Research ArticleSome Results on Generalized Quasi-Einstein Manifolds

D G Prakasha1 and H Venkatesha2

1 Department of Mathematics Karnatak University Dharwad 580 003 India2Department of Mathematics Kuvempu University Shankaraghatta 577 451 India

Correspondence should be addressed to D G Prakasha prakashadggmailcom

Received 19 September 2013 Accepted 31 October 2013 Published 21 January 2014

Academic Editors Y Fu N Tang and H Yan

Copyright copy 2014 D G Prakasha and H Venkatesha This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper deals with generalized quasi-Einstein manifold satisfying certain conditions on conharmonic curvature tensor Here westudy some geometric properties of generalized quasi-Einstein manifold and obtain results which reveal the nature of its associated1-forms

1 Introduction

It is well known that a Riemannian or a semi-Riemannianmanifold (119872119899 119892) (119899 ge 3) is an Einstein manifold if its Riccitensor 119878 of type (0 2) is of the form 119878 = (119903119899)119892 119903 being the(constant) scalar curvature of the manifold Let 119880

1198781

= 119909 isin

119872 119878 = (119903119899)119892 at 119909 Then the manifold (119872119899 119892) is said to bea quasi-Einstein manifold [1ndash7] if on 119880

1198781

sub 119872 we have

119878 = 120572119892 + 120573119860 otimes 119860 (1)

where119860 is 1-form on1198801198781

and 120572 120573 are some smooth functionson 1198801198781

It is clear that the function 120573 and the 1-form 119860

are nonzero at every point on 1198801198781

The scalars 120572 120573 are theassociated scalars of the manifold Also the 1-form119860 is calledassociated 1-form of the manifold defined by 119892(119883 120588) = 119860(119883)

for any vector field 119883 120588 being a unit vector field calledgenerator of the manifold Such an 119899-dimensional quasi-Einstein manifold is denoted by (119876119864)

119899 The quasi-Einstein

manifolds have also been studied in [8ndash11]Generalizing the notion of quasi-Einstein manifold in

[12] De and Ghosh introduced the notion of generalizedquasi-Einstein manifolds and studied its geometrical prop-erties with the existence of such notion by several nontrivialexamples A Riemannian manifold (119872

119899 119892) (119899 gt 3) is said tobe generalized quasi-Einstein manifold if its Ricci tensor 119878 oftype (0 2) is not identically zero and satisfies the condition

119878 (119883 119884) = 120572119892 (119883 119884) + 120573119860 (119883)119860 (119884) + 120574119861 (119883) 119861 (119884) (2)

where 120572 120573 and 120574 are scalars of which 120573 = 0 120574 = 0 and 119860 119861are nowhere vanishing 1-forms such that 119892(119883 120588) = 119860(119883)119892(119883 120583) = 119861(119883) for any vector field 119883 The unit vectors 120588

and 120583 corresponding to 1-forms 119860 and 119861 are orthogonal toeach other Also 120588 and 120583 are the generators of the manifoldSuch an 119899-dimensional manifold is denoted by 119866(119876119864)

119899 The

generalized quasi-Einstein manifolds have also been studiedin [13ndash16]

In 2008 De and Gazi [17] introduced the notion of nearlyquasi-Einstein manifolds A nonflat Riemannian manifold(119872119899 119892) (119899 gt 2) is called a nearly quasi-Einstein manifold if

its Ricci tensor of type (0 2) is not identically zero and satisfiesthe condition

119878 (119883 119884) = 120572119892 (119883 119884) + 120573119864 (119883 119884) (3)

where 120572 and 120573 are nonzero scalars and 119864 is a nonzerosymmetric tensor of type (0 2) An 119899-dimensional nearlyquasi-Einstein manifold was denoted by 119873(119876119864)

119899 The nearly

quasi-Einstein manifolds have also been studied by Prakashaand Bagewadi [18]

The present paper is organized as follows Section 2deals with the preliminaries Section 3 is concerned withconharmonic curvature tensor on 119866(119876119864)

119899 In this section

it is proved that a conharmonically flat 119866(119876119864)119899is one of

the manifold of generalized quasiconstant curvature Alsoit is proved that if in a 119866(119876119864)

119899(119899 ge 3) the associated

scalars are constants and the generators 120588 and 120583 are vector

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2014 Article ID 563803 5 pageshttpdxdoiorg1011552014563803

2 Chinese Journal of Mathematics

fields with the associated 1-forms 119860 and 119861 not being the 1-forms of recurrences then the manifold is conharmonicallyconservative In Section 4 we consider 119866(119876119864)

119899(119899 ge 3)

satisfying the conditionL sdot 119878 = 0 In the last section we studysome geometrical properties of a 119866(119876119864)

119899

2 Preliminaries

Consider a 119866(119876119864)119899with associated scalars 120572 120573 and 120574 and

associated 1-forms 119860 119861 Then from (2) we get

119903 = 119899120572 + 120573 + 120574 (4)

where 119903 is the scalar curvature of the manifold Since 120588 and120583 are orthogonal unit vector fields 119892(120588 120588) = 1 119892(120583 120583) = 1and 119892(120588 120583) = 0 Again from (2) we have

119878 (119883 120588) = (120572 + 120573)119860 (119883) 119878 (120588 120588) = 120572 + 120573

119878 (119883 120583) = (120572 + 120574) 119861 (119883) 119878 (120583 120583) = 120572 + 120574(5)

Let 119876 be the symmetric endomorphism of the tangentspace at each point of themanifold corresponding to the Riccitensor 119878 Then 119892(119876119883 119884) = 119878(119883 119884) for all 119883119884

The rank-four tensor L1015840 that remains invariant underconharmonic transformation for an 119899-dimensional Rieman-nian manifold is given by [19]

L1015840

(119883 119884 119885 119880)

= 1198771015840

(119883 119884 119885 119880) minus1

119899 minus 2

times [119878 (119884 119885) 119892 (119883119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119892 (119884 119885) 119878 (119883119880) minus 119892 (119883 119885) 119878 (119884 119880)]

(6)

where L1015840(119883 119884 119885 119880) = 119892(L(119883 119884)119885119880) and 1198771015840 denotesthe Riemannian curvature tensor of type (0 4) defined by1198771015840(119883 119884 119885 119880) = 119892(119877(119883 119884)119885119880) where 119877 is the Riemannian

curvature tensor of type (1 3)

3 Conharmonic Curvature Tensor on 119866(119876119864)119899

A manifold of generalized quasiconstant curvature tensoris 119866(119876119864)

119899 But the converse is not true in general In this

section we enquire under what conditions the converse willbe true

A manifold whose conharmonic curvature tensor van-ishes at every point of themanifold is called conharmonicallyflat manifold Thus this tensor represents the deviation ofthe manifold from conharmonic flatness It satisfies all thesymmetric properties of the Riemannian curvature tensor1198771015840 There are many physical applications of the tensor L

For example in [20] Abdussattar showed that sufficientcondition for a space-time to be conharmonic to a flat is eitherempty in which case it is flat or filled with a distributionrepresented by energy momentum tensor 119879 possessing thealgebraic structure of an electromagnetic field and conformalto a flat space-time [20]

Let us consider that the manifold under consideration isconharmonically flat Then from (6) we have

1198771015840

(119883 119884 119885 119880)

=1

119899 minus 2[119878 (119884 119885) 119892 (119883119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119892 (119884 119885) 119878 (119883119880) minus 119892 (119883 119885) 119878 (119884 119880)]

(7)

Using (2) in (7) we obtain

1198771015840

(119883 119884 119885 119880)

=2120572

119899 minus 2[119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+120573

119899 minus 2[119892 (119883119880)119860 (119884)119860 (119885) minus 119892 (119884 119880)119860 (119883)119860 (119885)

+119892 (119884 119885)119860 (119883)119860 (119880) minus 119892 (119883 119885)119860 (119884)119860 (119880)]

+120574

119899 minus 2[119892 (119883119880) 119861 (119884) 119861 (119885) minus 119892 (119884119880) 119861 (119883) 119861 (119885)

+119892 (119884 119885) 119861 (119883) 119861 (119880) minus 119892 (119883 119885) 119861 (119884) 119861 (119880)]

(8)

According toChen andYano [21] a Riemannianmanifold(119872119899 119892) (119899 gt 3) is said to be of quasiconstant curvature if it isconformally flat and its curvature tensor 1198771015840 of type (0 4) hasthe form

1198771015840

(119883 119884 119885 119880)

= 119901 [119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+ 119902 [119892 (119883119880) 119879 (119884) 119879 (119885) minus 119892 (119884 119880) 119879 (119883) 119879 (119885)

+ 119892 (119884 119885) 119879 (119883) 119879 (119880) minus 119892 (119883 119885) 119879 (119884) 119879 (119880)]

(9)

where 119879 is a form defined by 119892(119883 120588) = 119860(119883) with 120588 as a unitvector field and 119901 119902 are scalars of which 119902 = 0

In [12] De and Ghosh generalize the notion of quasicon-stant curvature and prove the existence of such a manifold ARiemannian manifold is said to be a manifold of generalizedquasiconstant curvature if the curvature tensor 119877

1015840 of type(0 4) satisfies the condition

1198771015840

(119883 119884 119885 119880)

= 119901 [119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+ 119902 [119892 (119883119880) 119879 (119884) 119879 (119885) minus 119892 (119884119880) 119879 (119883)119879 (119885)

+119892 (119884 119885) 119879 (119883) 119879 (119880) minus 119892 (119883 119885) 119879 (119884) 119879 (119880)]

+ 119904 [119892 (119883119880)119863 (119884)119863 (119885) minus 119892 (119884119880)119863 (119883)119863 (119885)

+ 119892 (119884 119885)119863 (119883)119863 (119880) minus 119892 (119883 119885)119863 (119884)119863 (119880)]

(10)

where119901 119902 and 119904 are scalars and119879 and119863 are nonzero 1-forms

Chinese Journal of Mathematics 3

We assume that the unit vector fields 120588 and 1205881defined by

119892(119883 120588) = 119879(119883) and 119892(119883 1205881) = 119863(119883) are orthogonal that is

119892(120588 1205881) = 0 Now the relation (8) can be written as

1198771015840

(119883 119884 119885 119880)

= 1199011[119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+ 1199021[119892 (119883119880) 119879 (119884) 119879 (119885) minus 119892 (119884119880) 119879 (119883) 119879 (119885)

+ 119892 (119884 119885) 119879 (119883)119879 (119880) minus 119892 (119883 119885) 119879 (119884) 119879 (119880)]

+ 1199041[119892 (119883119880)119863 (119884)119863 (119885) minus 119892 (119884119880)119863 (119883)119863 (119885)

+119892 (119884 119885)119863 (119883)119863 (119880) minus 119892 (119883 119885)119863 (119884)119863 (119880)]

(11)

where 1199011= (2120572(119899 minus 2)) 119902

1= (120573(119899 minus 2)) and 119904

1= (120574(119899 minus

2)) Comparing (10) and (11) it follows that the manifoldis of generalized quasiconstant curvature Thus we have thefollowing theorem

Theorem 1 A conharmonically flat 119866(119876119864)119899is one of general-

ized quasiconstant curvature

Next differentiating (6) covariantly and then contractingwe obtain(div sdotL) (119883 119884)119885

= (div sdot119877) (119883 119884)119885 minus1

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

minus1

2 (119899 minus 2)[(nabla119883119903) 119892 (119884 119885) minus (nabla

119884119903) 119892 (119883 119885)]

(12)

where div denotes the divergenceAgain it is known that in a Riemannianmanifold we have

(div sdot119877) (119883 119884)119885 = (nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885) (13)

Consequently by virtue of (13) the relation (12) takes theform

(div sdotL) (119883 119884)119885

=119899 minus 3

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

minus1

2 (119899 minus 2)[(nabla119883119903) 119892 (119884 119885) minus (nabla

119884119903) 119892 (119883 119885)]

(14)

Now consider the associated scalars 120572 120573 and 120574 as constantsthen (4) yields that the scalar curvature 119903 is constant andhence 119889119903(119883) = 0 for all 119883 Consequently (14) reduces to

(div sdotL) (119883 119884)119885 =119899 minus 3

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

(15)

Since 120572 120573 and 120574 are constants we have from (2) that

(nabla119883119878) (119884 119885) = 120573 [(nabla

119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)]

+ 120574 [(nabla119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)]

(16)

By virtue of (16) we get from (15) that

(div sdotL) (119883 119884)119885

=119899 minus 3

119899 minus 2[120573 (nabla

119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)

minus (nabla119884119860) (119883)119860 (119885) minus 119860 (119883) (nabla

119884119860) (119885)]

+ 120574 (nabla119883119861) (119884) 119861 (119885) + 119861 (119884) (nabla

119883119861) (119885)

minus (nabla119884119861) (119883) 119861 (119885) minus 119861 (119883) (nabla

119884119861) (119885)

(17)

Next if the generators 120588 and 120583 of the manifold underconsideration are recurrent vector fields [22] then we havenabla119883120588 = 120587

1(119883)120588 and nabla

119883120583 = 120587

2(119883)120583 where 120587

1and 120587

2are the 1-

forms of the recurrence such that 1205871and 120587

2are different from

119860 and 119861 Consequently we get

(nabla119883119860) (119884) = 119892 (nabla

119883120588 119884) = 119892 (120587

1(119883) 120588 119884) = 120587

1(119884)119860 (119885)

(nabla119883119861) (119884) = 119892 (nabla

119883120583 119884) = 119892 (120587

2(119883) 120583 119884) = 120587

2(119884) 119861 (119885)

(18)

In view of (18) relation (17) reduces to

(div sdotL) (119883 119884)119885

=2 (119899 minus 3)

(119899 minus 2)[120573 1205871(119883)119860 (119884) minus 120587

1(119884)119860 (119883)119860 (119885)

times120574 1205872(119883) 119861 (119884) minus 120587

2(119884) 119861 (119883) 119861 (119885)]

(19)

Also since 119892(120588 120588) = 119892(120583 120583) = 1 it follows that (nabla119883119860)(120588) =

119892(nabla119883120588 120588) = 0 and hence (18) reduces to 120587

1(119883) = 0 for

all 119883 Similarly we have 1205872(119883) = 0 Hence from (19)

we have (div sdotL)(119883 119884)119885 = 0 that is the manifold underconsideration is conharmonically conservative Hence wecan state the following theorem

Theorem 2 If in a 119866(119876119864)119899(119899 gt 3) the associated scalars

are constants and the generators 120588 and 120583 corresponding to theassociated 1-forms 119860 and 119861 are not being the recurrence 1-forms then the manifold is conharmonically conservative

4 119866(119876119864)119899(119899 gt 3) Satisfying

the Condition L sdot 119878 = 0

Let us consider a 119866(119876119864)119899(119899 gt 3) satisfying the condition

L sdot 119878 = 0 Then we have

119878 (L (119883 119884)119885119880) + 119878 (119885L (119883 119884)119880) = 0 (20)

Setting 119885 = 120588 and 119880 = 120583 in (20) we have

119878 (L (119883 119884) 120588 120583) + 119878 (120588L (119883 119884) 120583) = 0 (21)

By using (5) in (21) we obtain

(120572 + 120574) 119861 (L (119883 119884) 120588) + (120572 + 120573)119860 (L (119883 119884) 120583) = 0 (22)

4 Chinese Journal of Mathematics

Next putting119885 = 120588 in (6) and then taking inner product with120583 we get

L1015840

(119883 119884 120588 120583)

= 1198771015840

(119883 119884 120588 120583) minus1

119899 minus 2

times [119878 (119884 120583) 119892 (119883 120588) minus 119878 (119883 120583) 119892 (119884 120588)

+ 119892 (119884 120583) 119878 (119883 120588) minus119892 (119883 120583) 119878 (119884 120588)]

(23)

where 119892(L(119883 119884)120588 120583) = L1015840(119883 119884 120588 120583) = 119861(L(119883 119884)120588)Again from (5) we obtain from (23)

L1015840

(119883 119884 120588 120583) = 1198771015840

(119883 119884 120588 120583)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(24)

Again plugging 119885 = 120583 in (6) and then taking inner productwith 120588 we have

L1015840

(119883 119884 120583 120588) = 1198771015840

(119883 119884 120583 120588)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119883) 119861 (119884) minus 119860 (119884) 119861 (119883)]

(25)

where 119892(L(119883 119884)120583 120588) = L1015840(119883 119884 120583 120588) = 119860(L(119883 119884)120583)From (24) and (25) one can get

119860 (L (119883 119884) 120583) = minus119861 (L (119883 119884) 120588) (26)

By taking account of (26) in (22) we obtain

(120574 minus 120573) 119861 (L (119883 119884) 120583) = 0 (27)

This implies either 120574 = 120573 or 119861(L(119883 119884)120583) = 0Now if 120574 = 120573 then from (2) we have

119878 (119883 119884) = 120572119892 (119883 119884) + 120573 [119860 (119883)119860 (119884) + 119861 (119883) 119861 (119884)]

= 120572119892 (119883 119884) + 120573119864 (119883 119884) (28)

where119864(119883 119884) = 119860(119883)119860(119884)+119861(119883)119861(119884)That is themanifoldis a 119873(119876119864)

119899

On the other hand if 119861(119871(119883 119884)120588) = 0 then (24) gives

1198771015840

(119883 119884 120588 120583) =(2120572 + 120573 + 120574)

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(29)

Thus we can state the following theorem

Theorem 3 If119872 is a 119866(119876119864)119899(119899 gt 3) satisfying the condition

L sdot 119878 = 0 then either 119872 is 119873(119876119864)119899or the curvature tensor 119877

of the manifold satisfies the property (29)

5 Some Geometric Properties of 119866(119876119864)119899

In [23] Gray introduced two classes of Riemannian man-ifolds determined by the covariant differentiation of Riccitensor Class A consists of all Riemannian manifolds whoseRicci tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) = (nabla

119884119878) (119883 119885) (30)

that is Ricci tensor 119878 is a Codazzi type tensorClass B consists of all Riemannian manifolds whose Ricci

tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (31)

that is Ricci tensor 119878 is cyclic parallelFirst suppose that the associated scalars are constants and

the Ricci tensor 119878 is of Codazzi typeThen from (2) we obtain

(nabla119883119878) (119884 119885) = 120573 [(nabla

119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

(32)

Interchanging 119883 and 119884 in (32) we have

(nabla119884119878) (119883 119885) = 120573 [(nabla

119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(33)

Since 119878 is of Codazzi type we have from (32) and (33) that

120573 [(nabla119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

= 120573 [(nabla119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(34)

Putting 119885 = 120588 in (34) and using (nabla119883119860)(120588) = 0 and

(nabla119883119861)(120588) = 0 since 120588 is a unit vector we get

(nabla119883119861) (119884) minus (nabla

119884119861) (119883) = 0 (35)

This shows that 119889119861(119883 119884) = 0Again putting119885 = 120583 in (34) and using (nabla

119883119860)(120583) = 0 and

(nabla119883119861)(120583) = 0 since 120583 is a unit vector we get

(nabla119883119860) (119884) minus (nabla

119884119860) (119883) = 0 (36)

This shows that 119889119860(119883 119884) = 0Thus we can state the following theorem

Theorem 4 In a 119866(119876119864)119899 if the associated scalars are con-

stants and the Ricci tensor is of Codazzi type then theassociated 1-forms 119860 and 119861 are closed

Next suppose that the generators 120588 and 120583 are Killingvector fields in a 119866(119876119864)

119899and the associated scalars are

constants Then

(L119880119892) (119883 119884) = 0 (L

119881119892) (119883 119884) = 0 (37)

Chinese Journal of Mathematics 5

where L denotes the Lie derivative which implies that

119892 (nabla119883119880119884) + 119892 (119883 nabla

119884119880) = 0

119892 (nabla119883119881119884) + 119892 (119883 nabla

119884119881) = 0

(38)

From (38) it follows that

(nabla119883119860) (119884) + (nabla

119884119860) (119883) = 0

(nabla119883119861) (119884) + (nabla

119884119861) (119883) = 0

(39)

since 119892(nabla119883119880119884) = (nabla

119883119860)(119884) and 119892(nabla

119883119881119884) = (nabla

119883119861)(119884)

Similarly we have

(nabla119883119860) (119885) + (nabla

119885119860) (119883) = 0

(nabla119883119861) (119885) + (nabla

119885119861) (119883) = 0

(nabla119884119860) (119885) + (nabla

119885119860) (119884) = 0

(nabla119884119861) (119885) + (nabla

119885119861) (119884) = 0

(40)

for all119883119884 and119885 Now from (2) and using the relations (39)-(40) we have

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (41)

Therefore we can state the following theorem

Theorem 5 In a 119866(119876119864)119899 if the generators are Killing vector

fields and the associated scalars are constants then the Riccitensor of the manifold is cyclic parallel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

D G Prakasha is thankful to University Grants CommissionNew Delhi India for financial support in the form of MajorResearch Project (no 39-302010 (SR) dated December 232012)

References

[1] M C Chaki and R K Maity ldquoOn quasi Einstein manifoldsrdquoPublicationes Mathematicae vol 57 no 3-4 pp 297ndash306 2000

[2] F Defever R Deszcz M Hotlos M Kucharski and Z Sen-turk ldquoGeneralisations of Robertson walker space AnnalesUniversitatis Scientiarum Budapestinensis de Rolando EotvosNominataerdquo Sectio Mathematica vol 43 pp 13ndash24 2000

[3] R Deszcz F Dillen L Verstraelen and L Vrancken ldquoQuasi-Einstein totally real submanifolds of the nearly Kahler 6-sphererdquo Tohoku Mathematical Journal vol 51 no 4 pp 461ndash478 1999

[4] R Deszcz M Glogowska M Hotlorsquos and Z Senturk ldquoOncertain quasi-Einstein semi-symmetric hypersurfacesrdquo AnnalesUniversitatis Scientarium Budapestinensis vol 41 pp 153ndash1661998

[5] R Deszcz M Hotlorsquos and Z Senturk ldquoQuasi-Einstein hyper-surfaces in semi-Riemannian space formsrdquo Colloquium Mathe-maticum vol 81 pp 81ndash97 2001

[6] R Deszcz M Hotlorsquos and Z Senturk ldquoOn curvature propertiesof quasi-Einatein hypersurfaces in semi-Euclidean spacesrdquoSoochow Journal of Mathematics vol 27 no 4 pp 375ndash3892001

[7] M Głogowska ldquoOn quasi-Einstein Cartan type hypersurfacesrdquoJournal of Geometry and Physics vol 58 no 5 pp 599ndash6142008

[8] M C Chaki and P K Ghoshal ldquoSome global properties of quasiEinstein manifoldsrdquo Publicationes Mathematicae vol 63 no 4pp 635ndash641 2003

[9] U C De and G C Ghosh ldquoOn quasi Einstein manifoldsrdquoPeriodica Mathematica Hungarica vol 48 no 1-2 pp 223ndash2312004

[10] A A Shaikh and A Patra ldquoOn quasi-conformally at quasi-Einstein spacesrdquo Differential Geometry-Dynamical Systems vol12 pp 201ndash212 2010

[11] A A Shaikh D W Yoon and S K Hui ldquoOn quasi-Einsteinspacetimesrdquo Tsukuba Journal of Mathematics vol 33 no 2 pp305ndash326 2009

[12] U C De and G C Ghosh ldquoOn generalized quasi-Einsteinmanifoldsrdquo Kyungpook Mathematical Journal vol 44 pp 607ndash615 2004

[13] M C Chaki ldquoOn generalized quasi Einstein manifoldsrdquo Publi-cationes Mathematicae vol 58 no 4 pp 683ndash691 2001

[14] U C De and S Mallick ldquoOn the existence of generalized quasi-Einstein manifolds Archivum Mathematicum (Brno)rdquo Tomusvol 47 pp 279ndash291 2011

[15] S K Hui and R S Lemence ldquoOn generalized quasi-Einsteinmanifold admitting W2- curvature tensorrdquo International Jour-nal of Mathematical Analysis vol 6 no 23 pp 1115ndash1121 2012

[16] A A Shaikh and S K Hui ldquoOn some classes of general-ized quasi-Einstein manifoldsrdquo Communications of the KoreanMathematical Society vol 24 no 3 pp 415ndash424 2009

[17] U C De and A K Gazi ldquoOn nearly quasi-Einstein manifoldsrdquoNovi Sad Journal ofMathematics vol 38 no 2 pp 115ndash121 2008

[18] D G Prakasha and C S Bagewadi ldquoOn nearly quasi-EinsteinmanifoldsrdquoMathematica Pannonica vol 21 no 2 pp 265ndash2732010

[19] Y Ishii ldquoOn conharmonic transformationsrdquo Tensor NS vol 7pp 73ndash80 1957

[20] D B Abdussattar ldquoOn conharmonic transformations in generalrelativityrdquo Bulletin of Calcutta Mathematical Society vol 41 pp409ndash416 1966

[21] B Y Chen and K Yano ldquoHypersurfaces of conformally atspacesrdquo Tensor N S vol 26 pp 318ndash322 1972

[22] S Tanno ldquoRicci curvatures of Riemannian manifoldsrdquo TsukubaJournal of Mathematics vol 40 pp 441ndash448 1988

[23] A Gray ldquoEinstein-like manifolds which are not EinsteinrdquoGeometriae Dedicata vol 7 no 3 pp 259ndash280 1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Chinese Journal of Mathematics

fields with the associated 1-forms 119860 and 119861 not being the 1-forms of recurrences then the manifold is conharmonicallyconservative In Section 4 we consider 119866(119876119864)

119899(119899 ge 3)

satisfying the conditionL sdot 119878 = 0 In the last section we studysome geometrical properties of a 119866(119876119864)

119899

2 Preliminaries

Consider a 119866(119876119864)119899with associated scalars 120572 120573 and 120574 and

associated 1-forms 119860 119861 Then from (2) we get

119903 = 119899120572 + 120573 + 120574 (4)

where 119903 is the scalar curvature of the manifold Since 120588 and120583 are orthogonal unit vector fields 119892(120588 120588) = 1 119892(120583 120583) = 1and 119892(120588 120583) = 0 Again from (2) we have

119878 (119883 120588) = (120572 + 120573)119860 (119883) 119878 (120588 120588) = 120572 + 120573

119878 (119883 120583) = (120572 + 120574) 119861 (119883) 119878 (120583 120583) = 120572 + 120574(5)

Let 119876 be the symmetric endomorphism of the tangentspace at each point of themanifold corresponding to the Riccitensor 119878 Then 119892(119876119883 119884) = 119878(119883 119884) for all 119883119884

The rank-four tensor L1015840 that remains invariant underconharmonic transformation for an 119899-dimensional Rieman-nian manifold is given by [19]

L1015840

(119883 119884 119885 119880)

= 1198771015840

(119883 119884 119885 119880) minus1

119899 minus 2

times [119878 (119884 119885) 119892 (119883119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119892 (119884 119885) 119878 (119883119880) minus 119892 (119883 119885) 119878 (119884 119880)]

(6)

where L1015840(119883 119884 119885 119880) = 119892(L(119883 119884)119885119880) and 1198771015840 denotesthe Riemannian curvature tensor of type (0 4) defined by1198771015840(119883 119884 119885 119880) = 119892(119877(119883 119884)119885119880) where 119877 is the Riemannian

curvature tensor of type (1 3)

3 Conharmonic Curvature Tensor on 119866(119876119864)119899

A manifold of generalized quasiconstant curvature tensoris 119866(119876119864)

119899 But the converse is not true in general In this

section we enquire under what conditions the converse willbe true

A manifold whose conharmonic curvature tensor van-ishes at every point of themanifold is called conharmonicallyflat manifold Thus this tensor represents the deviation ofthe manifold from conharmonic flatness It satisfies all thesymmetric properties of the Riemannian curvature tensor1198771015840 There are many physical applications of the tensor L

For example in [20] Abdussattar showed that sufficientcondition for a space-time to be conharmonic to a flat is eitherempty in which case it is flat or filled with a distributionrepresented by energy momentum tensor 119879 possessing thealgebraic structure of an electromagnetic field and conformalto a flat space-time [20]

Let us consider that the manifold under consideration isconharmonically flat Then from (6) we have

1198771015840

(119883 119884 119885 119880)

=1

119899 minus 2[119878 (119884 119885) 119892 (119883119880) minus 119878 (119883 119885) 119892 (119884 119880)

+ 119892 (119884 119885) 119878 (119883119880) minus 119892 (119883 119885) 119878 (119884 119880)]

(7)

Using (2) in (7) we obtain

1198771015840

(119883 119884 119885 119880)

=2120572

119899 minus 2[119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+120573

119899 minus 2[119892 (119883119880)119860 (119884)119860 (119885) minus 119892 (119884 119880)119860 (119883)119860 (119885)

+119892 (119884 119885)119860 (119883)119860 (119880) minus 119892 (119883 119885)119860 (119884)119860 (119880)]

+120574

119899 minus 2[119892 (119883119880) 119861 (119884) 119861 (119885) minus 119892 (119884119880) 119861 (119883) 119861 (119885)

+119892 (119884 119885) 119861 (119883) 119861 (119880) minus 119892 (119883 119885) 119861 (119884) 119861 (119880)]

(8)

According toChen andYano [21] a Riemannianmanifold(119872119899 119892) (119899 gt 3) is said to be of quasiconstant curvature if it isconformally flat and its curvature tensor 1198771015840 of type (0 4) hasthe form

1198771015840

(119883 119884 119885 119880)

= 119901 [119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+ 119902 [119892 (119883119880) 119879 (119884) 119879 (119885) minus 119892 (119884 119880) 119879 (119883) 119879 (119885)

+ 119892 (119884 119885) 119879 (119883) 119879 (119880) minus 119892 (119883 119885) 119879 (119884) 119879 (119880)]

(9)

where 119879 is a form defined by 119892(119883 120588) = 119860(119883) with 120588 as a unitvector field and 119901 119902 are scalars of which 119902 = 0

In [12] De and Ghosh generalize the notion of quasicon-stant curvature and prove the existence of such a manifold ARiemannian manifold is said to be a manifold of generalizedquasiconstant curvature if the curvature tensor 119877

1015840 of type(0 4) satisfies the condition

1198771015840

(119883 119884 119885 119880)

= 119901 [119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+ 119902 [119892 (119883119880) 119879 (119884) 119879 (119885) minus 119892 (119884119880) 119879 (119883)119879 (119885)

+119892 (119884 119885) 119879 (119883) 119879 (119880) minus 119892 (119883 119885) 119879 (119884) 119879 (119880)]

+ 119904 [119892 (119883119880)119863 (119884)119863 (119885) minus 119892 (119884119880)119863 (119883)119863 (119885)

+ 119892 (119884 119885)119863 (119883)119863 (119880) minus 119892 (119883 119885)119863 (119884)119863 (119880)]

(10)

where119901 119902 and 119904 are scalars and119879 and119863 are nonzero 1-forms

Chinese Journal of Mathematics 3

We assume that the unit vector fields 120588 and 1205881defined by

119892(119883 120588) = 119879(119883) and 119892(119883 1205881) = 119863(119883) are orthogonal that is

119892(120588 1205881) = 0 Now the relation (8) can be written as

1198771015840

(119883 119884 119885 119880)

= 1199011[119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+ 1199021[119892 (119883119880) 119879 (119884) 119879 (119885) minus 119892 (119884119880) 119879 (119883) 119879 (119885)

+ 119892 (119884 119885) 119879 (119883)119879 (119880) minus 119892 (119883 119885) 119879 (119884) 119879 (119880)]

+ 1199041[119892 (119883119880)119863 (119884)119863 (119885) minus 119892 (119884119880)119863 (119883)119863 (119885)

+119892 (119884 119885)119863 (119883)119863 (119880) minus 119892 (119883 119885)119863 (119884)119863 (119880)]

(11)

where 1199011= (2120572(119899 minus 2)) 119902

1= (120573(119899 minus 2)) and 119904

1= (120574(119899 minus

2)) Comparing (10) and (11) it follows that the manifoldis of generalized quasiconstant curvature Thus we have thefollowing theorem

Theorem 1 A conharmonically flat 119866(119876119864)119899is one of general-

ized quasiconstant curvature

Next differentiating (6) covariantly and then contractingwe obtain(div sdotL) (119883 119884)119885

= (div sdot119877) (119883 119884)119885 minus1

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

minus1

2 (119899 minus 2)[(nabla119883119903) 119892 (119884 119885) minus (nabla

119884119903) 119892 (119883 119885)]

(12)

where div denotes the divergenceAgain it is known that in a Riemannianmanifold we have

(div sdot119877) (119883 119884)119885 = (nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885) (13)

Consequently by virtue of (13) the relation (12) takes theform

(div sdotL) (119883 119884)119885

=119899 minus 3

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

minus1

2 (119899 minus 2)[(nabla119883119903) 119892 (119884 119885) minus (nabla

119884119903) 119892 (119883 119885)]

(14)

Now consider the associated scalars 120572 120573 and 120574 as constantsthen (4) yields that the scalar curvature 119903 is constant andhence 119889119903(119883) = 0 for all 119883 Consequently (14) reduces to

(div sdotL) (119883 119884)119885 =119899 minus 3

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

(15)

Since 120572 120573 and 120574 are constants we have from (2) that

(nabla119883119878) (119884 119885) = 120573 [(nabla

119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)]

+ 120574 [(nabla119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)]

(16)

By virtue of (16) we get from (15) that

(div sdotL) (119883 119884)119885

=119899 minus 3

119899 minus 2[120573 (nabla

119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)

minus (nabla119884119860) (119883)119860 (119885) minus 119860 (119883) (nabla

119884119860) (119885)]

+ 120574 (nabla119883119861) (119884) 119861 (119885) + 119861 (119884) (nabla

119883119861) (119885)

minus (nabla119884119861) (119883) 119861 (119885) minus 119861 (119883) (nabla

119884119861) (119885)

(17)

Next if the generators 120588 and 120583 of the manifold underconsideration are recurrent vector fields [22] then we havenabla119883120588 = 120587

1(119883)120588 and nabla

119883120583 = 120587

2(119883)120583 where 120587

1and 120587

2are the 1-

forms of the recurrence such that 1205871and 120587

2are different from

119860 and 119861 Consequently we get

(nabla119883119860) (119884) = 119892 (nabla

119883120588 119884) = 119892 (120587

1(119883) 120588 119884) = 120587

1(119884)119860 (119885)

(nabla119883119861) (119884) = 119892 (nabla

119883120583 119884) = 119892 (120587

2(119883) 120583 119884) = 120587

2(119884) 119861 (119885)

(18)

In view of (18) relation (17) reduces to

(div sdotL) (119883 119884)119885

=2 (119899 minus 3)

(119899 minus 2)[120573 1205871(119883)119860 (119884) minus 120587

1(119884)119860 (119883)119860 (119885)

times120574 1205872(119883) 119861 (119884) minus 120587

2(119884) 119861 (119883) 119861 (119885)]

(19)

Also since 119892(120588 120588) = 119892(120583 120583) = 1 it follows that (nabla119883119860)(120588) =

119892(nabla119883120588 120588) = 0 and hence (18) reduces to 120587

1(119883) = 0 for

all 119883 Similarly we have 1205872(119883) = 0 Hence from (19)

we have (div sdotL)(119883 119884)119885 = 0 that is the manifold underconsideration is conharmonically conservative Hence wecan state the following theorem

Theorem 2 If in a 119866(119876119864)119899(119899 gt 3) the associated scalars

are constants and the generators 120588 and 120583 corresponding to theassociated 1-forms 119860 and 119861 are not being the recurrence 1-forms then the manifold is conharmonically conservative

4 119866(119876119864)119899(119899 gt 3) Satisfying

the Condition L sdot 119878 = 0

Let us consider a 119866(119876119864)119899(119899 gt 3) satisfying the condition

L sdot 119878 = 0 Then we have

119878 (L (119883 119884)119885119880) + 119878 (119885L (119883 119884)119880) = 0 (20)

Setting 119885 = 120588 and 119880 = 120583 in (20) we have

119878 (L (119883 119884) 120588 120583) + 119878 (120588L (119883 119884) 120583) = 0 (21)

By using (5) in (21) we obtain

(120572 + 120574) 119861 (L (119883 119884) 120588) + (120572 + 120573)119860 (L (119883 119884) 120583) = 0 (22)

4 Chinese Journal of Mathematics

Next putting119885 = 120588 in (6) and then taking inner product with120583 we get

L1015840

(119883 119884 120588 120583)

= 1198771015840

(119883 119884 120588 120583) minus1

119899 minus 2

times [119878 (119884 120583) 119892 (119883 120588) minus 119878 (119883 120583) 119892 (119884 120588)

+ 119892 (119884 120583) 119878 (119883 120588) minus119892 (119883 120583) 119878 (119884 120588)]

(23)

where 119892(L(119883 119884)120588 120583) = L1015840(119883 119884 120588 120583) = 119861(L(119883 119884)120588)Again from (5) we obtain from (23)

L1015840

(119883 119884 120588 120583) = 1198771015840

(119883 119884 120588 120583)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(24)

Again plugging 119885 = 120583 in (6) and then taking inner productwith 120588 we have

L1015840

(119883 119884 120583 120588) = 1198771015840

(119883 119884 120583 120588)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119883) 119861 (119884) minus 119860 (119884) 119861 (119883)]

(25)

where 119892(L(119883 119884)120583 120588) = L1015840(119883 119884 120583 120588) = 119860(L(119883 119884)120583)From (24) and (25) one can get

119860 (L (119883 119884) 120583) = minus119861 (L (119883 119884) 120588) (26)

By taking account of (26) in (22) we obtain

(120574 minus 120573) 119861 (L (119883 119884) 120583) = 0 (27)

This implies either 120574 = 120573 or 119861(L(119883 119884)120583) = 0Now if 120574 = 120573 then from (2) we have

119878 (119883 119884) = 120572119892 (119883 119884) + 120573 [119860 (119883)119860 (119884) + 119861 (119883) 119861 (119884)]

= 120572119892 (119883 119884) + 120573119864 (119883 119884) (28)

where119864(119883 119884) = 119860(119883)119860(119884)+119861(119883)119861(119884)That is themanifoldis a 119873(119876119864)

119899

On the other hand if 119861(119871(119883 119884)120588) = 0 then (24) gives

1198771015840

(119883 119884 120588 120583) =(2120572 + 120573 + 120574)

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(29)

Thus we can state the following theorem

Theorem 3 If119872 is a 119866(119876119864)119899(119899 gt 3) satisfying the condition

L sdot 119878 = 0 then either 119872 is 119873(119876119864)119899or the curvature tensor 119877

of the manifold satisfies the property (29)

5 Some Geometric Properties of 119866(119876119864)119899

In [23] Gray introduced two classes of Riemannian man-ifolds determined by the covariant differentiation of Riccitensor Class A consists of all Riemannian manifolds whoseRicci tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) = (nabla

119884119878) (119883 119885) (30)

that is Ricci tensor 119878 is a Codazzi type tensorClass B consists of all Riemannian manifolds whose Ricci

tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (31)

that is Ricci tensor 119878 is cyclic parallelFirst suppose that the associated scalars are constants and

the Ricci tensor 119878 is of Codazzi typeThen from (2) we obtain

(nabla119883119878) (119884 119885) = 120573 [(nabla

119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

(32)

Interchanging 119883 and 119884 in (32) we have

(nabla119884119878) (119883 119885) = 120573 [(nabla

119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(33)

Since 119878 is of Codazzi type we have from (32) and (33) that

120573 [(nabla119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

= 120573 [(nabla119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(34)

Putting 119885 = 120588 in (34) and using (nabla119883119860)(120588) = 0 and

(nabla119883119861)(120588) = 0 since 120588 is a unit vector we get

(nabla119883119861) (119884) minus (nabla

119884119861) (119883) = 0 (35)

This shows that 119889119861(119883 119884) = 0Again putting119885 = 120583 in (34) and using (nabla

119883119860)(120583) = 0 and

(nabla119883119861)(120583) = 0 since 120583 is a unit vector we get

(nabla119883119860) (119884) minus (nabla

119884119860) (119883) = 0 (36)

This shows that 119889119860(119883 119884) = 0Thus we can state the following theorem

Theorem 4 In a 119866(119876119864)119899 if the associated scalars are con-

stants and the Ricci tensor is of Codazzi type then theassociated 1-forms 119860 and 119861 are closed

Next suppose that the generators 120588 and 120583 are Killingvector fields in a 119866(119876119864)

119899and the associated scalars are

constants Then

(L119880119892) (119883 119884) = 0 (L

119881119892) (119883 119884) = 0 (37)

Chinese Journal of Mathematics 5

where L denotes the Lie derivative which implies that

119892 (nabla119883119880119884) + 119892 (119883 nabla

119884119880) = 0

119892 (nabla119883119881119884) + 119892 (119883 nabla

119884119881) = 0

(38)

From (38) it follows that

(nabla119883119860) (119884) + (nabla

119884119860) (119883) = 0

(nabla119883119861) (119884) + (nabla

119884119861) (119883) = 0

(39)

since 119892(nabla119883119880119884) = (nabla

119883119860)(119884) and 119892(nabla

119883119881119884) = (nabla

119883119861)(119884)

Similarly we have

(nabla119883119860) (119885) + (nabla

119885119860) (119883) = 0

(nabla119883119861) (119885) + (nabla

119885119861) (119883) = 0

(nabla119884119860) (119885) + (nabla

119885119860) (119884) = 0

(nabla119884119861) (119885) + (nabla

119885119861) (119884) = 0

(40)

for all119883119884 and119885 Now from (2) and using the relations (39)-(40) we have

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (41)

Therefore we can state the following theorem

Theorem 5 In a 119866(119876119864)119899 if the generators are Killing vector

fields and the associated scalars are constants then the Riccitensor of the manifold is cyclic parallel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

D G Prakasha is thankful to University Grants CommissionNew Delhi India for financial support in the form of MajorResearch Project (no 39-302010 (SR) dated December 232012)

References

[1] M C Chaki and R K Maity ldquoOn quasi Einstein manifoldsrdquoPublicationes Mathematicae vol 57 no 3-4 pp 297ndash306 2000

[2] F Defever R Deszcz M Hotlos M Kucharski and Z Sen-turk ldquoGeneralisations of Robertson walker space AnnalesUniversitatis Scientiarum Budapestinensis de Rolando EotvosNominataerdquo Sectio Mathematica vol 43 pp 13ndash24 2000

[3] R Deszcz F Dillen L Verstraelen and L Vrancken ldquoQuasi-Einstein totally real submanifolds of the nearly Kahler 6-sphererdquo Tohoku Mathematical Journal vol 51 no 4 pp 461ndash478 1999

[4] R Deszcz M Glogowska M Hotlorsquos and Z Senturk ldquoOncertain quasi-Einstein semi-symmetric hypersurfacesrdquo AnnalesUniversitatis Scientarium Budapestinensis vol 41 pp 153ndash1661998

[5] R Deszcz M Hotlorsquos and Z Senturk ldquoQuasi-Einstein hyper-surfaces in semi-Riemannian space formsrdquo Colloquium Mathe-maticum vol 81 pp 81ndash97 2001

[6] R Deszcz M Hotlorsquos and Z Senturk ldquoOn curvature propertiesof quasi-Einatein hypersurfaces in semi-Euclidean spacesrdquoSoochow Journal of Mathematics vol 27 no 4 pp 375ndash3892001

[7] M Głogowska ldquoOn quasi-Einstein Cartan type hypersurfacesrdquoJournal of Geometry and Physics vol 58 no 5 pp 599ndash6142008

[8] M C Chaki and P K Ghoshal ldquoSome global properties of quasiEinstein manifoldsrdquo Publicationes Mathematicae vol 63 no 4pp 635ndash641 2003

[9] U C De and G C Ghosh ldquoOn quasi Einstein manifoldsrdquoPeriodica Mathematica Hungarica vol 48 no 1-2 pp 223ndash2312004

[10] A A Shaikh and A Patra ldquoOn quasi-conformally at quasi-Einstein spacesrdquo Differential Geometry-Dynamical Systems vol12 pp 201ndash212 2010

[11] A A Shaikh D W Yoon and S K Hui ldquoOn quasi-Einsteinspacetimesrdquo Tsukuba Journal of Mathematics vol 33 no 2 pp305ndash326 2009

[12] U C De and G C Ghosh ldquoOn generalized quasi-Einsteinmanifoldsrdquo Kyungpook Mathematical Journal vol 44 pp 607ndash615 2004

[13] M C Chaki ldquoOn generalized quasi Einstein manifoldsrdquo Publi-cationes Mathematicae vol 58 no 4 pp 683ndash691 2001

[14] U C De and S Mallick ldquoOn the existence of generalized quasi-Einstein manifolds Archivum Mathematicum (Brno)rdquo Tomusvol 47 pp 279ndash291 2011

[15] S K Hui and R S Lemence ldquoOn generalized quasi-Einsteinmanifold admitting W2- curvature tensorrdquo International Jour-nal of Mathematical Analysis vol 6 no 23 pp 1115ndash1121 2012

[16] A A Shaikh and S K Hui ldquoOn some classes of general-ized quasi-Einstein manifoldsrdquo Communications of the KoreanMathematical Society vol 24 no 3 pp 415ndash424 2009

[17] U C De and A K Gazi ldquoOn nearly quasi-Einstein manifoldsrdquoNovi Sad Journal ofMathematics vol 38 no 2 pp 115ndash121 2008

[18] D G Prakasha and C S Bagewadi ldquoOn nearly quasi-EinsteinmanifoldsrdquoMathematica Pannonica vol 21 no 2 pp 265ndash2732010

[19] Y Ishii ldquoOn conharmonic transformationsrdquo Tensor NS vol 7pp 73ndash80 1957

[20] D B Abdussattar ldquoOn conharmonic transformations in generalrelativityrdquo Bulletin of Calcutta Mathematical Society vol 41 pp409ndash416 1966

[21] B Y Chen and K Yano ldquoHypersurfaces of conformally atspacesrdquo Tensor N S vol 26 pp 318ndash322 1972

[22] S Tanno ldquoRicci curvatures of Riemannian manifoldsrdquo TsukubaJournal of Mathematics vol 40 pp 441ndash448 1988

[23] A Gray ldquoEinstein-like manifolds which are not EinsteinrdquoGeometriae Dedicata vol 7 no 3 pp 259ndash280 1978

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

Chinese Journal of Mathematics 3

We assume that the unit vector fields 120588 and 1205881defined by

119892(119883 120588) = 119879(119883) and 119892(119883 1205881) = 119863(119883) are orthogonal that is

119892(120588 1205881) = 0 Now the relation (8) can be written as

1198771015840

(119883 119884 119885 119880)

= 1199011[119892 (119884 119885) 119892 (119883119880) minus 119892 (119883 119885) 119892 (119884 119880)]

+ 1199021[119892 (119883119880) 119879 (119884) 119879 (119885) minus 119892 (119884119880) 119879 (119883) 119879 (119885)

+ 119892 (119884 119885) 119879 (119883)119879 (119880) minus 119892 (119883 119885) 119879 (119884) 119879 (119880)]

+ 1199041[119892 (119883119880)119863 (119884)119863 (119885) minus 119892 (119884119880)119863 (119883)119863 (119885)

+119892 (119884 119885)119863 (119883)119863 (119880) minus 119892 (119883 119885)119863 (119884)119863 (119880)]

(11)

where 1199011= (2120572(119899 minus 2)) 119902

1= (120573(119899 minus 2)) and 119904

1= (120574(119899 minus

2)) Comparing (10) and (11) it follows that the manifoldis of generalized quasiconstant curvature Thus we have thefollowing theorem

Theorem 1 A conharmonically flat 119866(119876119864)119899is one of general-

ized quasiconstant curvature

Next differentiating (6) covariantly and then contractingwe obtain(div sdotL) (119883 119884)119885

= (div sdot119877) (119883 119884)119885 minus1

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

minus1

2 (119899 minus 2)[(nabla119883119903) 119892 (119884 119885) minus (nabla

119884119903) 119892 (119883 119885)]

(12)

where div denotes the divergenceAgain it is known that in a Riemannianmanifold we have

(div sdot119877) (119883 119884)119885 = (nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885) (13)

Consequently by virtue of (13) the relation (12) takes theform

(div sdotL) (119883 119884)119885

=119899 minus 3

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

minus1

2 (119899 minus 2)[(nabla119883119903) 119892 (119884 119885) minus (nabla

119884119903) 119892 (119883 119885)]

(14)

Now consider the associated scalars 120572 120573 and 120574 as constantsthen (4) yields that the scalar curvature 119903 is constant andhence 119889119903(119883) = 0 for all 119883 Consequently (14) reduces to

(div sdotL) (119883 119884)119885 =119899 minus 3

119899 minus 2[(nabla119883119878) (119884 119885) minus (nabla

119884119878) (119883 119885)]

(15)

Since 120572 120573 and 120574 are constants we have from (2) that

(nabla119883119878) (119884 119885) = 120573 [(nabla

119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)]

+ 120574 [(nabla119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)]

(16)

By virtue of (16) we get from (15) that

(div sdotL) (119883 119884)119885

=119899 minus 3

119899 minus 2[120573 (nabla

119883119860) (119884)119860 (119885) + 119860 (119884) (nabla

119883119860) (119885)

minus (nabla119884119860) (119883)119860 (119885) minus 119860 (119883) (nabla

119884119860) (119885)]

+ 120574 (nabla119883119861) (119884) 119861 (119885) + 119861 (119884) (nabla

119883119861) (119885)

minus (nabla119884119861) (119883) 119861 (119885) minus 119861 (119883) (nabla

119884119861) (119885)

(17)

Next if the generators 120588 and 120583 of the manifold underconsideration are recurrent vector fields [22] then we havenabla119883120588 = 120587

1(119883)120588 and nabla

119883120583 = 120587

2(119883)120583 where 120587

1and 120587

2are the 1-

forms of the recurrence such that 1205871and 120587

2are different from

119860 and 119861 Consequently we get

(nabla119883119860) (119884) = 119892 (nabla

119883120588 119884) = 119892 (120587

1(119883) 120588 119884) = 120587

1(119884)119860 (119885)

(nabla119883119861) (119884) = 119892 (nabla

119883120583 119884) = 119892 (120587

2(119883) 120583 119884) = 120587

2(119884) 119861 (119885)

(18)

In view of (18) relation (17) reduces to

(div sdotL) (119883 119884)119885

=2 (119899 minus 3)

(119899 minus 2)[120573 1205871(119883)119860 (119884) minus 120587

1(119884)119860 (119883)119860 (119885)

times120574 1205872(119883) 119861 (119884) minus 120587

2(119884) 119861 (119883) 119861 (119885)]

(19)

Also since 119892(120588 120588) = 119892(120583 120583) = 1 it follows that (nabla119883119860)(120588) =

119892(nabla119883120588 120588) = 0 and hence (18) reduces to 120587

1(119883) = 0 for

all 119883 Similarly we have 1205872(119883) = 0 Hence from (19)

we have (div sdotL)(119883 119884)119885 = 0 that is the manifold underconsideration is conharmonically conservative Hence wecan state the following theorem

Theorem 2 If in a 119866(119876119864)119899(119899 gt 3) the associated scalars

are constants and the generators 120588 and 120583 corresponding to theassociated 1-forms 119860 and 119861 are not being the recurrence 1-forms then the manifold is conharmonically conservative

4 119866(119876119864)119899(119899 gt 3) Satisfying

the Condition L sdot 119878 = 0

Let us consider a 119866(119876119864)119899(119899 gt 3) satisfying the condition

L sdot 119878 = 0 Then we have

119878 (L (119883 119884)119885119880) + 119878 (119885L (119883 119884)119880) = 0 (20)

Setting 119885 = 120588 and 119880 = 120583 in (20) we have

119878 (L (119883 119884) 120588 120583) + 119878 (120588L (119883 119884) 120583) = 0 (21)

By using (5) in (21) we obtain

(120572 + 120574) 119861 (L (119883 119884) 120588) + (120572 + 120573)119860 (L (119883 119884) 120583) = 0 (22)

4 Chinese Journal of Mathematics

Next putting119885 = 120588 in (6) and then taking inner product with120583 we get

L1015840

(119883 119884 120588 120583)

= 1198771015840

(119883 119884 120588 120583) minus1

119899 minus 2

times [119878 (119884 120583) 119892 (119883 120588) minus 119878 (119883 120583) 119892 (119884 120588)

+ 119892 (119884 120583) 119878 (119883 120588) minus119892 (119883 120583) 119878 (119884 120588)]

(23)

where 119892(L(119883 119884)120588 120583) = L1015840(119883 119884 120588 120583) = 119861(L(119883 119884)120588)Again from (5) we obtain from (23)

L1015840

(119883 119884 120588 120583) = 1198771015840

(119883 119884 120588 120583)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(24)

Again plugging 119885 = 120583 in (6) and then taking inner productwith 120588 we have

L1015840

(119883 119884 120583 120588) = 1198771015840

(119883 119884 120583 120588)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119883) 119861 (119884) minus 119860 (119884) 119861 (119883)]

(25)

where 119892(L(119883 119884)120583 120588) = L1015840(119883 119884 120583 120588) = 119860(L(119883 119884)120583)From (24) and (25) one can get

119860 (L (119883 119884) 120583) = minus119861 (L (119883 119884) 120588) (26)

By taking account of (26) in (22) we obtain

(120574 minus 120573) 119861 (L (119883 119884) 120583) = 0 (27)

This implies either 120574 = 120573 or 119861(L(119883 119884)120583) = 0Now if 120574 = 120573 then from (2) we have

119878 (119883 119884) = 120572119892 (119883 119884) + 120573 [119860 (119883)119860 (119884) + 119861 (119883) 119861 (119884)]

= 120572119892 (119883 119884) + 120573119864 (119883 119884) (28)

where119864(119883 119884) = 119860(119883)119860(119884)+119861(119883)119861(119884)That is themanifoldis a 119873(119876119864)

119899

On the other hand if 119861(119871(119883 119884)120588) = 0 then (24) gives

1198771015840

(119883 119884 120588 120583) =(2120572 + 120573 + 120574)

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(29)

Thus we can state the following theorem

Theorem 3 If119872 is a 119866(119876119864)119899(119899 gt 3) satisfying the condition

L sdot 119878 = 0 then either 119872 is 119873(119876119864)119899or the curvature tensor 119877

of the manifold satisfies the property (29)

5 Some Geometric Properties of 119866(119876119864)119899

In [23] Gray introduced two classes of Riemannian man-ifolds determined by the covariant differentiation of Riccitensor Class A consists of all Riemannian manifolds whoseRicci tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) = (nabla

119884119878) (119883 119885) (30)

that is Ricci tensor 119878 is a Codazzi type tensorClass B consists of all Riemannian manifolds whose Ricci

tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (31)

that is Ricci tensor 119878 is cyclic parallelFirst suppose that the associated scalars are constants and

the Ricci tensor 119878 is of Codazzi typeThen from (2) we obtain

(nabla119883119878) (119884 119885) = 120573 [(nabla

119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

(32)

Interchanging 119883 and 119884 in (32) we have

(nabla119884119878) (119883 119885) = 120573 [(nabla

119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(33)

Since 119878 is of Codazzi type we have from (32) and (33) that

120573 [(nabla119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

= 120573 [(nabla119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(34)

Putting 119885 = 120588 in (34) and using (nabla119883119860)(120588) = 0 and

(nabla119883119861)(120588) = 0 since 120588 is a unit vector we get

(nabla119883119861) (119884) minus (nabla

119884119861) (119883) = 0 (35)

This shows that 119889119861(119883 119884) = 0Again putting119885 = 120583 in (34) and using (nabla

119883119860)(120583) = 0 and

(nabla119883119861)(120583) = 0 since 120583 is a unit vector we get

(nabla119883119860) (119884) minus (nabla

119884119860) (119883) = 0 (36)

This shows that 119889119860(119883 119884) = 0Thus we can state the following theorem

Theorem 4 In a 119866(119876119864)119899 if the associated scalars are con-

stants and the Ricci tensor is of Codazzi type then theassociated 1-forms 119860 and 119861 are closed

Next suppose that the generators 120588 and 120583 are Killingvector fields in a 119866(119876119864)

119899and the associated scalars are

constants Then

(L119880119892) (119883 119884) = 0 (L

119881119892) (119883 119884) = 0 (37)

Chinese Journal of Mathematics 5

where L denotes the Lie derivative which implies that

119892 (nabla119883119880119884) + 119892 (119883 nabla

119884119880) = 0

119892 (nabla119883119881119884) + 119892 (119883 nabla

119884119881) = 0

(38)

From (38) it follows that

(nabla119883119860) (119884) + (nabla

119884119860) (119883) = 0

(nabla119883119861) (119884) + (nabla

119884119861) (119883) = 0

(39)

since 119892(nabla119883119880119884) = (nabla

119883119860)(119884) and 119892(nabla

119883119881119884) = (nabla

119883119861)(119884)

Similarly we have

(nabla119883119860) (119885) + (nabla

119885119860) (119883) = 0

(nabla119883119861) (119885) + (nabla

119885119861) (119883) = 0

(nabla119884119860) (119885) + (nabla

119885119860) (119884) = 0

(nabla119884119861) (119885) + (nabla

119885119861) (119884) = 0

(40)

for all119883119884 and119885 Now from (2) and using the relations (39)-(40) we have

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (41)

Therefore we can state the following theorem

Theorem 5 In a 119866(119876119864)119899 if the generators are Killing vector

fields and the associated scalars are constants then the Riccitensor of the manifold is cyclic parallel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

D G Prakasha is thankful to University Grants CommissionNew Delhi India for financial support in the form of MajorResearch Project (no 39-302010 (SR) dated December 232012)

References

[1] M C Chaki and R K Maity ldquoOn quasi Einstein manifoldsrdquoPublicationes Mathematicae vol 57 no 3-4 pp 297ndash306 2000

[2] F Defever R Deszcz M Hotlos M Kucharski and Z Sen-turk ldquoGeneralisations of Robertson walker space AnnalesUniversitatis Scientiarum Budapestinensis de Rolando EotvosNominataerdquo Sectio Mathematica vol 43 pp 13ndash24 2000

[3] R Deszcz F Dillen L Verstraelen and L Vrancken ldquoQuasi-Einstein totally real submanifolds of the nearly Kahler 6-sphererdquo Tohoku Mathematical Journal vol 51 no 4 pp 461ndash478 1999

[4] R Deszcz M Glogowska M Hotlorsquos and Z Senturk ldquoOncertain quasi-Einstein semi-symmetric hypersurfacesrdquo AnnalesUniversitatis Scientarium Budapestinensis vol 41 pp 153ndash1661998

[5] R Deszcz M Hotlorsquos and Z Senturk ldquoQuasi-Einstein hyper-surfaces in semi-Riemannian space formsrdquo Colloquium Mathe-maticum vol 81 pp 81ndash97 2001

[6] R Deszcz M Hotlorsquos and Z Senturk ldquoOn curvature propertiesof quasi-Einatein hypersurfaces in semi-Euclidean spacesrdquoSoochow Journal of Mathematics vol 27 no 4 pp 375ndash3892001

[7] M Głogowska ldquoOn quasi-Einstein Cartan type hypersurfacesrdquoJournal of Geometry and Physics vol 58 no 5 pp 599ndash6142008

[8] M C Chaki and P K Ghoshal ldquoSome global properties of quasiEinstein manifoldsrdquo Publicationes Mathematicae vol 63 no 4pp 635ndash641 2003

[9] U C De and G C Ghosh ldquoOn quasi Einstein manifoldsrdquoPeriodica Mathematica Hungarica vol 48 no 1-2 pp 223ndash2312004

[10] A A Shaikh and A Patra ldquoOn quasi-conformally at quasi-Einstein spacesrdquo Differential Geometry-Dynamical Systems vol12 pp 201ndash212 2010

[11] A A Shaikh D W Yoon and S K Hui ldquoOn quasi-Einsteinspacetimesrdquo Tsukuba Journal of Mathematics vol 33 no 2 pp305ndash326 2009

[12] U C De and G C Ghosh ldquoOn generalized quasi-Einsteinmanifoldsrdquo Kyungpook Mathematical Journal vol 44 pp 607ndash615 2004

[13] M C Chaki ldquoOn generalized quasi Einstein manifoldsrdquo Publi-cationes Mathematicae vol 58 no 4 pp 683ndash691 2001

[14] U C De and S Mallick ldquoOn the existence of generalized quasi-Einstein manifolds Archivum Mathematicum (Brno)rdquo Tomusvol 47 pp 279ndash291 2011

[15] S K Hui and R S Lemence ldquoOn generalized quasi-Einsteinmanifold admitting W2- curvature tensorrdquo International Jour-nal of Mathematical Analysis vol 6 no 23 pp 1115ndash1121 2012

[16] A A Shaikh and S K Hui ldquoOn some classes of general-ized quasi-Einstein manifoldsrdquo Communications of the KoreanMathematical Society vol 24 no 3 pp 415ndash424 2009

[17] U C De and A K Gazi ldquoOn nearly quasi-Einstein manifoldsrdquoNovi Sad Journal ofMathematics vol 38 no 2 pp 115ndash121 2008

[18] D G Prakasha and C S Bagewadi ldquoOn nearly quasi-EinsteinmanifoldsrdquoMathematica Pannonica vol 21 no 2 pp 265ndash2732010

[19] Y Ishii ldquoOn conharmonic transformationsrdquo Tensor NS vol 7pp 73ndash80 1957

[20] D B Abdussattar ldquoOn conharmonic transformations in generalrelativityrdquo Bulletin of Calcutta Mathematical Society vol 41 pp409ndash416 1966

[21] B Y Chen and K Yano ldquoHypersurfaces of conformally atspacesrdquo Tensor N S vol 26 pp 318ndash322 1972

[22] S Tanno ldquoRicci curvatures of Riemannian manifoldsrdquo TsukubaJournal of Mathematics vol 40 pp 441ndash448 1988

[23] A Gray ldquoEinstein-like manifolds which are not EinsteinrdquoGeometriae Dedicata vol 7 no 3 pp 259ndash280 1978

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

4 Chinese Journal of Mathematics

Next putting119885 = 120588 in (6) and then taking inner product with120583 we get

L1015840

(119883 119884 120588 120583)

= 1198771015840

(119883 119884 120588 120583) minus1

119899 minus 2

times [119878 (119884 120583) 119892 (119883 120588) minus 119878 (119883 120583) 119892 (119884 120588)

+ 119892 (119884 120583) 119878 (119883 120588) minus119892 (119883 120583) 119878 (119884 120588)]

(23)

where 119892(L(119883 119884)120588 120583) = L1015840(119883 119884 120588 120583) = 119861(L(119883 119884)120588)Again from (5) we obtain from (23)

L1015840

(119883 119884 120588 120583) = 1198771015840

(119883 119884 120588 120583)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(24)

Again plugging 119885 = 120583 in (6) and then taking inner productwith 120588 we have

L1015840

(119883 119884 120583 120588) = 1198771015840

(119883 119884 120583 120588)

minus2120572 + 120573 + 120574

(119899 minus 2)[119860 (119883) 119861 (119884) minus 119860 (119884) 119861 (119883)]

(25)

where 119892(L(119883 119884)120583 120588) = L1015840(119883 119884 120583 120588) = 119860(L(119883 119884)120583)From (24) and (25) one can get

119860 (L (119883 119884) 120583) = minus119861 (L (119883 119884) 120588) (26)

By taking account of (26) in (22) we obtain

(120574 minus 120573) 119861 (L (119883 119884) 120583) = 0 (27)

This implies either 120574 = 120573 or 119861(L(119883 119884)120583) = 0Now if 120574 = 120573 then from (2) we have

119878 (119883 119884) = 120572119892 (119883 119884) + 120573 [119860 (119883)119860 (119884) + 119861 (119883) 119861 (119884)]

= 120572119892 (119883 119884) + 120573119864 (119883 119884) (28)

where119864(119883 119884) = 119860(119883)119860(119884)+119861(119883)119861(119884)That is themanifoldis a 119873(119876119864)

119899

On the other hand if 119861(119871(119883 119884)120588) = 0 then (24) gives

1198771015840

(119883 119884 120588 120583) =(2120572 + 120573 + 120574)

(119899 minus 2)[119860 (119884) 119861 (119883) minus 119860 (119883) 119861 (119884)]

(29)

Thus we can state the following theorem

Theorem 3 If119872 is a 119866(119876119864)119899(119899 gt 3) satisfying the condition

L sdot 119878 = 0 then either 119872 is 119873(119876119864)119899or the curvature tensor 119877

of the manifold satisfies the property (29)

5 Some Geometric Properties of 119866(119876119864)119899

In [23] Gray introduced two classes of Riemannian man-ifolds determined by the covariant differentiation of Riccitensor Class A consists of all Riemannian manifolds whoseRicci tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) = (nabla

119884119878) (119883 119885) (30)

that is Ricci tensor 119878 is a Codazzi type tensorClass B consists of all Riemannian manifolds whose Ricci

tensor 119878 satisfies the condition

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (31)

that is Ricci tensor 119878 is cyclic parallelFirst suppose that the associated scalars are constants and

the Ricci tensor 119878 is of Codazzi typeThen from (2) we obtain

(nabla119883119878) (119884 119885) = 120573 [(nabla

119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

(32)

Interchanging 119883 and 119884 in (32) we have

(nabla119884119878) (119883 119885) = 120573 [(nabla

119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(33)

Since 119878 is of Codazzi type we have from (32) and (33) that

120573 [(nabla119883119860) (119884)119860 (119885) + (nabla

119883119860) (119885)119860 (119884)]

+ 120574 [(nabla119883119861) (119884) 119861 (119885) + (nabla

119883119861) (119885) 119861 (119884)]

= 120573 [(nabla119884119860) (119883)119860 (119885) + (nabla

119884119860) (119885)119860 (119883)]

+ 120574 [(nabla119884119861) (119883) 119861 (119885) + (nabla

119884119861) (119885) 119861 (119883)]

(34)

Putting 119885 = 120588 in (34) and using (nabla119883119860)(120588) = 0 and

(nabla119883119861)(120588) = 0 since 120588 is a unit vector we get

(nabla119883119861) (119884) minus (nabla

119884119861) (119883) = 0 (35)

This shows that 119889119861(119883 119884) = 0Again putting119885 = 120583 in (34) and using (nabla

119883119860)(120583) = 0 and

(nabla119883119861)(120583) = 0 since 120583 is a unit vector we get

(nabla119883119860) (119884) minus (nabla

119884119860) (119883) = 0 (36)

This shows that 119889119860(119883 119884) = 0Thus we can state the following theorem

Theorem 4 In a 119866(119876119864)119899 if the associated scalars are con-

stants and the Ricci tensor is of Codazzi type then theassociated 1-forms 119860 and 119861 are closed

Next suppose that the generators 120588 and 120583 are Killingvector fields in a 119866(119876119864)

119899and the associated scalars are

constants Then

(L119880119892) (119883 119884) = 0 (L

119881119892) (119883 119884) = 0 (37)

Chinese Journal of Mathematics 5

where L denotes the Lie derivative which implies that

119892 (nabla119883119880119884) + 119892 (119883 nabla

119884119880) = 0

119892 (nabla119883119881119884) + 119892 (119883 nabla

119884119881) = 0

(38)

From (38) it follows that

(nabla119883119860) (119884) + (nabla

119884119860) (119883) = 0

(nabla119883119861) (119884) + (nabla

119884119861) (119883) = 0

(39)

since 119892(nabla119883119880119884) = (nabla

119883119860)(119884) and 119892(nabla

119883119881119884) = (nabla

119883119861)(119884)

Similarly we have

(nabla119883119860) (119885) + (nabla

119885119860) (119883) = 0

(nabla119883119861) (119885) + (nabla

119885119861) (119883) = 0

(nabla119884119860) (119885) + (nabla

119885119860) (119884) = 0

(nabla119884119861) (119885) + (nabla

119885119861) (119884) = 0

(40)

for all119883119884 and119885 Now from (2) and using the relations (39)-(40) we have

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (41)

Therefore we can state the following theorem

Theorem 5 In a 119866(119876119864)119899 if the generators are Killing vector

fields and the associated scalars are constants then the Riccitensor of the manifold is cyclic parallel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

D G Prakasha is thankful to University Grants CommissionNew Delhi India for financial support in the form of MajorResearch Project (no 39-302010 (SR) dated December 232012)

References

[1] M C Chaki and R K Maity ldquoOn quasi Einstein manifoldsrdquoPublicationes Mathematicae vol 57 no 3-4 pp 297ndash306 2000

[2] F Defever R Deszcz M Hotlos M Kucharski and Z Sen-turk ldquoGeneralisations of Robertson walker space AnnalesUniversitatis Scientiarum Budapestinensis de Rolando EotvosNominataerdquo Sectio Mathematica vol 43 pp 13ndash24 2000

[3] R Deszcz F Dillen L Verstraelen and L Vrancken ldquoQuasi-Einstein totally real submanifolds of the nearly Kahler 6-sphererdquo Tohoku Mathematical Journal vol 51 no 4 pp 461ndash478 1999

[4] R Deszcz M Glogowska M Hotlorsquos and Z Senturk ldquoOncertain quasi-Einstein semi-symmetric hypersurfacesrdquo AnnalesUniversitatis Scientarium Budapestinensis vol 41 pp 153ndash1661998

[5] R Deszcz M Hotlorsquos and Z Senturk ldquoQuasi-Einstein hyper-surfaces in semi-Riemannian space formsrdquo Colloquium Mathe-maticum vol 81 pp 81ndash97 2001

[6] R Deszcz M Hotlorsquos and Z Senturk ldquoOn curvature propertiesof quasi-Einatein hypersurfaces in semi-Euclidean spacesrdquoSoochow Journal of Mathematics vol 27 no 4 pp 375ndash3892001

[7] M Głogowska ldquoOn quasi-Einstein Cartan type hypersurfacesrdquoJournal of Geometry and Physics vol 58 no 5 pp 599ndash6142008

[8] M C Chaki and P K Ghoshal ldquoSome global properties of quasiEinstein manifoldsrdquo Publicationes Mathematicae vol 63 no 4pp 635ndash641 2003

[9] U C De and G C Ghosh ldquoOn quasi Einstein manifoldsrdquoPeriodica Mathematica Hungarica vol 48 no 1-2 pp 223ndash2312004

[10] A A Shaikh and A Patra ldquoOn quasi-conformally at quasi-Einstein spacesrdquo Differential Geometry-Dynamical Systems vol12 pp 201ndash212 2010

[11] A A Shaikh D W Yoon and S K Hui ldquoOn quasi-Einsteinspacetimesrdquo Tsukuba Journal of Mathematics vol 33 no 2 pp305ndash326 2009

[12] U C De and G C Ghosh ldquoOn generalized quasi-Einsteinmanifoldsrdquo Kyungpook Mathematical Journal vol 44 pp 607ndash615 2004

[13] M C Chaki ldquoOn generalized quasi Einstein manifoldsrdquo Publi-cationes Mathematicae vol 58 no 4 pp 683ndash691 2001

[14] U C De and S Mallick ldquoOn the existence of generalized quasi-Einstein manifolds Archivum Mathematicum (Brno)rdquo Tomusvol 47 pp 279ndash291 2011

[15] S K Hui and R S Lemence ldquoOn generalized quasi-Einsteinmanifold admitting W2- curvature tensorrdquo International Jour-nal of Mathematical Analysis vol 6 no 23 pp 1115ndash1121 2012

[16] A A Shaikh and S K Hui ldquoOn some classes of general-ized quasi-Einstein manifoldsrdquo Communications of the KoreanMathematical Society vol 24 no 3 pp 415ndash424 2009

[17] U C De and A K Gazi ldquoOn nearly quasi-Einstein manifoldsrdquoNovi Sad Journal ofMathematics vol 38 no 2 pp 115ndash121 2008

[18] D G Prakasha and C S Bagewadi ldquoOn nearly quasi-EinsteinmanifoldsrdquoMathematica Pannonica vol 21 no 2 pp 265ndash2732010

[19] Y Ishii ldquoOn conharmonic transformationsrdquo Tensor NS vol 7pp 73ndash80 1957

[20] D B Abdussattar ldquoOn conharmonic transformations in generalrelativityrdquo Bulletin of Calcutta Mathematical Society vol 41 pp409ndash416 1966

[21] B Y Chen and K Yano ldquoHypersurfaces of conformally atspacesrdquo Tensor N S vol 26 pp 318ndash322 1972

[22] S Tanno ldquoRicci curvatures of Riemannian manifoldsrdquo TsukubaJournal of Mathematics vol 40 pp 441ndash448 1988

[23] A Gray ldquoEinstein-like manifolds which are not EinsteinrdquoGeometriae Dedicata vol 7 no 3 pp 259ndash280 1978

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

Chinese Journal of Mathematics 5

where L denotes the Lie derivative which implies that

119892 (nabla119883119880119884) + 119892 (119883 nabla

119884119880) = 0

119892 (nabla119883119881119884) + 119892 (119883 nabla

119884119881) = 0

(38)

From (38) it follows that

(nabla119883119860) (119884) + (nabla

119884119860) (119883) = 0

(nabla119883119861) (119884) + (nabla

119884119861) (119883) = 0

(39)

since 119892(nabla119883119880119884) = (nabla

119883119860)(119884) and 119892(nabla

119883119881119884) = (nabla

119883119861)(119884)

Similarly we have

(nabla119883119860) (119885) + (nabla

119885119860) (119883) = 0

(nabla119883119861) (119885) + (nabla

119885119861) (119883) = 0

(nabla119884119860) (119885) + (nabla

119885119860) (119884) = 0

(nabla119884119861) (119885) + (nabla

119885119861) (119884) = 0

(40)

for all119883119884 and119885 Now from (2) and using the relations (39)-(40) we have

(nabla119883119878) (119884 119885) + (nabla

119884119878) (119885119883) + (nabla

119885119878) (119883 119884) = 0 (41)

Therefore we can state the following theorem

Theorem 5 In a 119866(119876119864)119899 if the generators are Killing vector

fields and the associated scalars are constants then the Riccitensor of the manifold is cyclic parallel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

D G Prakasha is thankful to University Grants CommissionNew Delhi India for financial support in the form of MajorResearch Project (no 39-302010 (SR) dated December 232012)

References

[1] M C Chaki and R K Maity ldquoOn quasi Einstein manifoldsrdquoPublicationes Mathematicae vol 57 no 3-4 pp 297ndash306 2000

[2] F Defever R Deszcz M Hotlos M Kucharski and Z Sen-turk ldquoGeneralisations of Robertson walker space AnnalesUniversitatis Scientiarum Budapestinensis de Rolando EotvosNominataerdquo Sectio Mathematica vol 43 pp 13ndash24 2000

[3] R Deszcz F Dillen L Verstraelen and L Vrancken ldquoQuasi-Einstein totally real submanifolds of the nearly Kahler 6-sphererdquo Tohoku Mathematical Journal vol 51 no 4 pp 461ndash478 1999

[4] R Deszcz M Glogowska M Hotlorsquos and Z Senturk ldquoOncertain quasi-Einstein semi-symmetric hypersurfacesrdquo AnnalesUniversitatis Scientarium Budapestinensis vol 41 pp 153ndash1661998

[5] R Deszcz M Hotlorsquos and Z Senturk ldquoQuasi-Einstein hyper-surfaces in semi-Riemannian space formsrdquo Colloquium Mathe-maticum vol 81 pp 81ndash97 2001

[6] R Deszcz M Hotlorsquos and Z Senturk ldquoOn curvature propertiesof quasi-Einatein hypersurfaces in semi-Euclidean spacesrdquoSoochow Journal of Mathematics vol 27 no 4 pp 375ndash3892001

[7] M Głogowska ldquoOn quasi-Einstein Cartan type hypersurfacesrdquoJournal of Geometry and Physics vol 58 no 5 pp 599ndash6142008

[8] M C Chaki and P K Ghoshal ldquoSome global properties of quasiEinstein manifoldsrdquo Publicationes Mathematicae vol 63 no 4pp 635ndash641 2003

[9] U C De and G C Ghosh ldquoOn quasi Einstein manifoldsrdquoPeriodica Mathematica Hungarica vol 48 no 1-2 pp 223ndash2312004

[10] A A Shaikh and A Patra ldquoOn quasi-conformally at quasi-Einstein spacesrdquo Differential Geometry-Dynamical Systems vol12 pp 201ndash212 2010

[11] A A Shaikh D W Yoon and S K Hui ldquoOn quasi-Einsteinspacetimesrdquo Tsukuba Journal of Mathematics vol 33 no 2 pp305ndash326 2009

[12] U C De and G C Ghosh ldquoOn generalized quasi-Einsteinmanifoldsrdquo Kyungpook Mathematical Journal vol 44 pp 607ndash615 2004

[13] M C Chaki ldquoOn generalized quasi Einstein manifoldsrdquo Publi-cationes Mathematicae vol 58 no 4 pp 683ndash691 2001

[14] U C De and S Mallick ldquoOn the existence of generalized quasi-Einstein manifolds Archivum Mathematicum (Brno)rdquo Tomusvol 47 pp 279ndash291 2011

[15] S K Hui and R S Lemence ldquoOn generalized quasi-Einsteinmanifold admitting W2- curvature tensorrdquo International Jour-nal of Mathematical Analysis vol 6 no 23 pp 1115ndash1121 2012

[16] A A Shaikh and S K Hui ldquoOn some classes of general-ized quasi-Einstein manifoldsrdquo Communications of the KoreanMathematical Society vol 24 no 3 pp 415ndash424 2009

[17] U C De and A K Gazi ldquoOn nearly quasi-Einstein manifoldsrdquoNovi Sad Journal ofMathematics vol 38 no 2 pp 115ndash121 2008

[18] D G Prakasha and C S Bagewadi ldquoOn nearly quasi-EinsteinmanifoldsrdquoMathematica Pannonica vol 21 no 2 pp 265ndash2732010

[19] Y Ishii ldquoOn conharmonic transformationsrdquo Tensor NS vol 7pp 73ndash80 1957

[20] D B Abdussattar ldquoOn conharmonic transformations in generalrelativityrdquo Bulletin of Calcutta Mathematical Society vol 41 pp409ndash416 1966

[21] B Y Chen and K Yano ldquoHypersurfaces of conformally atspacesrdquo Tensor N S vol 26 pp 318ndash322 1972

[22] S Tanno ldquoRicci curvatures of Riemannian manifoldsrdquo TsukubaJournal of Mathematics vol 40 pp 441ndash448 1988

[23] A Gray ldquoEinstein-like manifolds which are not EinsteinrdquoGeometriae Dedicata vol 7 no 3 pp 259ndash280 1978

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