) 119883) + 119892 (120601119884 nabla119880119883)lowast120585

+ 119892 (nabla119880 (120595119884) 119883) + 119892 (120595119884 nabla119880119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (nabla119880 (120579119884) 119883) + 119892 (120579119884 nabla119880119883) + 119892 (119881 nabla119884119883)lowast120577

(60)

Similarly differentiating (38) covariantly in the direction of119881lowast and119882lowast respectively we have

1015840nabla119881lowast1015840nabla119884lowast119883

lowast

= (nabla119881nabla119884119883)lowast

minus 119892(120601119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119880lowast

+ 119892 (nabla119881 (120601119884) 119883) + 119892 (120601119884 nabla119881119883) + 119892 (119882nabla119884119883)lowast120585

+ 119892 (nabla119881 (120595119884) 119883) + 119892 (120595119884 nabla119881119883)lowast120578

+ 119892 (nabla119881 (120579119884) 119883) + 119892 (120579119884 nabla119881119883) minus 119892 (119880 nabla119884119883)lowast120577

(61)

1015840nabla119882lowast1015840nabla119884lowast119883

lowast

= (nabla119882nabla119884119883)lowast

minus 119892(120595119884119883)lowast119880lowast+ 119892(120601119884119883)

lowast119881lowast

+ 119892 (nabla119882 (120601119884) 119883) + 119892 (120601119884 nabla119882119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (nabla119882 (120595119884) 119883) + 119892 (120595119884 nabla119882119883) + 119892 (119880 nabla119884119883)lowast120578

+ 119892 (nabla119882 (120579119884) 119883) + 119892 (120579119884 nabla119882119883)lowast120577

(62)

On the other hand (38) and (39) with119883 = 119880 imply1015840nabla[119884lowast 119880lowast]119883

lowast= (nabla[119884119880]119883)

lowastminus 2119908 (119884) 120595119883 minus V (119884) 120579119883

lowast

+ 119892(120601 [119884 119880] 119883)lowast120585 + 119892(120595 [119884119880] 119883)

lowast120578

+ 119892(120579 [119884 119880] 119883)lowast120577

(63)

Similarly from (39) with 119883 = 119881 and (39) with 119883 = 119882respectively we find that1015840nabla[119884lowast 119881lowast]119883

lowast= (nabla[119884119881]119883)

lowast+ 2119908 (119884) 120601119883 minus 119906 (119884) 120579119883

lowast

+ 119892(120601 [119884 119881] 119883)lowast120585 + 119892(120595 [119884 119881] 119883)

lowast120578

+ 119892(120579 [119884 119881] 119883)lowast120577

(64)1015840nabla[119884lowast 119882lowast]119883

lowast= (nabla[119884119882]119883)

lowastminus 2V (119884) 120601119883 minus 119906 (119884) 120595119883

lowast

+ 119892(120601 [119884119882] 119883)lowast120585 + g(120595 [119884119882] 119883)

lowast120578

+ 119892(120579 [119884119882] 119883)lowast120577

(65)

Using (28)ndash(31) it follows from (57) (60) and (63) that1015840119877 (119884lowast 119880lowast)119883lowast= 119877 (119884 119880)119883

lowast

+ 119908 (119883)120595119884 minus V (119883) 120579119884

minus 119892 (120595119884119883)119882 + 119892 (120579119884119883)119881

+ 2119908 (119884) 120595119883 minus 2V (119884) 120579119883lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(66)

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Mathematics and Mathematical Sciences

2 Preliminaries

Let 119872 be a real (119899 + 119901)-dimensional quaternionic KahlermanifoldThen by definition there is a 3-dimensional vectorbundle 119881 consisting of tensor fields of type (1 1) over 119872satisfying the following conditions (a) (b) and (c)

(a) In any coordinate neighborhood U there is a localbasis 119865 119866119867 of 119881 such that

1198652= minus119868 119866

2= minus119868 119867

2= minus119868

119865119866 = minus119866119865 = 119867 119866119867 = minus119867119866 = 119865

119867119865 = minus119865119867 = 119866

(3)

(b) There is a Riemannianmetric119892which isHermitewithrespect to all of 119865 119866 and119867

(c) For the Riemannian connection nabla with respect to 119892

(

nabla119865

nabla119866

nabla119867

) = (

0 119903 minus119902

minus119903 0 119901

119902 minus119901 0

)(

119865

119866

119867

) (4)

where 119901 119902 and 119903 are local 1-forms defined inU Sucha local basis 119865 119866119867 is called a canonical local basisof the bundle 119881 inU (cf [10 19 20])

For canonical local bases 119865 119866119867 and 1015840119865 1015840119866 1015840119867 of 119881in coordinate neighborhoodsU and

1015840U it follows that inUcap

1015840U

(

101584011986510158401198661015840119867

) = (119904119909119910)(

119865

119866

119867

) (119909 119910 = 1 2 3) (5)

where 119904119909119910 are local differentiable functionswith (119904119909119910) isin SO(3)as a consequence of (3) As is well known (cf [19]) everyquaternionic Kahler manifold is orientable

Now let119872 be an 119899-dimensional QR-submanifold of (119901 minus1) QR-dimension isometrically immersed in 119872 Then bydefinition there is a unit normal vector field 119873 such that]perp119909 = Span119873 at each point 119909 in119872 We set

119880 = minus119865119873 119881 = minus119866119873 119882 = minus119867119873 (6)

Denoting by D119909 the maximal quaternionic invariant sub-space 119879119909119872 cap 119865119879119909119872 cap 119866119879119909119872 cap 119867119879119909119872 of 119879119909119872 we haveDperp119909 = Span119880 119881119882 where Dperp119909 means the complementaryorthogonal subspace toD119909 in 119879119909119872 (cf [1ndash3]) Thus we have

119879119909119872 = D119909 oplus Span 119880 119881119882 119909 isin 119872 (7)

which together with (3) and (6) implies

119865119879119909119872 119866119879119909119872 119867119879119909119872 sub 119879119909119872oplus Span 119873 (8)

Therefore for any tangent vector field 119883 and for a localorthonormal basis 119873120572120572=1119901 (1198731 = 119873) of normal vectorsto119872 we have

119865119883 = 120601119883 + 119906 (119883)119873

119866119883 = 120595119883 + V (119883)119873

119867119883 = 120579119883 + 119908 (119883)119873

(9)

119865119873120572 = minus119880120572 + 1198751119873120572

119866119873120572 = minus119881120572 + 1198752119873120572

119867119873120572 = minus119882120572 + 1198753119873120572

(10)

(120572 = 1 119901) Then it is easily seen that 120601 120595 120579 and1198751 1198752 1198753 are skew-symmetric endomorphisms acting on119879119909119872 and119879119909119872

perp respectivelyMoreover the Hermitian prop-erty of 119865 119866119867 implies

119892 (119883 120601119880120572) = minus119906 (119883) 119892 (1198731 1198751119873120572)

119892 (119883 120595119881120572) = minusV (119883) 119892 (1198731 1198752119873120572) 120572 = 1 119901

119892 (119883 120579119882120572) = minus119908 (119883) 119892 (1198731 1198753119873120572)

(11)

119892 (119880120572 119880120573) = 120575120572120573 minus 119892 (1198751119873120572 1198751119873120573)

119892 (119881120572 119881120573) = 120575120572120573 minus 119892 (1198752119873120572 1198752119873120573) 120572 120573 = 1 119901

119892 (119882120572119882120573) = 120575120572120573 minus 119892 (1198753119873120572 1198753119873120573)

(12)

Also from the hermitian properties 119892(119865119883119873120572) =

minus119892(119883 119865119873120572) 119892(119866119883119873120572) = minus119892(119883 119866119873120572) and 119892(119867119883119873120572) =

minus119892(119883119867119873120572) it follows that

119892 (119883119880120572) = 119906 (119883) 1205751120572 119892 (119883 119881120572) = V (119883) 1205751120572

119892 (119883119882120572) = 119908 (119883) 1205751120572(13)

and hence

119892 (1198801 119883) = 119906 (119883) 119892 (1198811 119883) = V (119883)

119892 (1198821 119883) = 119908 (119883) 119880120572 = 0

119881120572 = 0 119882120572 = 0 120572 = 2 119901

(14)

On the other hand comparing (6) and (10) with 120572 = 1 wehave 1198801 = 119880 1198811 = 119881 and1198821 = 119882 which together with (6)and (14) implies

119892 (119880119883) = 119906 (119883) 119892 (119881119883) = V (119883)

119892 (119882119883) = 119908 (119883) 119906 (119880) = 1 V (119881) = 1 119908 (119882) = 1

(15)

In the sequel we will use the notations 119880 119881 and119882 insteadof 1198801 1198811 and1198821

Next applying119865 to the first equation of (9) and using (10)(14) and (15) we have

1206012119883 = minus119883 + 119906 (119883)119880 119906 (119883) 1198751119873 = minus119906 (120601119883)119873 (16)

International Journal of Mathematics and Mathematical Sciences 3

Similarly we have

1206012119883 = minus119883 + 119906 (119883)119880 120595

2119883 = minus119883 + V (119883)119881

1205792119883 = minus119883 + 119908 (119883)119882

(17)

119906 (119883) 1198751119873 = minus119906 (120601119883)119873 V (119883) 1198752119873 = minusV (120595119883)119873

119908 (119883) 1198753119873 = minus119908 (120579119883)119873

(18)

from which taking account of the skew symmetry of 1198751 1198752and 1198753 and using (11) we also have

119906 (120601119883) = 0 V (120595119883) = 0 119908 (120579119883) = 0

120601119880 = 0 120595119881 = 0 120579119882 = 0 1198751119873 = 0

1198752119873 = 0 1198753119873 = 0

(19)

So (10) can be rewritten in the form

119865119873 = minus119880 119866119873 = minus119881 119867119873 = minus119882

119865119873120572 = 1198751119873120572 119866119873120572 = 1198752119873120572 119867119873120572 = 1198753119873120572

(20)

(120572 = 2 119901) Applying 119866 and119867 to the first equation of (9)and using (3) (9) and (20) we have

120579119883 + 119908 (119883)119873 = minus120595 (120601119883) minus V (120601119883)119873 + 119906 (119883)119881

120595119883 + V (119883)119873 = 120579 (120601119883) + 119908 (120601119883)119873 minus 119906 (119883)119882(21)

and consequently

120595 (120601119883) = minus120579119883 + 119906 (119883)119881 V (120601119883) = minus119908 (119883)

120579 (120601119883) = 120595119883 + 119906 (119883)119882 119908 (120601119883) = V (119883) (22)

Similarly the other equations of (9) yield

120601 (120595119883) = 120579119883 + V (119883)119880 119906 (120595119883) = 119908 (119883)

120579 (120595119883) = minus120601119883 + V (119883)119882 119908 (120595119883) = minus119906 (119883)

120601 (120579119883) = minus120595119883 + 119908 (119883)119880 119906 (120579119883) = minusV (119883)

120595 (120579119883) = 120601119883 + 119908 (119883)119881 V (120579119883) = 119906 (119883)

(23)

From the first three equations of (20) we also have

120595119880 = minus119882 V (119880) = 0 120579119880 = 119881

119908 (119880) = 0 120601119881 = 119882 119906 (119881) = 0

120579119881 = minus119880 119908 (119881) = 0 120601119882 = minus119881

119906 (119882) = 0 120595119882 = 119880 V (119882) = 0

(24)

Equations (14)ndash(17) (19) and (22)ndash(24) tell us that 119872admits the so-called almost contact 3-structure and conse-quently 119899 = 4119898 + 3 for some integer119898 (cf [12])

Now let nabla be the Levi-Civita connection on 119872 and letnablaperp be the normal connection induced from nabla in the normal

bundle of119872Then Gauss andWeingarten formulae are givenby

nabla119883119884 = nabla119883119884 + ℎ (119883 119884) (25)

nabla119883119873120572 = minus119860120572119883 + nablaperp

119883119873120572 120572 = 1 119901 (26)

for 119883 119884 tangent to 119872 Here ℎ denotes the second fun-damental form and 119860120572 the shape operator correspondingto 119873120572 They are related by ℎ(119883 119884) = sum

119901

120572=1 119892(119860120572119883119884)119873120572Furthermore put

nablaperp

119883119873120572 =

119901

sum120573=1

119904120572120573 (119883)119873120573 (27)

where (119904120572120573) is the skew-symmetric matrix of connectionforms of nablaperp

Differentiating the first equation of (9) covariantly andusing (4) (9) (10) (14) (25) and (26) we have

(nabla119884120601)119883 = 119903 (119884) 120595119883 minus 119902 (119884) 120579119883 + 119906 (119883)1198601119884

minus 119892 (1198601119884119883)119880

(nabla119884119906)119883 = 119903 (119884) V (119883) minus 119902 (119884)119908 (119883) + 119892 (1206011198601119884119883)

(28)

From the other equations of (9) we also have

(nabla119884120595)119883 = minus 119903 (119884) 120601119883 + 119901 (119884) 120579119883 + V (119883)1198601119884

minus 119892 (1198601119884119883)119881

(nabla119884V)119883 = minus119903 (119884) 119906 (119883) + 119901 (119884)119908 (119883) + 119892 (1205951198601119884119883)

(nabla119884120579)119883 = 119902 (119884) 120601119883 minus 119901 (119884)120595119883 + 119908 (119883)1198601119884

minus 119892 (1198601119884119883)119882

(nabla119884119908)119883 = 119902 (119884) 119906 (119883) minus 119901 (119884) V (119883) + 119892 (1205791198601119884119883)

(29)

Next differentiating the first equation of (20) covariantlyand comparing the tangential and normal parts we have

nabla119884119880 = 119903 (119884)119881 minus 119902 (119884)119882 + 1206011198601119884

119892 (119860120572119880119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198751120573120572 120572 = 2 119901(30)

From the other equations of (20) we have similarly

nabla119884119881 = minus119903 (119884)119880 + 119901 (119884)119882 + 1205951198601119884

119892 (119860120572119881119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198752120573120572 120572 = 2 119901

nabla119884119882 = 119902 (119884)119880 minus 119901 (119884)119881 + 1205791198601119884

119892 (119860120572119882119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198753120573120572 120572 = 2 119901

(31)

4 International Journal of Mathematics and Mathematical Sciences

Finally the equation of Gauss is given as follows (cf [21])

119892 (119877 (119883 119884)119885119882)

= 119892 (119877 (119883 119884)119885119882)

+sum120572

119892 (119860120572119883119885) 119892 (119860120572119884119882)

minus119892 (119860120572119884119885) 119892 (119860120572119883119882)

(32)

for 119883119884 and 119885 tangent to 119872 where 119877 and 119877 denote theRiemannian curvature tensor of119872 and119872 respectively

In the rest of this paper we assume that the distinguishednormal vector field 1198731 = 119873 is parallel with respect to thenormal connection nablaperp Then it follows from (27) that 1199041120573 = 0and consequently (30)-(31) imply

119860120572119880 = 0 119860120572119881 = 0 119860120572119882 = 0 120572 = 2 119901

(33)

On the other hand since the curvature tensor 119877 ofQP(119899+119901)4 is of the form

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883119885)119884

+ 119892 (119865119884 119885) 119865119883 minus 119892 (119865119883119885)119865119884

minus 2119892 (119865119883 119884) 119865119885 + 119892 (119866119884119885)119866119883

minus 119892 (119866119883119885)119866119884 minus 2119892 (119866119883 119884)119866119885

+ 119892 (119867119884119885)119867119883 minus 119892 (119867119883119885)119867119884

minus 2119892 (119867119883119884)119867119885

(34)

for119883119884 and 119885 tangent to QP(119899+119901)4 (32) reduces to

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883 119885) 119884

+ 119892 (120601119884 119885) 120601119883 minus 119892 (120601119883119885) 120601119884

minus 2119892 (120601119883 119884) 120601119885 + 119892 (120595119884 119885)120595119883

minus 119892 (120595119883119885)120595119884 minus 2119892 (120595119883 119884)120595119885

+ 119892 (120579119884 119885) 120579119883 minus 119892 (120579119883 119885) 120579119884

minus 2119892 (120579119883 119884) 120579119885

+sum120572

119892 (119860120572119884119885)119860120572119883 minus 119892 (119860120572119883119885)119860120572119884

(35)

3 Fibrations and Immersions

Fromnowon 119899-dimensionalQR-submanifolds of (119901minus1) QR-dimension isometrically immersed in QP(119899+119901)4 only will beconsidered Moreover we will use the assumption and thenotations as in Section 2

Let 119878119899+119901+3(119886) be the hypersphere of radius 119886 (gt0) in119876(119899+119901+4)4 the quaternionic space of quaternionic dimension

(119899 + 119901 + 4)4 which is identified with the Euclidean (119899 +

119901 + 4)-spaceR119899+119901+4 The unit sphere 119878119899+119901+3(1) will be brieflydenoted by 119878119899+119901+3 Let 119878119899+119901+3 rarr QP(119899+119901)4 be thenatural projection of 119878119899+119901+3 onto QP(119899+119901)4 defined by theHopf fibration 1198783 rarr 119878

119899+119901+3rarr QP(119899+119901)4 As is well known

(cf [10 11 20]) 119878119899+119901+3 admits a Sasakian 3-structure whereby120585 120578 and 120577 are mutually orthogonal unit Killing vector fieldsThus it follows that

nabla120585120585 = 0 nabla120578120578 = 0 nabla

120577120577 = 0

nabla120577120578 = minusnabla120578120577 = 120585 nabla

120585120577 = minusnabla

120577120585 = 120578

nabla120578120585 = minusnabla120585120578 = 120577

(36)

where nabla denotes the Riemannian connection with respect tothe canonical metric 119892 on 119878119899+119901+3 (cf [6 9ndash13]) Moreovereach fibre minus1(119909) of 119909 in QP(119899+119901)4 is a maximal integralsubmanifold of the distribution spanned by 120585 120578 and 120577 Thusthe base space QP(119899+119901)4 admits the induced quaternionicKahler structure of constant 119876-sectional curvature 4 (cf[10 11]) We have especially a fibration 120587 120587

minus1(119872) rarr

119872 which is compatible with the Hopf fibration Moreprecisely speaking 120587 120587

minus1(119872) rarr 119872 is a fibration withtotally geodesic fibers such that the following diagram iscommutative

120587minus1(M) S

n+p+3

M QP(n+p)4i

i998400

120587 (37)

where 1198941015840 120587minus1(119872) rarr 119878119899+119901+3 and 119894 119872 rarr QP(119899+119901)4 areisometric immersions

Now let 120585 120578 and 120577 be the unit vector fields tangent to thefibers of 120587minus1(119872) such that 1198941015840lowast120585 = 120585 1198941015840lowast120578 = 120578 and 1198941015840lowast120577 = 120577(In what follows we will again delete the 1198941015840 and 1198941015840lowast in ournotation) Furthermore we denote by 119883lowast the horizontal liftof a vector field 119883 tangent to 119872 Then the horizontal lifts119873lowast120572 (120572 = 1 119901) of the normal vectors 119873120572 to 119872 form an

orthonormal basis of normal vectors to 120587minus1(119872) in 119878119899+119901+3 Let1198601015840120572 and 119904

1015840120572120573 be the corresponding shape operators and normal

connection forms respectively Then as shown in [3 9 22]the fundamental equations for the submersion 120587 are given by

1015840nabla119883lowast119884

lowast= (nabla119883119884)

lowast+ 1198921015840((120601119883)

lowast 119884lowast) 120585 + 119892

1015840((120595119883)

lowast 119884lowast) 120578

+ 1198921015840((120579119883)

lowast 119884lowast) 120577

(38)

International Journal of Mathematics and Mathematical Sciences 5

[119883lowast 119884lowast] = [119883 119884]

lowast+ 21198921015840((120601119883)

lowast 119884lowast) 120585

+ 21198921015840((120595119883)

lowast 119884lowast) 120578 + 2119892

1015840((120579119883)

lowast 119884lowast) 120577

(39)

1015840nabla119883lowast120585 =

1015840nabla120585119883lowast= minus(120601119883)

lowast

1015840nabla119883lowast120578 =

1015840nabla120578119883lowast= minus(120595119883)

lowast

1015840nabla119883lowast120577 =

1015840nabla120577119883lowast= minus(120579119883)

lowast

(40)

[119883lowast 120585] = 0 [119883

lowast 120578] = 0 [119883

lowast 120577] = 0 (41)

where 1198921015840 denotes the Riemannian metric of 120587minus1(119872) inducedfrom 119892 in 119878119899+119901+3 and 1015840nabla the Levi-Civita connection withrespect to1198921015840The same equations are valid for the submersion by replacing 120601 120595 and 120579 (resp 120585 120578 and 120577) with 119865119866 and119867(resp 120585 120578 and 120577) respectivelyWe denote by 1015840nabla

perp the normalconnection of 120587minus1(119872) induced from nabla Since the diagram iscommutative nabla119883lowast119873

lowast120572 implies

1015840nablaperp

119883lowast119873lowast

120572 minus 1198601015840

120572119883lowast= (nabla119883119873120572)

lowast+ 119892 ((119865119883)

lowast 119873lowast

120572 ) 120585

+ 119892 ((119866119883)lowast 119873lowast

120572 ) 120578 + 119892 ((119867119883)lowast 119873lowast

120572 ) 120577

= minus (119860120572119883)lowast+ 119892(119880120572 119883)

lowast120585 + 119892(119881120572 119883)

lowast120578

+ 119892(119882120572 119883)lowast120577 + (nabla

perp

119883119873120572)lowast

(42)

because of (10) (26) and (38) from which comparing thetangential part we have

1198601015840

120572119883lowast= (119860120572119883)

lowastminus 119892(119880120572 119883)

lowast120585

minus 119892(119881120572 119883)lowast120578 minus 119892(119882120572 119883)

lowast120577

(43)

Next calculating nabla120585119873lowast120572 and using (10) (26) and (40) we have

1015840nablaperp

120585119873lowast

120572 minus 1198601015840

120572120585 = minus(119865119873120572)lowast= 119880lowast

120572 minus (1198751119873120572)lowast (44)

which yields

1198601015840

120572120585 = minus119880lowast

120572(45)

and similarly

1198601015840

120572120585 = minus119880lowast

120572 1198601015840

120572120578 = minus119881lowast

120572 1198601015840

120572120577 = minus119882lowast

120572 (46)

Hence (43) and (46) with 120572 = 1 imply

1198601015840

1119883lowast= (1198601119883)

lowastminus 119892(119880119883)

lowast120585 minus 119892(119881119883)

lowast120578 minus 119892(119882119883)

lowast120577

1198601015840

1120585 = minus119880lowast 119860

1015840

1120578 = minus119881lowast 119860

1015840

1120577 = minus119882lowast

(47)

4 Co-Gauss Equations for the Submersion120587120587minus1(119872) rarr 119872

In this section we derive the co-Gauss and co-Codazziequations of the submersion 120587 120587minus1(119872) rarr 119872 for later use

Differentiating (38) with119884 = 119880 covariantly along120587minus1(119872)

and using (24) (38) and (39) we have

1015840nabla119884lowast1015840nabla119883lowast119880

lowast

= (nabla119884nabla119883119880)lowast+ V (119883) 120579119884 minus 119908 (119883)120595119884

lowast

+ 119892(120601119884 nabla119883119880)lowast120585

+ 119892 (120595119884 nabla119883119880) + 119892 (nabla119884119883119882) + 119892 (119883 nabla119884119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (nabla119884119883119881) minus 119892 (119883 nabla119884119881)lowast120577

(48)

Similarly (38) with 119884 = 119881 and (38) with 119884 = 119882 give

1015840nabla119884lowast1015840nabla119883lowast119881

lowast

= (nabla119884nabla119883119881)lowast+ 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892 (120601119884 nabla119883119881) minus 119892 (nabla119884119883119882) minus 119892 (119883 nabla119884119882)lowast120585

+ 119892(120595119884 nabla119883119881)lowast120578

+ 119892 (120579119884 nabla119883119881) + 119892 (nabla119884119883119880) + 119892 (119883 nabla119884119880)lowast120577

(49)1015840nabla119884lowast1015840nabla119883lowast119882

lowast

= (nabla119884nabla119883119882)lowast

minus V (119883) 120601119884 minus 119906 (119883)120595119884lowast

+ 119892 (120601119884 nabla119883119882) + 119892 (nabla119884119883119881) + 119892 (119883 nabla119884119881)lowast120585

+ 119892 (120595119884 nabla119883119882) minus 119892 (nabla119884119883119880) minus 119892 (119883 nabla119884119880)lowast120578

+ 119892(120579119884 nabla119883119882)lowast120577

(50)

respectively On the other hand it follows from (19) (24)(38) and (39) that

1015840nabla[119884lowast 119883lowast]119880

lowast= (nabla[119884119883]119880)

lowast+ 2119892(120595119884119883)

lowast119882lowast

minus 2119892(120579119884119883)lowast119881lowast+ 119892([119884119883]119882)

lowast120578

minus 119892([119884119883] 119881)lowast120577

(51)

6 International Journal of Mathematics and Mathematical Sciences

1015840nabla[119884lowast119883lowast]119881

lowast= (nabla[119884119883]119881)

lowastminus 2119892(120601119884119883)

lowast119882lowast

+ 2119892(120579119884119883)lowast119880lowastminus 119892([119884119883] 119882)

lowast120585

+ 119892([119884119883] 119880)lowast120577

(52)

1015840nabla[119884lowast119883lowast]119882

lowast= (nabla[119884119883]119882)

lowast+ 2119892(120601119884119883)

lowast119881lowast

minus 2119892(120595119884119883)lowast119880lowast+ 119892([119884119883] 119881)

lowast120585

minus 119892([119884119883] 119880)lowast120578

(53)

By means of (48) and (51) we have

1015840119877 (119884lowast 119883lowast) 119880lowast= 119877 (119884119883)119880

lowast

+ 119908 (119884) 120595119883 minus 119908 (119883)120595119884 minus V (119884) 120579119883

+ V (119883) 120579119884 + 2119892 (120579119884119883)119881

minus 2119892 (120595119884119883)119882lowast

+ 119892 (120601119884 nabla119883119880) minus 119892 (120601119883 nabla119884119880)lowast120585

+ 119892 (120595119884 nabla119883119880) minus 119892 (120595119883 nabla119884119880)

+119892 (119883 nabla119884119882) minus 119892 (119884 nabla119883119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (120579119883 nabla119884119880)

minus119892(119883 nabla119884119881) + 119892(119884 nabla119883119881)lowast120577

(54)

where 1015840119877 denotes the curvature tensor of 120587minus1(119872) withrespect to the connection 1015840nabla Using (30) (31) (33) and (35)we can easily see that

1015840119877 (119884lowast 119883lowast) 119880lowast= 119906 (119883)119884 minus 119906 (119884)119883

+ 119906 (1198601119883)1198601119884 minus 119906 (1198601119884)1198601119883lowast

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119906 (119883) 119906 (1198601119884) minus 119906 (119884) 119906 (1198601119883)lowast120585

+ 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V (119883) 119906 (1198601119884) minus V (119884) 119906 (1198601119883)lowast120578

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908(119883)119906(1198601119884) minus 119908(119884)119906(1198601119883)lowast120577

(55)

By the same method we can easily verify that (49) (50) (52)and (53) yield

1015840119877 (119884lowast 119883lowast) 119881lowast= V (119883) 119884 minus V (119884)119883 + V (1198601119883)1198601119884

minus V (1198601119884)1198601119883lowast

+ 119902 (119883) 119906 (119884) minus 119902 (119884) 119906 (119883)

minus 119906 (119884) V (1198601119883) + 119906 (119883) V (1198601119884)lowast120585

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ V (119883) V (1198601119884) minus V (119884) V (1198601119883)lowast120578

+ 119902 (119883)119908 (119884) minus 119902 (119884)119908 (119883)

minus 119908 (119884) V (1198601119883) + 119908 (119883) V (1198601119884)lowast120577

1015840119877 (119884lowast 119883lowast)119882lowast= 119908 (119883)119884 minus 119908 (119884)119883 + 119908 (1198601119883)1198601119884

minus 119908 (1198601119884)1198601119883lowast

+ 119903 (119883) 119906 (119884) minus 119903 (119884) 119906 (119883)

minus 119906 (119884)119908 (1198601119883) + 119906 (119883)119908 (1198601119884)lowast120585

+ 119903 (119883) V (119884) minus 119903 (119884) V (119883)

minus V (119884)119908 (1198601119883) + V (119883)119908 (1198601119884)lowast120578

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ 119908(119883)119908(1198601119884) minus 119908(119884)119908(1198601119883)lowast120577

(56)

Differentiating (38)with119883 = 119880 covariantly along120587minus1(119872)

and using (24) we have

1015840nabla119884lowast1015840nabla119880lowast119883

lowast

= (nabla119884nabla119880119883)lowast+ 119908 (119883)120595119884 minus V (119883) 120579119884

lowast

+ 119892(120601119884 nabla119880119883)lowast120585

+ 119892 (120595119884 nabla119880119883) minus 119892 (nabla119884119882119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (120579119884 nabla119880119883) + 119892 (nabla119884119881119883) + 119892 (119881 nabla119884119883)lowast120577

(57)

International Journal of Mathematics and Mathematical Sciences 7

Similarly (38) with119883 = 119881 and (38) with119883 = 119882 respectivelygive

1015840nabla119884lowast1015840nabla119881lowast119883

lowast

= (nabla119884nabla119881119883)lowastminus 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892(120595119884 nabla119881119883)lowast120578

+ 119892(120601119884 nabla119881119883) + 119892(nabla119884119882119883) + 119892(119882 nabla119884119883)lowast120585

+ 119892(120579119884 nabla119881119883) minus 119892(nabla119884119880119883) minus 119892(119880 nabla119884119883)lowast120577

(58)

1015840nabla119884lowast1015840nabla119882lowast119883

lowast

= (nabla119884nabla119882119883)lowast+ V (119883) 120601119884 minus 119906 (119883)120595119884

lowast

+ 119892(120579119884 nabla119882119883)lowast120577

+ 119892 (120601119884 nabla119882119883) minus 119892 (nabla119884119881119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (120595119884 nabla119882119883) + 119892 (nabla119884119880119883) + 119892 (119880 nabla119884119883)lowast120578

(59)

Differentiating (38) also covariantly in the direction of119880lowast andusing (24) we have

1015840nabla119880lowast1015840nabla119884lowast119883

lowast

= (nabla119880nabla119884119883)lowast

+ 119892(120595119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119881lowast

+ 119892 (nabla119880 (120601119884) 119883) + 119892 (120601119884 nabla119880119883)lowast120585

+ 119892 (nabla119880 (120595119884) 119883) + 119892 (120595119884 nabla119880119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (nabla119880 (120579119884) 119883) + 119892 (120579119884 nabla119880119883) + 119892 (119881 nabla119884119883)lowast120577

(60)

Similarly differentiating (38) covariantly in the direction of119881lowast and119882lowast respectively we have

1015840nabla119881lowast1015840nabla119884lowast119883

lowast

= (nabla119881nabla119884119883)lowast

minus 119892(120601119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119880lowast

+ 119892 (nabla119881 (120601119884) 119883) + 119892 (120601119884 nabla119881119883) + 119892 (119882nabla119884119883)lowast120585

+ 119892 (nabla119881 (120595119884) 119883) + 119892 (120595119884 nabla119881119883)lowast120578

+ 119892 (nabla119881 (120579119884) 119883) + 119892 (120579119884 nabla119881119883) minus 119892 (119880 nabla119884119883)lowast120577

(61)

1015840nabla119882lowast1015840nabla119884lowast119883

lowast

= (nabla119882nabla119884119883)lowast

minus 119892(120595119884119883)lowast119880lowast+ 119892(120601119884119883)

lowast119881lowast

+ 119892 (nabla119882 (120601119884) 119883) + 119892 (120601119884 nabla119882119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (nabla119882 (120595119884) 119883) + 119892 (120595119884 nabla119882119883) + 119892 (119880 nabla119884119883)lowast120578

+ 119892 (nabla119882 (120579119884) 119883) + 119892 (120579119884 nabla119882119883)lowast120577

(62)

On the other hand (38) and (39) with119883 = 119880 imply1015840nabla[119884lowast 119880lowast]119883

lowast= (nabla[119884119880]119883)

lowastminus 2119908 (119884) 120595119883 minus V (119884) 120579119883

lowast

+ 119892(120601 [119884 119880] 119883)lowast120585 + 119892(120595 [119884119880] 119883)

lowast120578

+ 119892(120579 [119884 119880] 119883)lowast120577

(63)

Similarly from (39) with 119883 = 119881 and (39) with 119883 = 119882respectively we find that1015840nabla[119884lowast 119881lowast]119883

lowast= (nabla[119884119881]119883)

lowast+ 2119908 (119884) 120601119883 minus 119906 (119884) 120579119883

lowast

+ 119892(120601 [119884 119881] 119883)lowast120585 + 119892(120595 [119884 119881] 119883)

lowast120578

+ 119892(120579 [119884 119881] 119883)lowast120577

(64)1015840nabla[119884lowast 119882lowast]119883

lowast= (nabla[119884119882]119883)

lowastminus 2V (119884) 120601119883 minus 119906 (119884) 120595119883

lowast

+ 119892(120601 [119884119882] 119883)lowast120585 + g(120595 [119884119882] 119883)

lowast120578

+ 119892(120579 [119884119882] 119883)lowast120577

(65)

Using (28)ndash(31) it follows from (57) (60) and (63) that1015840119877 (119884lowast 119880lowast)119883lowast= 119877 (119884 119880)119883

lowast

+ 119908 (119883)120595119884 minus V (119883) 120579119884

minus 119892 (120595119884119883)119882 + 119892 (120579119884119883)119881

+ 2119908 (119884) 120595119883 minus 2V (119884) 120579119883lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(66)

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 3

Similarly we have

1206012119883 = minus119883 + 119906 (119883)119880 120595

2119883 = minus119883 + V (119883)119881

1205792119883 = minus119883 + 119908 (119883)119882

(17)

119906 (119883) 1198751119873 = minus119906 (120601119883)119873 V (119883) 1198752119873 = minusV (120595119883)119873

119908 (119883) 1198753119873 = minus119908 (120579119883)119873

(18)

from which taking account of the skew symmetry of 1198751 1198752and 1198753 and using (11) we also have

119906 (120601119883) = 0 V (120595119883) = 0 119908 (120579119883) = 0

120601119880 = 0 120595119881 = 0 120579119882 = 0 1198751119873 = 0

1198752119873 = 0 1198753119873 = 0

(19)

So (10) can be rewritten in the form

119865119873 = minus119880 119866119873 = minus119881 119867119873 = minus119882

119865119873120572 = 1198751119873120572 119866119873120572 = 1198752119873120572 119867119873120572 = 1198753119873120572

(20)

(120572 = 2 119901) Applying 119866 and119867 to the first equation of (9)and using (3) (9) and (20) we have

120579119883 + 119908 (119883)119873 = minus120595 (120601119883) minus V (120601119883)119873 + 119906 (119883)119881

120595119883 + V (119883)119873 = 120579 (120601119883) + 119908 (120601119883)119873 minus 119906 (119883)119882(21)

and consequently

120595 (120601119883) = minus120579119883 + 119906 (119883)119881 V (120601119883) = minus119908 (119883)

120579 (120601119883) = 120595119883 + 119906 (119883)119882 119908 (120601119883) = V (119883) (22)

Similarly the other equations of (9) yield

120601 (120595119883) = 120579119883 + V (119883)119880 119906 (120595119883) = 119908 (119883)

120579 (120595119883) = minus120601119883 + V (119883)119882 119908 (120595119883) = minus119906 (119883)

120601 (120579119883) = minus120595119883 + 119908 (119883)119880 119906 (120579119883) = minusV (119883)

120595 (120579119883) = 120601119883 + 119908 (119883)119881 V (120579119883) = 119906 (119883)

(23)

From the first three equations of (20) we also have

120595119880 = minus119882 V (119880) = 0 120579119880 = 119881

119908 (119880) = 0 120601119881 = 119882 119906 (119881) = 0

120579119881 = minus119880 119908 (119881) = 0 120601119882 = minus119881

119906 (119882) = 0 120595119882 = 119880 V (119882) = 0

(24)

Equations (14)ndash(17) (19) and (22)ndash(24) tell us that 119872admits the so-called almost contact 3-structure and conse-quently 119899 = 4119898 + 3 for some integer119898 (cf [12])

Now let nabla be the Levi-Civita connection on 119872 and letnablaperp be the normal connection induced from nabla in the normal

bundle of119872Then Gauss andWeingarten formulae are givenby

nabla119883119884 = nabla119883119884 + ℎ (119883 119884) (25)

nabla119883119873120572 = minus119860120572119883 + nablaperp

119883119873120572 120572 = 1 119901 (26)

for 119883 119884 tangent to 119872 Here ℎ denotes the second fun-damental form and 119860120572 the shape operator correspondingto 119873120572 They are related by ℎ(119883 119884) = sum

119901

120572=1 119892(119860120572119883119884)119873120572Furthermore put

nablaperp

119883119873120572 =

119901

sum120573=1

119904120572120573 (119883)119873120573 (27)

where (119904120572120573) is the skew-symmetric matrix of connectionforms of nablaperp

Differentiating the first equation of (9) covariantly andusing (4) (9) (10) (14) (25) and (26) we have

(nabla119884120601)119883 = 119903 (119884) 120595119883 minus 119902 (119884) 120579119883 + 119906 (119883)1198601119884

minus 119892 (1198601119884119883)119880

(nabla119884119906)119883 = 119903 (119884) V (119883) minus 119902 (119884)119908 (119883) + 119892 (1206011198601119884119883)

(28)

From the other equations of (9) we also have

(nabla119884120595)119883 = minus 119903 (119884) 120601119883 + 119901 (119884) 120579119883 + V (119883)1198601119884

minus 119892 (1198601119884119883)119881

(nabla119884V)119883 = minus119903 (119884) 119906 (119883) + 119901 (119884)119908 (119883) + 119892 (1205951198601119884119883)

(nabla119884120579)119883 = 119902 (119884) 120601119883 minus 119901 (119884)120595119883 + 119908 (119883)1198601119884

minus 119892 (1198601119884119883)119882

(nabla119884119908)119883 = 119902 (119884) 119906 (119883) minus 119901 (119884) V (119883) + 119892 (1205791198601119884119883)

(29)

Next differentiating the first equation of (20) covariantlyand comparing the tangential and normal parts we have

nabla119884119880 = 119903 (119884)119881 minus 119902 (119884)119882 + 1206011198601119884

119892 (119860120572119880119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198751120573120572 120572 = 2 119901(30)

From the other equations of (20) we have similarly

nabla119884119881 = minus119903 (119884)119880 + 119901 (119884)119882 + 1205951198601119884

119892 (119860120572119881119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198752120573120572 120572 = 2 119901

nabla119884119882 = 119902 (119884)119880 minus 119901 (119884)119881 + 1205791198601119884

119892 (119860120572119882119884) = minus

119901

sum120573=2

1199041120573 (119884) 1198753120573120572 120572 = 2 119901

(31)

4 International Journal of Mathematics and Mathematical Sciences

Finally the equation of Gauss is given as follows (cf [21])

119892 (119877 (119883 119884)119885119882)

= 119892 (119877 (119883 119884)119885119882)

+sum120572

119892 (119860120572119883119885) 119892 (119860120572119884119882)

minus119892 (119860120572119884119885) 119892 (119860120572119883119882)

(32)

for 119883119884 and 119885 tangent to 119872 where 119877 and 119877 denote theRiemannian curvature tensor of119872 and119872 respectively

In the rest of this paper we assume that the distinguishednormal vector field 1198731 = 119873 is parallel with respect to thenormal connection nablaperp Then it follows from (27) that 1199041120573 = 0and consequently (30)-(31) imply

119860120572119880 = 0 119860120572119881 = 0 119860120572119882 = 0 120572 = 2 119901

(33)

On the other hand since the curvature tensor 119877 ofQP(119899+119901)4 is of the form

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883119885)119884

+ 119892 (119865119884 119885) 119865119883 minus 119892 (119865119883119885)119865119884

minus 2119892 (119865119883 119884) 119865119885 + 119892 (119866119884119885)119866119883

minus 119892 (119866119883119885)119866119884 minus 2119892 (119866119883 119884)119866119885

+ 119892 (119867119884119885)119867119883 minus 119892 (119867119883119885)119867119884

minus 2119892 (119867119883119884)119867119885

(34)

for119883119884 and 119885 tangent to QP(119899+119901)4 (32) reduces to

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883 119885) 119884

+ 119892 (120601119884 119885) 120601119883 minus 119892 (120601119883119885) 120601119884

minus 2119892 (120601119883 119884) 120601119885 + 119892 (120595119884 119885)120595119883

minus 119892 (120595119883119885)120595119884 minus 2119892 (120595119883 119884)120595119885

+ 119892 (120579119884 119885) 120579119883 minus 119892 (120579119883 119885) 120579119884

minus 2119892 (120579119883 119884) 120579119885

+sum120572

119892 (119860120572119884119885)119860120572119883 minus 119892 (119860120572119883119885)119860120572119884

(35)

3 Fibrations and Immersions

Fromnowon 119899-dimensionalQR-submanifolds of (119901minus1) QR-dimension isometrically immersed in QP(119899+119901)4 only will beconsidered Moreover we will use the assumption and thenotations as in Section 2

Let 119878119899+119901+3(119886) be the hypersphere of radius 119886 (gt0) in119876(119899+119901+4)4 the quaternionic space of quaternionic dimension

(119899 + 119901 + 4)4 which is identified with the Euclidean (119899 +

119901 + 4)-spaceR119899+119901+4 The unit sphere 119878119899+119901+3(1) will be brieflydenoted by 119878119899+119901+3 Let 119878119899+119901+3 rarr QP(119899+119901)4 be thenatural projection of 119878119899+119901+3 onto QP(119899+119901)4 defined by theHopf fibration 1198783 rarr 119878

119899+119901+3rarr QP(119899+119901)4 As is well known

(cf [10 11 20]) 119878119899+119901+3 admits a Sasakian 3-structure whereby120585 120578 and 120577 are mutually orthogonal unit Killing vector fieldsThus it follows that

nabla120585120585 = 0 nabla120578120578 = 0 nabla

120577120577 = 0

nabla120577120578 = minusnabla120578120577 = 120585 nabla

120585120577 = minusnabla

120577120585 = 120578

nabla120578120585 = minusnabla120585120578 = 120577

(36)

where nabla denotes the Riemannian connection with respect tothe canonical metric 119892 on 119878119899+119901+3 (cf [6 9ndash13]) Moreovereach fibre minus1(119909) of 119909 in QP(119899+119901)4 is a maximal integralsubmanifold of the distribution spanned by 120585 120578 and 120577 Thusthe base space QP(119899+119901)4 admits the induced quaternionicKahler structure of constant 119876-sectional curvature 4 (cf[10 11]) We have especially a fibration 120587 120587

minus1(119872) rarr

119872 which is compatible with the Hopf fibration Moreprecisely speaking 120587 120587

minus1(119872) rarr 119872 is a fibration withtotally geodesic fibers such that the following diagram iscommutative

120587minus1(M) S

n+p+3

M QP(n+p)4i

i998400

120587 (37)

where 1198941015840 120587minus1(119872) rarr 119878119899+119901+3 and 119894 119872 rarr QP(119899+119901)4 areisometric immersions

Now let 120585 120578 and 120577 be the unit vector fields tangent to thefibers of 120587minus1(119872) such that 1198941015840lowast120585 = 120585 1198941015840lowast120578 = 120578 and 1198941015840lowast120577 = 120577(In what follows we will again delete the 1198941015840 and 1198941015840lowast in ournotation) Furthermore we denote by 119883lowast the horizontal liftof a vector field 119883 tangent to 119872 Then the horizontal lifts119873lowast120572 (120572 = 1 119901) of the normal vectors 119873120572 to 119872 form an

orthonormal basis of normal vectors to 120587minus1(119872) in 119878119899+119901+3 Let1198601015840120572 and 119904

1015840120572120573 be the corresponding shape operators and normal

connection forms respectively Then as shown in [3 9 22]the fundamental equations for the submersion 120587 are given by

1015840nabla119883lowast119884

lowast= (nabla119883119884)

lowast+ 1198921015840((120601119883)

lowast 119884lowast) 120585 + 119892

1015840((120595119883)

lowast 119884lowast) 120578

+ 1198921015840((120579119883)

lowast 119884lowast) 120577

(38)

International Journal of Mathematics and Mathematical Sciences 5

[119883lowast 119884lowast] = [119883 119884]

lowast+ 21198921015840((120601119883)

lowast 119884lowast) 120585

+ 21198921015840((120595119883)

lowast 119884lowast) 120578 + 2119892

1015840((120579119883)

lowast 119884lowast) 120577

(39)

1015840nabla119883lowast120585 =

1015840nabla120585119883lowast= minus(120601119883)

lowast

1015840nabla119883lowast120578 =

1015840nabla120578119883lowast= minus(120595119883)

lowast

1015840nabla119883lowast120577 =

1015840nabla120577119883lowast= minus(120579119883)

lowast

(40)

[119883lowast 120585] = 0 [119883

lowast 120578] = 0 [119883

lowast 120577] = 0 (41)

where 1198921015840 denotes the Riemannian metric of 120587minus1(119872) inducedfrom 119892 in 119878119899+119901+3 and 1015840nabla the Levi-Civita connection withrespect to1198921015840The same equations are valid for the submersion by replacing 120601 120595 and 120579 (resp 120585 120578 and 120577) with 119865119866 and119867(resp 120585 120578 and 120577) respectivelyWe denote by 1015840nabla

perp the normalconnection of 120587minus1(119872) induced from nabla Since the diagram iscommutative nabla119883lowast119873

lowast120572 implies

1015840nablaperp

119883lowast119873lowast

120572 minus 1198601015840

120572119883lowast= (nabla119883119873120572)

lowast+ 119892 ((119865119883)

lowast 119873lowast

120572 ) 120585

+ 119892 ((119866119883)lowast 119873lowast

120572 ) 120578 + 119892 ((119867119883)lowast 119873lowast

120572 ) 120577

= minus (119860120572119883)lowast+ 119892(119880120572 119883)

lowast120585 + 119892(119881120572 119883)

lowast120578

+ 119892(119882120572 119883)lowast120577 + (nabla

perp

119883119873120572)lowast

(42)

because of (10) (26) and (38) from which comparing thetangential part we have

1198601015840

120572119883lowast= (119860120572119883)

lowastminus 119892(119880120572 119883)

lowast120585

minus 119892(119881120572 119883)lowast120578 minus 119892(119882120572 119883)

lowast120577

(43)

Next calculating nabla120585119873lowast120572 and using (10) (26) and (40) we have

1015840nablaperp

120585119873lowast

120572 minus 1198601015840

120572120585 = minus(119865119873120572)lowast= 119880lowast

120572 minus (1198751119873120572)lowast (44)

which yields

1198601015840

120572120585 = minus119880lowast

120572(45)

and similarly

1198601015840

120572120585 = minus119880lowast

120572 1198601015840

120572120578 = minus119881lowast

120572 1198601015840

120572120577 = minus119882lowast

120572 (46)

Hence (43) and (46) with 120572 = 1 imply

1198601015840

1119883lowast= (1198601119883)

lowastminus 119892(119880119883)

lowast120585 minus 119892(119881119883)

lowast120578 minus 119892(119882119883)

lowast120577

1198601015840

1120585 = minus119880lowast 119860

1015840

1120578 = minus119881lowast 119860

1015840

1120577 = minus119882lowast

(47)

4 Co-Gauss Equations for the Submersion120587120587minus1(119872) rarr 119872

In this section we derive the co-Gauss and co-Codazziequations of the submersion 120587 120587minus1(119872) rarr 119872 for later use

Differentiating (38) with119884 = 119880 covariantly along120587minus1(119872)

and using (24) (38) and (39) we have

1015840nabla119884lowast1015840nabla119883lowast119880

lowast

= (nabla119884nabla119883119880)lowast+ V (119883) 120579119884 minus 119908 (119883)120595119884

lowast

+ 119892(120601119884 nabla119883119880)lowast120585

+ 119892 (120595119884 nabla119883119880) + 119892 (nabla119884119883119882) + 119892 (119883 nabla119884119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (nabla119884119883119881) minus 119892 (119883 nabla119884119881)lowast120577

(48)

Similarly (38) with 119884 = 119881 and (38) with 119884 = 119882 give

1015840nabla119884lowast1015840nabla119883lowast119881

lowast

= (nabla119884nabla119883119881)lowast+ 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892 (120601119884 nabla119883119881) minus 119892 (nabla119884119883119882) minus 119892 (119883 nabla119884119882)lowast120585

+ 119892(120595119884 nabla119883119881)lowast120578

+ 119892 (120579119884 nabla119883119881) + 119892 (nabla119884119883119880) + 119892 (119883 nabla119884119880)lowast120577

(49)1015840nabla119884lowast1015840nabla119883lowast119882

lowast

= (nabla119884nabla119883119882)lowast

minus V (119883) 120601119884 minus 119906 (119883)120595119884lowast

+ 119892 (120601119884 nabla119883119882) + 119892 (nabla119884119883119881) + 119892 (119883 nabla119884119881)lowast120585

+ 119892 (120595119884 nabla119883119882) minus 119892 (nabla119884119883119880) minus 119892 (119883 nabla119884119880)lowast120578

+ 119892(120579119884 nabla119883119882)lowast120577

(50)

respectively On the other hand it follows from (19) (24)(38) and (39) that

1015840nabla[119884lowast 119883lowast]119880

lowast= (nabla[119884119883]119880)

lowast+ 2119892(120595119884119883)

lowast119882lowast

minus 2119892(120579119884119883)lowast119881lowast+ 119892([119884119883]119882)

lowast120578

minus 119892([119884119883] 119881)lowast120577

(51)

6 International Journal of Mathematics and Mathematical Sciences

1015840nabla[119884lowast119883lowast]119881

lowast= (nabla[119884119883]119881)

lowastminus 2119892(120601119884119883)

lowast119882lowast

+ 2119892(120579119884119883)lowast119880lowastminus 119892([119884119883] 119882)

lowast120585

+ 119892([119884119883] 119880)lowast120577

(52)

1015840nabla[119884lowast119883lowast]119882

lowast= (nabla[119884119883]119882)

lowast+ 2119892(120601119884119883)

lowast119881lowast

minus 2119892(120595119884119883)lowast119880lowast+ 119892([119884119883] 119881)

lowast120585

minus 119892([119884119883] 119880)lowast120578

(53)

By means of (48) and (51) we have

1015840119877 (119884lowast 119883lowast) 119880lowast= 119877 (119884119883)119880

lowast

+ 119908 (119884) 120595119883 minus 119908 (119883)120595119884 minus V (119884) 120579119883

+ V (119883) 120579119884 + 2119892 (120579119884119883)119881

minus 2119892 (120595119884119883)119882lowast

+ 119892 (120601119884 nabla119883119880) minus 119892 (120601119883 nabla119884119880)lowast120585

+ 119892 (120595119884 nabla119883119880) minus 119892 (120595119883 nabla119884119880)

+119892 (119883 nabla119884119882) minus 119892 (119884 nabla119883119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (120579119883 nabla119884119880)

minus119892(119883 nabla119884119881) + 119892(119884 nabla119883119881)lowast120577

(54)

where 1015840119877 denotes the curvature tensor of 120587minus1(119872) withrespect to the connection 1015840nabla Using (30) (31) (33) and (35)we can easily see that

1015840119877 (119884lowast 119883lowast) 119880lowast= 119906 (119883)119884 minus 119906 (119884)119883

+ 119906 (1198601119883)1198601119884 minus 119906 (1198601119884)1198601119883lowast

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119906 (119883) 119906 (1198601119884) minus 119906 (119884) 119906 (1198601119883)lowast120585

+ 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V (119883) 119906 (1198601119884) minus V (119884) 119906 (1198601119883)lowast120578

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908(119883)119906(1198601119884) minus 119908(119884)119906(1198601119883)lowast120577

(55)

By the same method we can easily verify that (49) (50) (52)and (53) yield

1015840119877 (119884lowast 119883lowast) 119881lowast= V (119883) 119884 minus V (119884)119883 + V (1198601119883)1198601119884

minus V (1198601119884)1198601119883lowast

+ 119902 (119883) 119906 (119884) minus 119902 (119884) 119906 (119883)

minus 119906 (119884) V (1198601119883) + 119906 (119883) V (1198601119884)lowast120585

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ V (119883) V (1198601119884) minus V (119884) V (1198601119883)lowast120578

+ 119902 (119883)119908 (119884) minus 119902 (119884)119908 (119883)

minus 119908 (119884) V (1198601119883) + 119908 (119883) V (1198601119884)lowast120577

1015840119877 (119884lowast 119883lowast)119882lowast= 119908 (119883)119884 minus 119908 (119884)119883 + 119908 (1198601119883)1198601119884

minus 119908 (1198601119884)1198601119883lowast

+ 119903 (119883) 119906 (119884) minus 119903 (119884) 119906 (119883)

minus 119906 (119884)119908 (1198601119883) + 119906 (119883)119908 (1198601119884)lowast120585

+ 119903 (119883) V (119884) minus 119903 (119884) V (119883)

minus V (119884)119908 (1198601119883) + V (119883)119908 (1198601119884)lowast120578

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ 119908(119883)119908(1198601119884) minus 119908(119884)119908(1198601119883)lowast120577

(56)

Differentiating (38)with119883 = 119880 covariantly along120587minus1(119872)

and using (24) we have

1015840nabla119884lowast1015840nabla119880lowast119883

lowast

= (nabla119884nabla119880119883)lowast+ 119908 (119883)120595119884 minus V (119883) 120579119884

lowast

+ 119892(120601119884 nabla119880119883)lowast120585

+ 119892 (120595119884 nabla119880119883) minus 119892 (nabla119884119882119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (120579119884 nabla119880119883) + 119892 (nabla119884119881119883) + 119892 (119881 nabla119884119883)lowast120577

(57)

International Journal of Mathematics and Mathematical Sciences 7

Similarly (38) with119883 = 119881 and (38) with119883 = 119882 respectivelygive

1015840nabla119884lowast1015840nabla119881lowast119883

lowast

= (nabla119884nabla119881119883)lowastminus 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892(120595119884 nabla119881119883)lowast120578

+ 119892(120601119884 nabla119881119883) + 119892(nabla119884119882119883) + 119892(119882 nabla119884119883)lowast120585

+ 119892(120579119884 nabla119881119883) minus 119892(nabla119884119880119883) minus 119892(119880 nabla119884119883)lowast120577

(58)

1015840nabla119884lowast1015840nabla119882lowast119883

lowast

= (nabla119884nabla119882119883)lowast+ V (119883) 120601119884 minus 119906 (119883)120595119884

lowast

+ 119892(120579119884 nabla119882119883)lowast120577

+ 119892 (120601119884 nabla119882119883) minus 119892 (nabla119884119881119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (120595119884 nabla119882119883) + 119892 (nabla119884119880119883) + 119892 (119880 nabla119884119883)lowast120578

(59)

Differentiating (38) also covariantly in the direction of119880lowast andusing (24) we have

1015840nabla119880lowast1015840nabla119884lowast119883

lowast

= (nabla119880nabla119884119883)lowast

+ 119892(120595119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119881lowast

+ 119892 (nabla119880 (120601119884) 119883) + 119892 (120601119884 nabla119880119883)lowast120585

+ 119892 (nabla119880 (120595119884) 119883) + 119892 (120595119884 nabla119880119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (nabla119880 (120579119884) 119883) + 119892 (120579119884 nabla119880119883) + 119892 (119881 nabla119884119883)lowast120577

(60)

Similarly differentiating (38) covariantly in the direction of119881lowast and119882lowast respectively we have

1015840nabla119881lowast1015840nabla119884lowast119883

lowast

= (nabla119881nabla119884119883)lowast

minus 119892(120601119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119880lowast

+ 119892 (nabla119881 (120601119884) 119883) + 119892 (120601119884 nabla119881119883) + 119892 (119882nabla119884119883)lowast120585

+ 119892 (nabla119881 (120595119884) 119883) + 119892 (120595119884 nabla119881119883)lowast120578

+ 119892 (nabla119881 (120579119884) 119883) + 119892 (120579119884 nabla119881119883) minus 119892 (119880 nabla119884119883)lowast120577

(61)

1015840nabla119882lowast1015840nabla119884lowast119883

lowast

= (nabla119882nabla119884119883)lowast

minus 119892(120595119884119883)lowast119880lowast+ 119892(120601119884119883)

lowast119881lowast

+ 119892 (nabla119882 (120601119884) 119883) + 119892 (120601119884 nabla119882119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (nabla119882 (120595119884) 119883) + 119892 (120595119884 nabla119882119883) + 119892 (119880 nabla119884119883)lowast120578

+ 119892 (nabla119882 (120579119884) 119883) + 119892 (120579119884 nabla119882119883)lowast120577

(62)

On the other hand (38) and (39) with119883 = 119880 imply1015840nabla[119884lowast 119880lowast]119883

lowast= (nabla[119884119880]119883)

lowastminus 2119908 (119884) 120595119883 minus V (119884) 120579119883

lowast

+ 119892(120601 [119884 119880] 119883)lowast120585 + 119892(120595 [119884119880] 119883)

lowast120578

+ 119892(120579 [119884 119880] 119883)lowast120577

(63)

Similarly from (39) with 119883 = 119881 and (39) with 119883 = 119882respectively we find that1015840nabla[119884lowast 119881lowast]119883

lowast= (nabla[119884119881]119883)

lowast+ 2119908 (119884) 120601119883 minus 119906 (119884) 120579119883

lowast

+ 119892(120601 [119884 119881] 119883)lowast120585 + 119892(120595 [119884 119881] 119883)

lowast120578

+ 119892(120579 [119884 119881] 119883)lowast120577

(64)1015840nabla[119884lowast 119882lowast]119883

lowast= (nabla[119884119882]119883)

lowastminus 2V (119884) 120601119883 minus 119906 (119884) 120595119883

lowast

+ 119892(120601 [119884119882] 119883)lowast120585 + g(120595 [119884119882] 119883)

lowast120578

+ 119892(120579 [119884119882] 119883)lowast120577

(65)

Using (28)ndash(31) it follows from (57) (60) and (63) that1015840119877 (119884lowast 119880lowast)119883lowast= 119877 (119884 119880)119883

lowast

+ 119908 (119883)120595119884 minus V (119883) 120579119884

minus 119892 (120595119884119883)119882 + 119892 (120579119884119883)119881

+ 2119908 (119884) 120595119883 minus 2V (119884) 120579119883lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(66)

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Mathematics and Mathematical Sciences

Finally the equation of Gauss is given as follows (cf [21])

119892 (119877 (119883 119884)119885119882)

= 119892 (119877 (119883 119884)119885119882)

+sum120572

119892 (119860120572119883119885) 119892 (119860120572119884119882)

minus119892 (119860120572119884119885) 119892 (119860120572119883119882)

(32)

for 119883119884 and 119885 tangent to 119872 where 119877 and 119877 denote theRiemannian curvature tensor of119872 and119872 respectively

In the rest of this paper we assume that the distinguishednormal vector field 1198731 = 119873 is parallel with respect to thenormal connection nablaperp Then it follows from (27) that 1199041120573 = 0and consequently (30)-(31) imply

119860120572119880 = 0 119860120572119881 = 0 119860120572119882 = 0 120572 = 2 119901

(33)

On the other hand since the curvature tensor 119877 ofQP(119899+119901)4 is of the form

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883119885)119884

+ 119892 (119865119884 119885) 119865119883 minus 119892 (119865119883119885)119865119884

minus 2119892 (119865119883 119884) 119865119885 + 119892 (119866119884119885)119866119883

minus 119892 (119866119883119885)119866119884 minus 2119892 (119866119883 119884)119866119885

+ 119892 (119867119884119885)119867119883 minus 119892 (119867119883119885)119867119884

minus 2119892 (119867119883119884)119867119885

(34)

for119883119884 and 119885 tangent to QP(119899+119901)4 (32) reduces to

119877 (119883 119884)119885 = 119892 (119884 119885)119883 minus 119892 (119883 119885) 119884

+ 119892 (120601119884 119885) 120601119883 minus 119892 (120601119883119885) 120601119884

minus 2119892 (120601119883 119884) 120601119885 + 119892 (120595119884 119885)120595119883

minus 119892 (120595119883119885)120595119884 minus 2119892 (120595119883 119884)120595119885

+ 119892 (120579119884 119885) 120579119883 minus 119892 (120579119883 119885) 120579119884

minus 2119892 (120579119883 119884) 120579119885

+sum120572

119892 (119860120572119884119885)119860120572119883 minus 119892 (119860120572119883119885)119860120572119884

(35)

3 Fibrations and Immersions

Fromnowon 119899-dimensionalQR-submanifolds of (119901minus1) QR-dimension isometrically immersed in QP(119899+119901)4 only will beconsidered Moreover we will use the assumption and thenotations as in Section 2

Let 119878119899+119901+3(119886) be the hypersphere of radius 119886 (gt0) in119876(119899+119901+4)4 the quaternionic space of quaternionic dimension

(119899 + 119901 + 4)4 which is identified with the Euclidean (119899 +

119901 + 4)-spaceR119899+119901+4 The unit sphere 119878119899+119901+3(1) will be brieflydenoted by 119878119899+119901+3 Let 119878119899+119901+3 rarr QP(119899+119901)4 be thenatural projection of 119878119899+119901+3 onto QP(119899+119901)4 defined by theHopf fibration 1198783 rarr 119878

119899+119901+3rarr QP(119899+119901)4 As is well known

(cf [10 11 20]) 119878119899+119901+3 admits a Sasakian 3-structure whereby120585 120578 and 120577 are mutually orthogonal unit Killing vector fieldsThus it follows that

nabla120585120585 = 0 nabla120578120578 = 0 nabla

120577120577 = 0

nabla120577120578 = minusnabla120578120577 = 120585 nabla

120585120577 = minusnabla

120577120585 = 120578

nabla120578120585 = minusnabla120585120578 = 120577

(36)

where nabla denotes the Riemannian connection with respect tothe canonical metric 119892 on 119878119899+119901+3 (cf [6 9ndash13]) Moreovereach fibre minus1(119909) of 119909 in QP(119899+119901)4 is a maximal integralsubmanifold of the distribution spanned by 120585 120578 and 120577 Thusthe base space QP(119899+119901)4 admits the induced quaternionicKahler structure of constant 119876-sectional curvature 4 (cf[10 11]) We have especially a fibration 120587 120587

minus1(119872) rarr

119872 which is compatible with the Hopf fibration Moreprecisely speaking 120587 120587

minus1(119872) rarr 119872 is a fibration withtotally geodesic fibers such that the following diagram iscommutative

120587minus1(M) S

n+p+3

M QP(n+p)4i

i998400

120587 (37)

where 1198941015840 120587minus1(119872) rarr 119878119899+119901+3 and 119894 119872 rarr QP(119899+119901)4 areisometric immersions

Now let 120585 120578 and 120577 be the unit vector fields tangent to thefibers of 120587minus1(119872) such that 1198941015840lowast120585 = 120585 1198941015840lowast120578 = 120578 and 1198941015840lowast120577 = 120577(In what follows we will again delete the 1198941015840 and 1198941015840lowast in ournotation) Furthermore we denote by 119883lowast the horizontal liftof a vector field 119883 tangent to 119872 Then the horizontal lifts119873lowast120572 (120572 = 1 119901) of the normal vectors 119873120572 to 119872 form an

orthonormal basis of normal vectors to 120587minus1(119872) in 119878119899+119901+3 Let1198601015840120572 and 119904

1015840120572120573 be the corresponding shape operators and normal

connection forms respectively Then as shown in [3 9 22]the fundamental equations for the submersion 120587 are given by

1015840nabla119883lowast119884

lowast= (nabla119883119884)

lowast+ 1198921015840((120601119883)

lowast 119884lowast) 120585 + 119892

1015840((120595119883)

lowast 119884lowast) 120578

+ 1198921015840((120579119883)

lowast 119884lowast) 120577

(38)

International Journal of Mathematics and Mathematical Sciences 5

[119883lowast 119884lowast] = [119883 119884]

lowast+ 21198921015840((120601119883)

lowast 119884lowast) 120585

+ 21198921015840((120595119883)

lowast 119884lowast) 120578 + 2119892

1015840((120579119883)

lowast 119884lowast) 120577

(39)

1015840nabla119883lowast120585 =

1015840nabla120585119883lowast= minus(120601119883)

lowast

1015840nabla119883lowast120578 =

1015840nabla120578119883lowast= minus(120595119883)

lowast

1015840nabla119883lowast120577 =

1015840nabla120577119883lowast= minus(120579119883)

lowast

(40)

[119883lowast 120585] = 0 [119883

lowast 120578] = 0 [119883

lowast 120577] = 0 (41)

where 1198921015840 denotes the Riemannian metric of 120587minus1(119872) inducedfrom 119892 in 119878119899+119901+3 and 1015840nabla the Levi-Civita connection withrespect to1198921015840The same equations are valid for the submersion by replacing 120601 120595 and 120579 (resp 120585 120578 and 120577) with 119865119866 and119867(resp 120585 120578 and 120577) respectivelyWe denote by 1015840nabla

perp the normalconnection of 120587minus1(119872) induced from nabla Since the diagram iscommutative nabla119883lowast119873

lowast120572 implies

1015840nablaperp

119883lowast119873lowast

120572 minus 1198601015840

120572119883lowast= (nabla119883119873120572)

lowast+ 119892 ((119865119883)

lowast 119873lowast

120572 ) 120585

+ 119892 ((119866119883)lowast 119873lowast

120572 ) 120578 + 119892 ((119867119883)lowast 119873lowast

120572 ) 120577

= minus (119860120572119883)lowast+ 119892(119880120572 119883)

lowast120585 + 119892(119881120572 119883)

lowast120578

+ 119892(119882120572 119883)lowast120577 + (nabla

perp

119883119873120572)lowast

(42)

because of (10) (26) and (38) from which comparing thetangential part we have

1198601015840

120572119883lowast= (119860120572119883)

lowastminus 119892(119880120572 119883)

lowast120585

minus 119892(119881120572 119883)lowast120578 minus 119892(119882120572 119883)

lowast120577

(43)

Next calculating nabla120585119873lowast120572 and using (10) (26) and (40) we have

1015840nablaperp

120585119873lowast

120572 minus 1198601015840

120572120585 = minus(119865119873120572)lowast= 119880lowast

120572 minus (1198751119873120572)lowast (44)

which yields

1198601015840

120572120585 = minus119880lowast

120572(45)

and similarly

1198601015840

120572120585 = minus119880lowast

120572 1198601015840

120572120578 = minus119881lowast

120572 1198601015840

120572120577 = minus119882lowast

120572 (46)

Hence (43) and (46) with 120572 = 1 imply

1198601015840

1119883lowast= (1198601119883)

lowastminus 119892(119880119883)

lowast120585 minus 119892(119881119883)

lowast120578 minus 119892(119882119883)

lowast120577

1198601015840

1120585 = minus119880lowast 119860

1015840

1120578 = minus119881lowast 119860

1015840

1120577 = minus119882lowast

(47)

4 Co-Gauss Equations for the Submersion120587120587minus1(119872) rarr 119872

In this section we derive the co-Gauss and co-Codazziequations of the submersion 120587 120587minus1(119872) rarr 119872 for later use

Differentiating (38) with119884 = 119880 covariantly along120587minus1(119872)

and using (24) (38) and (39) we have

1015840nabla119884lowast1015840nabla119883lowast119880

lowast

= (nabla119884nabla119883119880)lowast+ V (119883) 120579119884 minus 119908 (119883)120595119884

lowast

+ 119892(120601119884 nabla119883119880)lowast120585

+ 119892 (120595119884 nabla119883119880) + 119892 (nabla119884119883119882) + 119892 (119883 nabla119884119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (nabla119884119883119881) minus 119892 (119883 nabla119884119881)lowast120577

(48)

Similarly (38) with 119884 = 119881 and (38) with 119884 = 119882 give

1015840nabla119884lowast1015840nabla119883lowast119881

lowast

= (nabla119884nabla119883119881)lowast+ 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892 (120601119884 nabla119883119881) minus 119892 (nabla119884119883119882) minus 119892 (119883 nabla119884119882)lowast120585

+ 119892(120595119884 nabla119883119881)lowast120578

+ 119892 (120579119884 nabla119883119881) + 119892 (nabla119884119883119880) + 119892 (119883 nabla119884119880)lowast120577

(49)1015840nabla119884lowast1015840nabla119883lowast119882

lowast

= (nabla119884nabla119883119882)lowast

minus V (119883) 120601119884 minus 119906 (119883)120595119884lowast

+ 119892 (120601119884 nabla119883119882) + 119892 (nabla119884119883119881) + 119892 (119883 nabla119884119881)lowast120585

+ 119892 (120595119884 nabla119883119882) minus 119892 (nabla119884119883119880) minus 119892 (119883 nabla119884119880)lowast120578

+ 119892(120579119884 nabla119883119882)lowast120577

(50)

respectively On the other hand it follows from (19) (24)(38) and (39) that

1015840nabla[119884lowast 119883lowast]119880

lowast= (nabla[119884119883]119880)

lowast+ 2119892(120595119884119883)

lowast119882lowast

minus 2119892(120579119884119883)lowast119881lowast+ 119892([119884119883]119882)

lowast120578

minus 119892([119884119883] 119881)lowast120577

(51)

6 International Journal of Mathematics and Mathematical Sciences

1015840nabla[119884lowast119883lowast]119881

lowast= (nabla[119884119883]119881)

lowastminus 2119892(120601119884119883)

lowast119882lowast

+ 2119892(120579119884119883)lowast119880lowastminus 119892([119884119883] 119882)

lowast120585

+ 119892([119884119883] 119880)lowast120577

(52)

1015840nabla[119884lowast119883lowast]119882

lowast= (nabla[119884119883]119882)

lowast+ 2119892(120601119884119883)

lowast119881lowast

minus 2119892(120595119884119883)lowast119880lowast+ 119892([119884119883] 119881)

lowast120585

minus 119892([119884119883] 119880)lowast120578

(53)

By means of (48) and (51) we have

1015840119877 (119884lowast 119883lowast) 119880lowast= 119877 (119884119883)119880

lowast

+ 119908 (119884) 120595119883 minus 119908 (119883)120595119884 minus V (119884) 120579119883

+ V (119883) 120579119884 + 2119892 (120579119884119883)119881

minus 2119892 (120595119884119883)119882lowast

+ 119892 (120601119884 nabla119883119880) minus 119892 (120601119883 nabla119884119880)lowast120585

+ 119892 (120595119884 nabla119883119880) minus 119892 (120595119883 nabla119884119880)

+119892 (119883 nabla119884119882) minus 119892 (119884 nabla119883119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (120579119883 nabla119884119880)

minus119892(119883 nabla119884119881) + 119892(119884 nabla119883119881)lowast120577

(54)

where 1015840119877 denotes the curvature tensor of 120587minus1(119872) withrespect to the connection 1015840nabla Using (30) (31) (33) and (35)we can easily see that

1015840119877 (119884lowast 119883lowast) 119880lowast= 119906 (119883)119884 minus 119906 (119884)119883

+ 119906 (1198601119883)1198601119884 minus 119906 (1198601119884)1198601119883lowast

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119906 (119883) 119906 (1198601119884) minus 119906 (119884) 119906 (1198601119883)lowast120585

+ 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V (119883) 119906 (1198601119884) minus V (119884) 119906 (1198601119883)lowast120578

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908(119883)119906(1198601119884) minus 119908(119884)119906(1198601119883)lowast120577

(55)

By the same method we can easily verify that (49) (50) (52)and (53) yield

1015840119877 (119884lowast 119883lowast) 119881lowast= V (119883) 119884 minus V (119884)119883 + V (1198601119883)1198601119884

minus V (1198601119884)1198601119883lowast

+ 119902 (119883) 119906 (119884) minus 119902 (119884) 119906 (119883)

minus 119906 (119884) V (1198601119883) + 119906 (119883) V (1198601119884)lowast120585

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ V (119883) V (1198601119884) minus V (119884) V (1198601119883)lowast120578

+ 119902 (119883)119908 (119884) minus 119902 (119884)119908 (119883)

minus 119908 (119884) V (1198601119883) + 119908 (119883) V (1198601119884)lowast120577

1015840119877 (119884lowast 119883lowast)119882lowast= 119908 (119883)119884 minus 119908 (119884)119883 + 119908 (1198601119883)1198601119884

minus 119908 (1198601119884)1198601119883lowast

+ 119903 (119883) 119906 (119884) minus 119903 (119884) 119906 (119883)

minus 119906 (119884)119908 (1198601119883) + 119906 (119883)119908 (1198601119884)lowast120585

+ 119903 (119883) V (119884) minus 119903 (119884) V (119883)

minus V (119884)119908 (1198601119883) + V (119883)119908 (1198601119884)lowast120578

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ 119908(119883)119908(1198601119884) minus 119908(119884)119908(1198601119883)lowast120577

(56)

Differentiating (38)with119883 = 119880 covariantly along120587minus1(119872)

and using (24) we have

1015840nabla119884lowast1015840nabla119880lowast119883

lowast

= (nabla119884nabla119880119883)lowast+ 119908 (119883)120595119884 minus V (119883) 120579119884

lowast

+ 119892(120601119884 nabla119880119883)lowast120585

+ 119892 (120595119884 nabla119880119883) minus 119892 (nabla119884119882119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (120579119884 nabla119880119883) + 119892 (nabla119884119881119883) + 119892 (119881 nabla119884119883)lowast120577

(57)

International Journal of Mathematics and Mathematical Sciences 7

Similarly (38) with119883 = 119881 and (38) with119883 = 119882 respectivelygive

1015840nabla119884lowast1015840nabla119881lowast119883

lowast

= (nabla119884nabla119881119883)lowastminus 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892(120595119884 nabla119881119883)lowast120578

+ 119892(120601119884 nabla119881119883) + 119892(nabla119884119882119883) + 119892(119882 nabla119884119883)lowast120585

+ 119892(120579119884 nabla119881119883) minus 119892(nabla119884119880119883) minus 119892(119880 nabla119884119883)lowast120577

(58)

1015840nabla119884lowast1015840nabla119882lowast119883

lowast

= (nabla119884nabla119882119883)lowast+ V (119883) 120601119884 minus 119906 (119883)120595119884

lowast

+ 119892(120579119884 nabla119882119883)lowast120577

+ 119892 (120601119884 nabla119882119883) minus 119892 (nabla119884119881119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (120595119884 nabla119882119883) + 119892 (nabla119884119880119883) + 119892 (119880 nabla119884119883)lowast120578

(59)

Differentiating (38) also covariantly in the direction of119880lowast andusing (24) we have

1015840nabla119880lowast1015840nabla119884lowast119883

lowast

= (nabla119880nabla119884119883)lowast

+ 119892(120595119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119881lowast

+ 119892 (nabla119880 (120601119884) 119883) + 119892 (120601119884 nabla119880119883)lowast120585

+ 119892 (nabla119880 (120595119884) 119883) + 119892 (120595119884 nabla119880119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (nabla119880 (120579119884) 119883) + 119892 (120579119884 nabla119880119883) + 119892 (119881 nabla119884119883)lowast120577

(60)

Similarly differentiating (38) covariantly in the direction of119881lowast and119882lowast respectively we have

1015840nabla119881lowast1015840nabla119884lowast119883

lowast

= (nabla119881nabla119884119883)lowast

minus 119892(120601119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119880lowast

+ 119892 (nabla119881 (120601119884) 119883) + 119892 (120601119884 nabla119881119883) + 119892 (119882nabla119884119883)lowast120585

+ 119892 (nabla119881 (120595119884) 119883) + 119892 (120595119884 nabla119881119883)lowast120578

+ 119892 (nabla119881 (120579119884) 119883) + 119892 (120579119884 nabla119881119883) minus 119892 (119880 nabla119884119883)lowast120577

(61)

1015840nabla119882lowast1015840nabla119884lowast119883

lowast

= (nabla119882nabla119884119883)lowast

minus 119892(120595119884119883)lowast119880lowast+ 119892(120601119884119883)

lowast119881lowast

+ 119892 (nabla119882 (120601119884) 119883) + 119892 (120601119884 nabla119882119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (nabla119882 (120595119884) 119883) + 119892 (120595119884 nabla119882119883) + 119892 (119880 nabla119884119883)lowast120578

+ 119892 (nabla119882 (120579119884) 119883) + 119892 (120579119884 nabla119882119883)lowast120577

(62)

On the other hand (38) and (39) with119883 = 119880 imply1015840nabla[119884lowast 119880lowast]119883

lowast= (nabla[119884119880]119883)

lowastminus 2119908 (119884) 120595119883 minus V (119884) 120579119883

lowast

+ 119892(120601 [119884 119880] 119883)lowast120585 + 119892(120595 [119884119880] 119883)

lowast120578

+ 119892(120579 [119884 119880] 119883)lowast120577

(63)

Similarly from (39) with 119883 = 119881 and (39) with 119883 = 119882respectively we find that1015840nabla[119884lowast 119881lowast]119883

lowast= (nabla[119884119881]119883)

lowast+ 2119908 (119884) 120601119883 minus 119906 (119884) 120579119883

lowast

+ 119892(120601 [119884 119881] 119883)lowast120585 + 119892(120595 [119884 119881] 119883)

lowast120578

+ 119892(120579 [119884 119881] 119883)lowast120577

(64)1015840nabla[119884lowast 119882lowast]119883

lowast= (nabla[119884119882]119883)

lowastminus 2V (119884) 120601119883 minus 119906 (119884) 120595119883

lowast

+ 119892(120601 [119884119882] 119883)lowast120585 + g(120595 [119884119882] 119883)

lowast120578

+ 119892(120579 [119884119882] 119883)lowast120577

(65)

Using (28)ndash(31) it follows from (57) (60) and (63) that1015840119877 (119884lowast 119880lowast)119883lowast= 119877 (119884 119880)119883

lowast

+ 119908 (119883)120595119884 minus V (119883) 120579119884

minus 119892 (120595119884119883)119882 + 119892 (120579119884119883)119881

+ 2119908 (119884) 120595119883 minus 2V (119884) 120579119883lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(66)

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 5

[119883lowast 119884lowast] = [119883 119884]

lowast+ 21198921015840((120601119883)

lowast 119884lowast) 120585

+ 21198921015840((120595119883)

lowast 119884lowast) 120578 + 2119892

1015840((120579119883)

lowast 119884lowast) 120577

(39)

1015840nabla119883lowast120585 =

1015840nabla120585119883lowast= minus(120601119883)

lowast

1015840nabla119883lowast120578 =

1015840nabla120578119883lowast= minus(120595119883)

lowast

1015840nabla119883lowast120577 =

1015840nabla120577119883lowast= minus(120579119883)

lowast

(40)

[119883lowast 120585] = 0 [119883

lowast 120578] = 0 [119883

lowast 120577] = 0 (41)

where 1198921015840 denotes the Riemannian metric of 120587minus1(119872) inducedfrom 119892 in 119878119899+119901+3 and 1015840nabla the Levi-Civita connection withrespect to1198921015840The same equations are valid for the submersion by replacing 120601 120595 and 120579 (resp 120585 120578 and 120577) with 119865119866 and119867(resp 120585 120578 and 120577) respectivelyWe denote by 1015840nabla

perp the normalconnection of 120587minus1(119872) induced from nabla Since the diagram iscommutative nabla119883lowast119873

lowast120572 implies

1015840nablaperp

119883lowast119873lowast

120572 minus 1198601015840

120572119883lowast= (nabla119883119873120572)

lowast+ 119892 ((119865119883)

lowast 119873lowast

120572 ) 120585

+ 119892 ((119866119883)lowast 119873lowast

120572 ) 120578 + 119892 ((119867119883)lowast 119873lowast

120572 ) 120577

= minus (119860120572119883)lowast+ 119892(119880120572 119883)

lowast120585 + 119892(119881120572 119883)

lowast120578

+ 119892(119882120572 119883)lowast120577 + (nabla

perp

119883119873120572)lowast

(42)

because of (10) (26) and (38) from which comparing thetangential part we have

1198601015840

120572119883lowast= (119860120572119883)

lowastminus 119892(119880120572 119883)

lowast120585

minus 119892(119881120572 119883)lowast120578 minus 119892(119882120572 119883)

lowast120577

(43)

Next calculating nabla120585119873lowast120572 and using (10) (26) and (40) we have

1015840nablaperp

120585119873lowast

120572 minus 1198601015840

120572120585 = minus(119865119873120572)lowast= 119880lowast

120572 minus (1198751119873120572)lowast (44)

which yields

1198601015840

120572120585 = minus119880lowast

120572(45)

and similarly

1198601015840

120572120585 = minus119880lowast

120572 1198601015840

120572120578 = minus119881lowast

120572 1198601015840

120572120577 = minus119882lowast

120572 (46)

Hence (43) and (46) with 120572 = 1 imply

1198601015840

1119883lowast= (1198601119883)

lowastminus 119892(119880119883)

lowast120585 minus 119892(119881119883)

lowast120578 minus 119892(119882119883)

lowast120577

1198601015840

1120585 = minus119880lowast 119860

1015840

1120578 = minus119881lowast 119860

1015840

1120577 = minus119882lowast

(47)

4 Co-Gauss Equations for the Submersion120587120587minus1(119872) rarr 119872

In this section we derive the co-Gauss and co-Codazziequations of the submersion 120587 120587minus1(119872) rarr 119872 for later use

Differentiating (38) with119884 = 119880 covariantly along120587minus1(119872)

and using (24) (38) and (39) we have

1015840nabla119884lowast1015840nabla119883lowast119880

lowast

= (nabla119884nabla119883119880)lowast+ V (119883) 120579119884 minus 119908 (119883)120595119884

lowast

+ 119892(120601119884 nabla119883119880)lowast120585

+ 119892 (120595119884 nabla119883119880) + 119892 (nabla119884119883119882) + 119892 (119883 nabla119884119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (nabla119884119883119881) minus 119892 (119883 nabla119884119881)lowast120577

(48)

Similarly (38) with 119884 = 119881 and (38) with 119884 = 119882 give

1015840nabla119884lowast1015840nabla119883lowast119881

lowast

= (nabla119884nabla119883119881)lowast+ 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892 (120601119884 nabla119883119881) minus 119892 (nabla119884119883119882) minus 119892 (119883 nabla119884119882)lowast120585

+ 119892(120595119884 nabla119883119881)lowast120578

+ 119892 (120579119884 nabla119883119881) + 119892 (nabla119884119883119880) + 119892 (119883 nabla119884119880)lowast120577

(49)1015840nabla119884lowast1015840nabla119883lowast119882

lowast

= (nabla119884nabla119883119882)lowast

minus V (119883) 120601119884 minus 119906 (119883)120595119884lowast

+ 119892 (120601119884 nabla119883119882) + 119892 (nabla119884119883119881) + 119892 (119883 nabla119884119881)lowast120585

+ 119892 (120595119884 nabla119883119882) minus 119892 (nabla119884119883119880) minus 119892 (119883 nabla119884119880)lowast120578

+ 119892(120579119884 nabla119883119882)lowast120577

(50)

respectively On the other hand it follows from (19) (24)(38) and (39) that

1015840nabla[119884lowast 119883lowast]119880

lowast= (nabla[119884119883]119880)

lowast+ 2119892(120595119884119883)

lowast119882lowast

minus 2119892(120579119884119883)lowast119881lowast+ 119892([119884119883]119882)

lowast120578

minus 119892([119884119883] 119881)lowast120577

(51)

6 International Journal of Mathematics and Mathematical Sciences

1015840nabla[119884lowast119883lowast]119881

lowast= (nabla[119884119883]119881)

lowastminus 2119892(120601119884119883)

lowast119882lowast

+ 2119892(120579119884119883)lowast119880lowastminus 119892([119884119883] 119882)

lowast120585

+ 119892([119884119883] 119880)lowast120577

(52)

1015840nabla[119884lowast119883lowast]119882

lowast= (nabla[119884119883]119882)

lowast+ 2119892(120601119884119883)

lowast119881lowast

minus 2119892(120595119884119883)lowast119880lowast+ 119892([119884119883] 119881)

lowast120585

minus 119892([119884119883] 119880)lowast120578

(53)

By means of (48) and (51) we have

1015840119877 (119884lowast 119883lowast) 119880lowast= 119877 (119884119883)119880

lowast

+ 119908 (119884) 120595119883 minus 119908 (119883)120595119884 minus V (119884) 120579119883

+ V (119883) 120579119884 + 2119892 (120579119884119883)119881

minus 2119892 (120595119884119883)119882lowast

+ 119892 (120601119884 nabla119883119880) minus 119892 (120601119883 nabla119884119880)lowast120585

+ 119892 (120595119884 nabla119883119880) minus 119892 (120595119883 nabla119884119880)

+119892 (119883 nabla119884119882) minus 119892 (119884 nabla119883119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (120579119883 nabla119884119880)

minus119892(119883 nabla119884119881) + 119892(119884 nabla119883119881)lowast120577

(54)

where 1015840119877 denotes the curvature tensor of 120587minus1(119872) withrespect to the connection 1015840nabla Using (30) (31) (33) and (35)we can easily see that

1015840119877 (119884lowast 119883lowast) 119880lowast= 119906 (119883)119884 minus 119906 (119884)119883

+ 119906 (1198601119883)1198601119884 minus 119906 (1198601119884)1198601119883lowast

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119906 (119883) 119906 (1198601119884) minus 119906 (119884) 119906 (1198601119883)lowast120585

+ 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V (119883) 119906 (1198601119884) minus V (119884) 119906 (1198601119883)lowast120578

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908(119883)119906(1198601119884) minus 119908(119884)119906(1198601119883)lowast120577

(55)

By the same method we can easily verify that (49) (50) (52)and (53) yield

1015840119877 (119884lowast 119883lowast) 119881lowast= V (119883) 119884 minus V (119884)119883 + V (1198601119883)1198601119884

minus V (1198601119884)1198601119883lowast

+ 119902 (119883) 119906 (119884) minus 119902 (119884) 119906 (119883)

minus 119906 (119884) V (1198601119883) + 119906 (119883) V (1198601119884)lowast120585

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ V (119883) V (1198601119884) minus V (119884) V (1198601119883)lowast120578

+ 119902 (119883)119908 (119884) minus 119902 (119884)119908 (119883)

minus 119908 (119884) V (1198601119883) + 119908 (119883) V (1198601119884)lowast120577

1015840119877 (119884lowast 119883lowast)119882lowast= 119908 (119883)119884 minus 119908 (119884)119883 + 119908 (1198601119883)1198601119884

minus 119908 (1198601119884)1198601119883lowast

+ 119903 (119883) 119906 (119884) minus 119903 (119884) 119906 (119883)

minus 119906 (119884)119908 (1198601119883) + 119906 (119883)119908 (1198601119884)lowast120585

+ 119903 (119883) V (119884) minus 119903 (119884) V (119883)

minus V (119884)119908 (1198601119883) + V (119883)119908 (1198601119884)lowast120578

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ 119908(119883)119908(1198601119884) minus 119908(119884)119908(1198601119883)lowast120577

(56)

Differentiating (38)with119883 = 119880 covariantly along120587minus1(119872)

and using (24) we have

1015840nabla119884lowast1015840nabla119880lowast119883

lowast

= (nabla119884nabla119880119883)lowast+ 119908 (119883)120595119884 minus V (119883) 120579119884

lowast

+ 119892(120601119884 nabla119880119883)lowast120585

+ 119892 (120595119884 nabla119880119883) minus 119892 (nabla119884119882119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (120579119884 nabla119880119883) + 119892 (nabla119884119881119883) + 119892 (119881 nabla119884119883)lowast120577

(57)

International Journal of Mathematics and Mathematical Sciences 7

Similarly (38) with119883 = 119881 and (38) with119883 = 119882 respectivelygive

1015840nabla119884lowast1015840nabla119881lowast119883

lowast

= (nabla119884nabla119881119883)lowastminus 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892(120595119884 nabla119881119883)lowast120578

+ 119892(120601119884 nabla119881119883) + 119892(nabla119884119882119883) + 119892(119882 nabla119884119883)lowast120585

+ 119892(120579119884 nabla119881119883) minus 119892(nabla119884119880119883) minus 119892(119880 nabla119884119883)lowast120577

(58)

1015840nabla119884lowast1015840nabla119882lowast119883

lowast

= (nabla119884nabla119882119883)lowast+ V (119883) 120601119884 minus 119906 (119883)120595119884

lowast

+ 119892(120579119884 nabla119882119883)lowast120577

+ 119892 (120601119884 nabla119882119883) minus 119892 (nabla119884119881119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (120595119884 nabla119882119883) + 119892 (nabla119884119880119883) + 119892 (119880 nabla119884119883)lowast120578

(59)

Differentiating (38) also covariantly in the direction of119880lowast andusing (24) we have

1015840nabla119880lowast1015840nabla119884lowast119883

lowast

= (nabla119880nabla119884119883)lowast

+ 119892(120595119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119881lowast

+ 119892 (nabla119880 (120601119884) 119883) + 119892 (120601119884 nabla119880119883)lowast120585

+ 119892 (nabla119880 (120595119884) 119883) + 119892 (120595119884 nabla119880119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (nabla119880 (120579119884) 119883) + 119892 (120579119884 nabla119880119883) + 119892 (119881 nabla119884119883)lowast120577

(60)

Similarly differentiating (38) covariantly in the direction of119881lowast and119882lowast respectively we have

1015840nabla119881lowast1015840nabla119884lowast119883

lowast

= (nabla119881nabla119884119883)lowast

minus 119892(120601119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119880lowast

+ 119892 (nabla119881 (120601119884) 119883) + 119892 (120601119884 nabla119881119883) + 119892 (119882nabla119884119883)lowast120585

+ 119892 (nabla119881 (120595119884) 119883) + 119892 (120595119884 nabla119881119883)lowast120578

+ 119892 (nabla119881 (120579119884) 119883) + 119892 (120579119884 nabla119881119883) minus 119892 (119880 nabla119884119883)lowast120577

(61)

1015840nabla119882lowast1015840nabla119884lowast119883

lowast

= (nabla119882nabla119884119883)lowast

minus 119892(120595119884119883)lowast119880lowast+ 119892(120601119884119883)

lowast119881lowast

+ 119892 (nabla119882 (120601119884) 119883) + 119892 (120601119884 nabla119882119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (nabla119882 (120595119884) 119883) + 119892 (120595119884 nabla119882119883) + 119892 (119880 nabla119884119883)lowast120578

+ 119892 (nabla119882 (120579119884) 119883) + 119892 (120579119884 nabla119882119883)lowast120577

(62)

On the other hand (38) and (39) with119883 = 119880 imply1015840nabla[119884lowast 119880lowast]119883

lowast= (nabla[119884119880]119883)

lowastminus 2119908 (119884) 120595119883 minus V (119884) 120579119883

lowast

+ 119892(120601 [119884 119880] 119883)lowast120585 + 119892(120595 [119884119880] 119883)

lowast120578

+ 119892(120579 [119884 119880] 119883)lowast120577

(63)

Similarly from (39) with 119883 = 119881 and (39) with 119883 = 119882respectively we find that1015840nabla[119884lowast 119881lowast]119883

lowast= (nabla[119884119881]119883)

lowast+ 2119908 (119884) 120601119883 minus 119906 (119884) 120579119883

lowast

+ 119892(120601 [119884 119881] 119883)lowast120585 + 119892(120595 [119884 119881] 119883)

lowast120578

+ 119892(120579 [119884 119881] 119883)lowast120577

(64)1015840nabla[119884lowast 119882lowast]119883

lowast= (nabla[119884119882]119883)

lowastminus 2V (119884) 120601119883 minus 119906 (119884) 120595119883

lowast

+ 119892(120601 [119884119882] 119883)lowast120585 + g(120595 [119884119882] 119883)

lowast120578

+ 119892(120579 [119884119882] 119883)lowast120577

(65)

Using (28)ndash(31) it follows from (57) (60) and (63) that1015840119877 (119884lowast 119880lowast)119883lowast= 119877 (119884 119880)119883

lowast

+ 119908 (119883)120595119884 minus V (119883) 120579119884

minus 119892 (120595119884119883)119882 + 119892 (120579119884119883)119881

+ 2119908 (119884) 120595119883 minus 2V (119884) 120579119883lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(66)

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Mathematics and Mathematical Sciences

1015840nabla[119884lowast119883lowast]119881

lowast= (nabla[119884119883]119881)

lowastminus 2119892(120601119884119883)

lowast119882lowast

+ 2119892(120579119884119883)lowast119880lowastminus 119892([119884119883] 119882)

lowast120585

+ 119892([119884119883] 119880)lowast120577

(52)

1015840nabla[119884lowast119883lowast]119882

lowast= (nabla[119884119883]119882)

lowast+ 2119892(120601119884119883)

lowast119881lowast

minus 2119892(120595119884119883)lowast119880lowast+ 119892([119884119883] 119881)

lowast120585

minus 119892([119884119883] 119880)lowast120578

(53)

By means of (48) and (51) we have

1015840119877 (119884lowast 119883lowast) 119880lowast= 119877 (119884119883)119880

lowast

+ 119908 (119884) 120595119883 minus 119908 (119883)120595119884 minus V (119884) 120579119883

+ V (119883) 120579119884 + 2119892 (120579119884119883)119881

minus 2119892 (120595119884119883)119882lowast

+ 119892 (120601119884 nabla119883119880) minus 119892 (120601119883 nabla119884119880)lowast120585

+ 119892 (120595119884 nabla119883119880) minus 119892 (120595119883 nabla119884119880)

+119892 (119883 nabla119884119882) minus 119892 (119884 nabla119883119882)lowast120578

+ 119892 (120579119884 nabla119883119880) minus 119892 (120579119883 nabla119884119880)

minus119892(119883 nabla119884119881) + 119892(119884 nabla119883119881)lowast120577

(54)

where 1015840119877 denotes the curvature tensor of 120587minus1(119872) withrespect to the connection 1015840nabla Using (30) (31) (33) and (35)we can easily see that

1015840119877 (119884lowast 119883lowast) 119880lowast= 119906 (119883)119884 minus 119906 (119884)119883

+ 119906 (1198601119883)1198601119884 minus 119906 (1198601119884)1198601119883lowast

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119906 (119883) 119906 (1198601119884) minus 119906 (119884) 119906 (1198601119883)lowast120585

+ 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V (119883) 119906 (1198601119884) minus V (119884) 119906 (1198601119883)lowast120578

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908(119883)119906(1198601119884) minus 119908(119884)119906(1198601119883)lowast120577

(55)

By the same method we can easily verify that (49) (50) (52)and (53) yield

1015840119877 (119884lowast 119883lowast) 119881lowast= V (119883) 119884 minus V (119884)119883 + V (1198601119883)1198601119884

minus V (1198601119884)1198601119883lowast

+ 119902 (119883) 119906 (119884) minus 119902 (119884) 119906 (119883)

minus 119906 (119884) V (1198601119883) + 119906 (119883) V (1198601119884)lowast120585

+ 119903 (119884)119908 (119883) minus 119903 (119883)119908 (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ V (119883) V (1198601119884) minus V (119884) V (1198601119883)lowast120578

+ 119902 (119883)119908 (119884) minus 119902 (119884)119908 (119883)

minus 119908 (119884) V (1198601119883) + 119908 (119883) V (1198601119884)lowast120577

1015840119877 (119884lowast 119883lowast)119882lowast= 119908 (119883)119884 minus 119908 (119884)119883 + 119908 (1198601119883)1198601119884

minus 119908 (1198601119884)1198601119883lowast

+ 119903 (119883) 119906 (119884) minus 119903 (119884) 119906 (119883)

minus 119906 (119884)119908 (1198601119883) + 119906 (119883)119908 (1198601119884)lowast120585

+ 119903 (119883) V (119884) minus 119903 (119884) V (119883)

minus V (119884)119908 (1198601119883) + V (119883)119908 (1198601119884)lowast120578

+ 119902 (119884) V (119883) minus 119902 (119883) V (119884)

+ 119901 (119884) 119906 (119883) minus 119901 (119883) 119906 (119884)

+ 119908(119883)119908(1198601119884) minus 119908(119884)119908(1198601119883)lowast120577

(56)

Differentiating (38)with119883 = 119880 covariantly along120587minus1(119872)

and using (24) we have

1015840nabla119884lowast1015840nabla119880lowast119883

lowast

= (nabla119884nabla119880119883)lowast+ 119908 (119883)120595119884 minus V (119883) 120579119884

lowast

+ 119892(120601119884 nabla119880119883)lowast120585

+ 119892 (120595119884 nabla119880119883) minus 119892 (nabla119884119882119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (120579119884 nabla119880119883) + 119892 (nabla119884119881119883) + 119892 (119881 nabla119884119883)lowast120577

(57)

International Journal of Mathematics and Mathematical Sciences 7

Similarly (38) with119883 = 119881 and (38) with119883 = 119882 respectivelygive

1015840nabla119884lowast1015840nabla119881lowast119883

lowast

= (nabla119884nabla119881119883)lowastminus 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892(120595119884 nabla119881119883)lowast120578

+ 119892(120601119884 nabla119881119883) + 119892(nabla119884119882119883) + 119892(119882 nabla119884119883)lowast120585

+ 119892(120579119884 nabla119881119883) minus 119892(nabla119884119880119883) minus 119892(119880 nabla119884119883)lowast120577

(58)

1015840nabla119884lowast1015840nabla119882lowast119883

lowast

= (nabla119884nabla119882119883)lowast+ V (119883) 120601119884 minus 119906 (119883)120595119884

lowast

+ 119892(120579119884 nabla119882119883)lowast120577

+ 119892 (120601119884 nabla119882119883) minus 119892 (nabla119884119881119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (120595119884 nabla119882119883) + 119892 (nabla119884119880119883) + 119892 (119880 nabla119884119883)lowast120578

(59)

Differentiating (38) also covariantly in the direction of119880lowast andusing (24) we have

1015840nabla119880lowast1015840nabla119884lowast119883

lowast

= (nabla119880nabla119884119883)lowast

+ 119892(120595119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119881lowast

+ 119892 (nabla119880 (120601119884) 119883) + 119892 (120601119884 nabla119880119883)lowast120585

+ 119892 (nabla119880 (120595119884) 119883) + 119892 (120595119884 nabla119880119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (nabla119880 (120579119884) 119883) + 119892 (120579119884 nabla119880119883) + 119892 (119881 nabla119884119883)lowast120577

(60)

Similarly differentiating (38) covariantly in the direction of119881lowast and119882lowast respectively we have

1015840nabla119881lowast1015840nabla119884lowast119883

lowast

= (nabla119881nabla119884119883)lowast

minus 119892(120601119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119880lowast

+ 119892 (nabla119881 (120601119884) 119883) + 119892 (120601119884 nabla119881119883) + 119892 (119882nabla119884119883)lowast120585

+ 119892 (nabla119881 (120595119884) 119883) + 119892 (120595119884 nabla119881119883)lowast120578

+ 119892 (nabla119881 (120579119884) 119883) + 119892 (120579119884 nabla119881119883) minus 119892 (119880 nabla119884119883)lowast120577

(61)

1015840nabla119882lowast1015840nabla119884lowast119883

lowast

= (nabla119882nabla119884119883)lowast

minus 119892(120595119884119883)lowast119880lowast+ 119892(120601119884119883)

lowast119881lowast

+ 119892 (nabla119882 (120601119884) 119883) + 119892 (120601119884 nabla119882119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (nabla119882 (120595119884) 119883) + 119892 (120595119884 nabla119882119883) + 119892 (119880 nabla119884119883)lowast120578

+ 119892 (nabla119882 (120579119884) 119883) + 119892 (120579119884 nabla119882119883)lowast120577

(62)

On the other hand (38) and (39) with119883 = 119880 imply1015840nabla[119884lowast 119880lowast]119883

lowast= (nabla[119884119880]119883)

lowastminus 2119908 (119884) 120595119883 minus V (119884) 120579119883

lowast

+ 119892(120601 [119884 119880] 119883)lowast120585 + 119892(120595 [119884119880] 119883)

lowast120578

+ 119892(120579 [119884 119880] 119883)lowast120577

(63)

Similarly from (39) with 119883 = 119881 and (39) with 119883 = 119882respectively we find that1015840nabla[119884lowast 119881lowast]119883

lowast= (nabla[119884119881]119883)

lowast+ 2119908 (119884) 120601119883 minus 119906 (119884) 120579119883

lowast

+ 119892(120601 [119884 119881] 119883)lowast120585 + 119892(120595 [119884 119881] 119883)

lowast120578

+ 119892(120579 [119884 119881] 119883)lowast120577

(64)1015840nabla[119884lowast 119882lowast]119883

lowast= (nabla[119884119882]119883)

lowastminus 2V (119884) 120601119883 minus 119906 (119884) 120595119883

lowast

+ 119892(120601 [119884119882] 119883)lowast120585 + g(120595 [119884119882] 119883)

lowast120578

+ 119892(120579 [119884119882] 119883)lowast120577

(65)

Using (28)ndash(31) it follows from (57) (60) and (63) that1015840119877 (119884lowast 119880lowast)119883lowast= 119877 (119884 119880)119883

lowast

+ 119908 (119883)120595119884 minus V (119883) 120579119884

minus 119892 (120595119884119883)119882 + 119892 (120579119884119883)119881

+ 2119908 (119884) 120595119883 minus 2V (119884) 120579119883lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(66)

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 7

Similarly (38) with119883 = 119881 and (38) with119883 = 119882 respectivelygive

1015840nabla119884lowast1015840nabla119881lowast119883

lowast

= (nabla119884nabla119881119883)lowastminus 119908 (119883) 120601119884 minus 119906 (119883) 120579119884

lowast

+ 119892(120595119884 nabla119881119883)lowast120578

+ 119892(120601119884 nabla119881119883) + 119892(nabla119884119882119883) + 119892(119882 nabla119884119883)lowast120585

+ 119892(120579119884 nabla119881119883) minus 119892(nabla119884119880119883) minus 119892(119880 nabla119884119883)lowast120577

(58)

1015840nabla119884lowast1015840nabla119882lowast119883

lowast

= (nabla119884nabla119882119883)lowast+ V (119883) 120601119884 minus 119906 (119883)120595119884

lowast

+ 119892(120579119884 nabla119882119883)lowast120577

+ 119892 (120601119884 nabla119882119883) minus 119892 (nabla119884119881119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (120595119884 nabla119882119883) + 119892 (nabla119884119880119883) + 119892 (119880 nabla119884119883)lowast120578

(59)

Differentiating (38) also covariantly in the direction of119880lowast andusing (24) we have

1015840nabla119880lowast1015840nabla119884lowast119883

lowast

= (nabla119880nabla119884119883)lowast

+ 119892(120595119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119881lowast

+ 119892 (nabla119880 (120601119884) 119883) + 119892 (120601119884 nabla119880119883)lowast120585

+ 119892 (nabla119880 (120595119884) 119883) + 119892 (120595119884 nabla119880119883) minus 119892 (119882nabla119884119883)lowast120578

+ 119892 (nabla119880 (120579119884) 119883) + 119892 (120579119884 nabla119880119883) + 119892 (119881 nabla119884119883)lowast120577

(60)

Similarly differentiating (38) covariantly in the direction of119881lowast and119882lowast respectively we have

1015840nabla119881lowast1015840nabla119884lowast119883

lowast

= (nabla119881nabla119884119883)lowast

minus 119892(120601119884119883)lowast119882lowastminus 119892(120579119884119883)

lowast119880lowast

+ 119892 (nabla119881 (120601119884) 119883) + 119892 (120601119884 nabla119881119883) + 119892 (119882nabla119884119883)lowast120585

+ 119892 (nabla119881 (120595119884) 119883) + 119892 (120595119884 nabla119881119883)lowast120578

+ 119892 (nabla119881 (120579119884) 119883) + 119892 (120579119884 nabla119881119883) minus 119892 (119880 nabla119884119883)lowast120577

(61)

1015840nabla119882lowast1015840nabla119884lowast119883

lowast

= (nabla119882nabla119884119883)lowast

minus 119892(120595119884119883)lowast119880lowast+ 119892(120601119884119883)

lowast119881lowast

+ 119892 (nabla119882 (120601119884) 119883) + 119892 (120601119884 nabla119882119883) minus 119892 (119881 nabla119884119883)lowast120585

+ 119892 (nabla119882 (120595119884) 119883) + 119892 (120595119884 nabla119882119883) + 119892 (119880 nabla119884119883)lowast120578

+ 119892 (nabla119882 (120579119884) 119883) + 119892 (120579119884 nabla119882119883)lowast120577

(62)

On the other hand (38) and (39) with119883 = 119880 imply1015840nabla[119884lowast 119880lowast]119883

lowast= (nabla[119884119880]119883)

lowastminus 2119908 (119884) 120595119883 minus V (119884) 120579119883

lowast

+ 119892(120601 [119884 119880] 119883)lowast120585 + 119892(120595 [119884119880] 119883)

lowast120578

+ 119892(120579 [119884 119880] 119883)lowast120577

(63)

Similarly from (39) with 119883 = 119881 and (39) with 119883 = 119882respectively we find that1015840nabla[119884lowast 119881lowast]119883

lowast= (nabla[119884119881]119883)

lowast+ 2119908 (119884) 120601119883 minus 119906 (119884) 120579119883

lowast

+ 119892(120601 [119884 119881] 119883)lowast120585 + 119892(120595 [119884 119881] 119883)

lowast120578

+ 119892(120579 [119884 119881] 119883)lowast120577

(64)1015840nabla[119884lowast 119882lowast]119883

lowast= (nabla[119884119882]119883)

lowastminus 2V (119884) 120601119883 minus 119906 (119884) 120595119883

lowast

+ 119892(120601 [119884119882] 119883)lowast120585 + g(120595 [119884119882] 119883)

lowast120578

+ 119892(120579 [119884119882] 119883)lowast120577

(65)

Using (28)ndash(31) it follows from (57) (60) and (63) that1015840119877 (119884lowast 119880lowast)119883lowast= 119877 (119884 119880)119883

lowast

+ 119908 (119883)120595119884 minus V (119883) 120579119884

minus 119892 (120595119884119883)119882 + 119892 (120579119884119883)119881

+ 2119908 (119884) 120595119883 minus 2V (119884) 120579119883lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(66)

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Mathematics and Mathematical Sciences

from which taking account of (35) and using (24) and (33)we obtain

1015840119877 (119884lowast 119880lowast)119883lowast= 119906 (119883)119884 minus 119892 (119884119883)119880

+ 119906 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119880lowast

minus 119903 (119880) 119892 (120595119884119883) minus 119902 (119880) 119892 (120579119884119883)

+ 119906 (119884) 119906 (1198601119883) + 119903 (119884)119908 (119883)

+ 119902 (119884) V (119883) minus 119892 (1198601119884119883)lowast120585

minus minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+ 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (1198601119883)lowast120578

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus 119901(119880)119892(120595119884119883) + 119908(119884)119906(1198601119883)lowast120577

(67)

Similarly by using (58) (59) (61) (62) (64) and (65) we caneasily obtain

1015840119877 (119884lowast 119881lowast)119883lowast= V (119883) 119884 minus 119892 (119884119883)119881

+V (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119881lowast

minus minus119903 (119881) 119892 (120601119884119883) + 119901 (119881) 119892 (120579119884119883)

+ V (119884) V (1198601119883) + 119903 (119884)119908 (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120578

+ 119902 (119884) 119906 (119883) minus 119903 (119881) 119892 (120595119884119883)

+119902 (119881) 119892 (120579119884119883) minus 119906 (119884) V (1198601119883)lowast120585

minus minus119902 (119884)119908 (119883) + 119902 (119881) 119892 (120601119884119883)

minus119901(119881)119892(120595119884119883) + 119908(119884)V(1198601119883)lowast120577

1015840119877 (119884lowast119882lowast)119883lowast= 119908 (119883)119884 minus 119892 (119884119883)119882

+119908 (1198601119883)1198601119884 minus 119892 (1198601119884119883)1198601119882lowast

minus 119902 (119882) 119892 (120601119884119883) minus 119901 (119882) 119892 (120595119884119883)

+ 119908 (119884)119908 (1198601119883) + 119902 (119884) V (119883)

+119901 (119884) 119906 (119883) minus 119892 (1198601119884119883)lowast120577

minus minus119903 (119884) 119906 (119883) + 119903 (119882) 119892 (120595119884119883)

minus119902 (119882) 119892 (120579119884119883) + 119906 (119884)119908 (1198601119883)lowast120585

+ 119903 (119884) V (119883) + 119903 (119882) 119892 (120601119884119883)

minus119901(119882)119892(120579119884119883) minus V(119884)119908(1198601119883)lowast120578

(68)

5 Main Results

It is well known [3] that if 120587minus1(119872) is locally symmetric then1015840nabla1198601 = 0 which implies identities (2) in Theorem K-P Inthis point of view we consider the following assumptions in(69) which are weaker conditions than the locally symmetryof 120587minus1(119872)

In order to obtain our main results let 119872 be 119899-dimensional QR-submanifolds of (119901 minus 1) QR-dimension inQP(119899+119901)4 with the assumptions

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119883lowast) 119881lowast= 0

(1015840nabla1205771015840119877) (119884

lowast 119883lowast)119882lowast= 0

(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (

1015840nabla1205781015840119877) (119884

lowast 119881lowast)119883lowast= 0

(1015840nabla1205771015840119877) (119884

lowast119882lowast)119883lowast= 0

(69)

Wenotice here that the above curvature conditions in (69)are different from those in [18] due to Pak and Sohn

We first consider the assumption

(1015840nabla1205851015840119877) (119884

lowast 119883lowast) 119880lowast= 0 (70)

Differentiating (55) covariantly in the direction of120585 and using (19) (36) and (40) and the assumption(1015840nabla1205851015840119877) (119884

lowast 119883lowast)119880lowast = 0 we have

minus1015840119877 ((120601119884)

lowast 119883lowast)119880lowastminus1015840119877 (119884lowast (120601119883)

lowast)119880lowast

= minus119906 (119883) 120601119884 + 119906 (119884) 120601119883

minus119906 (1198601119883)1206011198601119884 + 119906 (1198601119884) 1206011198601119883lowast

+ 119901 (119883)119908 (119884) minus 119901 (119884)119908 (119883)

+ 119908 (119883) 119906 (1198601119884) minus 119908 (119884) 119906 (1198601119883)lowast120578

minus 119901 (119883) V (119884) minus 119901 (119884) V (119883)

+ V(119883)119906(1198601119884) minus V(119884)119906(1198601119883)lowast120577

(71)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (55) itself we can get

119903 (119883) V (119884) minus 119903 (119884) V (119883) minus 119902 (119883)119908 (119884)

+ 119902 (119884)119908 (119883) minus 119903 (120601119884)119908 (119883) + 119903 (120601119883)119908 (119884)

minus 119902 (120601119884) V (119883) + 119902 (120601119883) V (119884)

minus 119906 (119883) 119906 (1198601120601119884) + 119906 (119884) 119906 (1198601120601119883) = 0

(72)

minus 119906 (1198601120601119884) V (119883) + 119906 (1198601120601119883) V (119884) + 119901 (120601119884) V (119883)

minus 119901 (120601119883) V (119884) = 0(73)

minus 119906 (1198601120601119884)119908 (119883) + 119906 (1198601120601119883)119908 (119884) + 119901 (120601119884)119908 (119883)

minus 119901 (120601119883)119908 (119884) = 0(74)

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 9

Putting 119884 = 119880 in (72) and using (19) and (24) we have1206011198601119880 + 119903 (119880)119881 minus 119902 (119880)119882 = 0 (75)

and consequently

119903 (119880) = 119908 (1198601119880) = 119906 (1198601119882)

119902 (119880) = V (1198601119880) = 119906 (1198601119881) (76)

Putting 119884 = 119882 and 119883 = 119881 in (73) and using (15) and (24)yield

119901 (119881) = V (1198601119880) = 119906 (1198601119881) (77)Also putting 119884 = 119881 and 119883 = 119882 in (74) and using (15) and(24) we have

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) (78)Summing up we have

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

(79)

Thus we get the following lemma

Lemma 1 Let119872 be an 119899-dimensional QR-submanifold of (119901minus1) QR-dimension in a quaternionic projective space QP(119899+119901)4and let the normal vector field 1198731 be parallel with respect tothe normal connection If the equalities in (69) are establishedthen

1198601119880 = 119906 (1198601119880)119880 + 119901 (119881)119881 + 119901 (119882)119882

1198601119881 = 119902 (119880)119880 + V (1198601119881)119881 + 119902 (119882)119882

1198601119882 = 119903 (119880)119880 + 119903 (119881)119881 + 119908 (1198601119882)119882

119901 (119881) = V (1198601119880) = 119906 (1198601119881) = 119902 (119880)

119901 (119882) = 119908 (1198601119880) = 119906 (1198601119882) = 119903 (119880)

119902 (119882) = 119908 (1198601119881) = V (1198601119882) = 119903 (119881)

(80)

Next we assume the additional condition(1015840nabla1205851015840119877) (119884

lowast 119880lowast)119883lowast= 0 (81)

Differentiating (67) covariantly in the direction of 120585 andusing (36) (40) and the assumption ( 1015840nabla120585

1015840119877) (119884

lowast 119880lowast)119883lowast =

0 we have

minus1015840119877 ((120601119884)

lowast 119880lowast)119883lowastminus1015840119877 (119884lowast 119880lowast) (120601119883)

lowast

= minus119906 (119883) 120601119884 minus 119906 (1198601119883)1206011198601119884

+119892(1198601119884119883)1206011198601119880lowast

minus minus119901 (119884)119908 (119883) + 119902 (119880) 119892 (120601119884119883)

minus119901 (119880) 119892 (120595119884119883) + 119908 (119884) 119906 (1198601119883)lowast120578

+ minus119901 (119884) V (119883) minus 119903 (119880) 119892 (120601119884119883)

+119901(119880)119892(120579119884119883) + V(119884)119906(1198601119883)lowast120577

(82)

from which taking the vertical component of 120585 120578 and 120577respectively and using (22)ndash(24) and (67) itself we can find

119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) + 119903 (120601119884)119908 (119883) + 119903 (119884) V (119883)

+ 119902 (120601119884) V (119883) minus 119902 (119884)119908 (119883)

+ 119892 (1206011198601119884 minus 1198601120601119884119883) = 0

(83)

minus 119902 (119880) 119892 (120601119884119883) + 119901 (120601119884) V (119883) minus V (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120595119884119883) + 119906 (119884)119908 (119883)

minus119908 (119884) 119906 (119883) = 0

(84)

minus 119903 (119880) 119892 (120601119884119883) + 119901 (120601119884)119908 (119883) minus 119908 (119884) 119906 (1198601120601119883)

minus 119901 (119880) 119892 (120579119884119883) + V (119884) 119906 (119883) minus 119906 (119884) V (119883) = 0

(85)

Taking the skew-symmetric part of (83) with respect to119883and 119884 we have

2119903 (119880) minus2119892 (120579119884119883) + 119906 (119884) V (119883) minus V (119884) 119906 (119883)

minus 2119902 (119880) 2119892 (120595119884119883) + 119906 (119884)119908 (119883) minus 119908 (119884) 119906 (119883)

+ 119906 (119884) 119906 (1198601120601119883) minus 119906 (119883) 119906 (1198601120601119884) + 119903 (120601119884)119908 (119883)

minus 119903 (120601119883)119908 (119884) + 119903 (119884) V (119883) minus 119903 (119883) V (119884)

+ 119902 (120601119884) V (119883) minus 119902 (120601119883) V (119884)

minus 119902 (119884)119908 (119883) + 119902 (119883)119908 (119884) = 0

(86)

Replacing 119884 with 120579119884 in (86) and using (19) and (22)ndash(24) wehave

119903 (119880) 4119892 (119884119883) minus 3119908 (119884)119908 (119883)

minus2119906 (119884) 119906 (119883) minus 2V (119884) V (119883)

minus 119902 (119880) 4119892 (120601119884119883) + 3119908 (119884) V (119883) minus 2119908 (119883) V (119884)

minus 119902 (120579119884)119908 (119883) minus 119902 (120595119884) V (119883) minus 119902 (120601119883) 119906 (119884)

+ 119903 (120579119884) V (119883) minus 119903 (119883) 119906 (119884) minus 119903 (120595119884)119908 (119883)

minus V (119884) 119906 (1198601120601119883) + 119906 (119883) 119906 (1198601120595119884)

minus 119906 (1198601119880) 119906 (119883)119908 (119884) = 0

(87)

Now we consider the following orthonormal basis

119880 119881119882 1198901 119890119898 120601 (1198901) 120601 (119890119898)

120595 (1198901) 120595 (119890119898) 120579 (1198901) 120579 (119890119898) (88)

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Mathematics and Mathematical Sciences

which will be called 119876-basis where 4119898 + 3 = dim119872 Takingthe trace of the above equation with respect to the 119876-basisand using (76) we can easily see 4119898119903(119880) = 0 that is

119903 (119880) = 0 (89)

Replacing also119884with120595119884 in (86) and using (19) and (22)ndash(24) we have

119902 (119880) = 0 (90)

Substituting (89) and (90) into (75) we have 1206011198601119880 = 0 andhence

1198601119880 = 119906 (1198601119880)119880 (91)

On the other hand replacing119884with120595119884 in (84) and using(19) (22) (23) and (90) we obtain

119901 (119880) 119892 (119884119883) minus 119906 (119884) 119906 (119883) minus 119908 (119884)119908 (119883)

+ 119901 (120579119884) V (119883) = 0(92)

from which taking the trace with respect to the 119876-basis andusing (15) and (24) we find 4119898119901(119880) = 0 that is

119901 (119880) = 0 (93)

which together with (84) and (90) implies

119901 (120601119884) = 0 (94)

Replacing119884with 120601119884 in the above equation and using (17) and(93) we can easily see that

119901 (119884) = 0 (95)

for any vector 119884 tangent to119872Summing up we have the following lemma

Lemma 2 Let119872 be as in Lemma 1 and let the normal vectorfield1198731 be parallel with respect to the normal connection If theequalities in (69) and (52) are established then

1198601119880 = 119906 (1198601119880)119880 1198601119881 = V (1198601119881)119881

1198601119882 = 119908(1198601119882)119882 119901 = 119902 = 119903 = 0(96)

Finally we will prove our main theorem

Theorem 3 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the equalities in (69) and(52) are established then 120587minus1(119872) is locally a product of1198721 times1198722 where1198721 and1198722 belong to some (41198991 +3)- and (41198992 +3)-dimensional spheres (120587 is the Hopf fibration 119878119899+119901+3(1) rarr

119876119875(119899+119901)4)

Proof By means of (96) it follows easily from (83) that

1206011198601 = 1198601120601 (97)

By the quite same method we can obtain

1205951198601 = 1198601120595 1205791198601 = 1198601120579 (98)

Combining with those equalities and Theorem K-P wecomplete the proof

Corollary 4 Let 119872 be an 119899-dimensional 119876119877-submanifoldof (119901 minus 1) 119876119877-dimension in a quaternionic projective space119876119875(119899+119901)4 and let the normal vector field 1198731 be parallel withrespect to the normal connection If the following equalities

1015840nabla1205851015840119877 = 0

1015840nabla1205781015840119877 = 0

1015840nabla1205771015840119877 = 0 (99)

are established then 120587minus1(119872) is locally a product of 1198721 times 1198722where 1198721 and 1198722 belong to some (41198991 + 3)- and (41198992 + 3)-dimensional spheres

Acknowledgment

This work was supported by the 2013 Inje University researchgrant

References

[1] A BejancuGeometry of CR-Submanifolds D Reidel PublishingCompany Dordrecht The Netherlands 1986

[2] J-H Kwon and J S Pak ldquoScalar curvature of QR-submanifoldsimmersed in a quaternionic projective spacerdquo Saitama Mathe-matical Journal vol 17 pp 89ndash116 1999

[3] J-H Kwon and J S Pak ldquoQR-submanifolds of119876QR-dimensionin a quaternionic projective space119876119875(119899+119901)4rdquo Acta MathematicaHungarica vol 86 no 1-2 pp 89ndash116 2000

[4] H B Lawson Jr ldquoRigidity theorems in rank-(119899 minus 1) symmetricspacesrdquo Journal of Differential Geometry vol 4 pp 349ndash3571970

[5] AMartınez and J D Perez ldquoReal hypersurfaces in quaternionicprojective spacerdquo Annali di Matematica Pura ed Applicata vol145 pp 355ndash384 1986

[6] J S Pak ldquoReal hypersurfaces in quaternionic Kaehlerianmanifolds with constant 119876119875(119899+119901)4-sectional curvaturerdquo KodaiMathematical Seminar Reports vol 29 no 1-2 pp 22ndash61 1977

[7] J D Perez and F G Santos ldquoOn pseudo-Einstein real hypersur-faces of the quaternionic projective spacerdquo Kyungpook Mathe-matical Journal vol 25 no 1 pp 15ndash28 1985

[8] J D Perez and F G Santos ldquoOn real hypersurfaces withharmonic curvature of a quaternionic projective spacerdquo Journalof Geometry vol 40 no 1-2 pp 165ndash169 1991

[9] Y Shibuya ldquoReal submanifolds in a quaternionic projectivespacerdquo Kodai Mathematical Journal vol 1 no 3 pp 421ndash4391978

[10] S Ishihara ldquoQuaternion Kahlerian manifolds and fibred Rie-mannian spaces with Sasakian (119901 minus 1)-structurerdquo Kodai Math-ematical Seminar Reports vol 25 pp 321ndash329 1973

[11] S Ishihara and M Konishi ldquoFibred riemannian space withsasakian 3-structurerdquo in Differential Geometry in Honor of KYano pp 179ndash194 Kinokuniya Tokyo Japan 1972

[12] Y-Y Kuo ldquoOn almost contact (119901 minus 1)-structurerdquo The TohokuMathematical Journal vol 22 pp 325ndash332 1970

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 11

[13] Y Y Kuo and S-I Tachibana ldquoOn the distribution appeared incontact (119901 minus 1)-structurerdquo Taiwanese Journal of Mathematicsvol 2 pp 17ndash24 1970

[14] Y-W Choe andMOkumura ldquoScalar curvature of a certain CR-submanifold of complex projective spacerdquo Archiv der Mathe-matik vol 68 no 4 pp 340ndash346 1997

[15] M Okumura and L Vanhecke ldquo119899-dimensional real submani-folds with (119901 minus 1)-dimensional maximal holomorphic tangentsubspace in complex projective spacesrdquo Rendiconti del CircoloMatematico di Palermo vol 43 no 2 pp 233ndash249 1994

[16] J S Pak and W-H Sohn ldquoSome curvature conditions of 119899-dimensional CR-submanifolds of (119901 minus 1) CR-dimension in acomplex projective spacerdquo Bulletin of the Korean MathematicalSociety vol 38 no 3 pp 575ndash586 2001

[17] W-H Sohn ldquoSome curvature conditions of n-dimensional CR-submanifolds of (nminus1) CR-dimension in a complex projectivespacerdquo Communications of the Korean Mathematical Societyvol 25 pp 15ndash28 2000

[18] J S Pak and W-H Sohn ldquoSome curvature conditions of119899-dimensional QR-submanifolds of 119876 QR-dimension in aquaternionic projective space 119876119875(119899+119901)4rdquo Bulletin of the KoreanMathematical Society vol 40 no 4 pp 613ndash631 2003

[19] S Ishihara ldquoQuaternion Kahlerian manifoldsrdquo Journal of Dif-ferential Geometry vol 9 pp 483ndash500 1974

[20] K Yano and S Ishihara ldquoFibred spaces with invariant Rieman-nian metricrdquo Kodai Mathematical Seminar Reports vol 19 pp317ndash360 1967

[21] B Y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973

[22] B OrsquoNeill ldquoThe fundamental equations of a submersionrdquo TheMichigan Mathematical Journal vol 13 pp 459ndash469 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of