Kyungpook Math. J Yolumc Z7. Number 2 December. 1987

TOTALLY REAL SUBMANIFOLDS WITH HARMONIC CURVATURE

By U Hang Ki‘ and Hisao I\akaga“ a

O. Introduction

Tota lJ y real submanifo lds in a complex projcctivc spacc arc st udied from

various points of vicw [2. 14.15]. ln particu lar. H. Naitoh [7] has complctely

c1assificd lhosc submani fold s if thc sccond fundamcntal form is para llel. On thc

other hand . compacl tota lJ y rcal submanifolds of positivc curvaturc Or non ­

ncgatlvc cur‘ aturc arc vcry rcccnlly invcstigatcd by F. U rbano [12] or Y

Ohnila [9] . respectivcly ‘

A I~iemannîan curvaturc is said to bc harmonic if thc Ricci tcnsor Rij satisfies

lhc so ca lJ cd Codazzi cqualion

\7 .R ,,= \7,R ij - v r~ JtI

I~ iemannian submanifo lds with harmonic curvaturc in a Ricmannian manifold

of a conslant curvalurc are in vestigated by E. Omachi [8]. M. Umehara [11]

and lhe authors [3.4]. ln particular. lhe aulhors [3J c1assificd completcly

submanifoJds with harmonic curvature in a Ricmannian manifold of constant

cUI'\'alurc if the normal connect ion in thc normal bundlc is f1at and the mean

curvalure vector is para lJcl. The purpose of lhis paper is lo study totally real

submanifoJds with harmonic curvaturc in a Kachlerian manifold of constant

holomorphic curvaturc.

J. Totally real submanifolds of a KaehJerian manifoJd

Lcl ( .14. g) be a Kach lcrian manifoJd of a real dimension 2m cquippcd with

an almost complex struclure 1 and a Hcrmitian metric if. Manifolds and

submanifolds which are discussed in this papcr arc assumcd to bc connected.

and all geometric objccls arc a lso assumed to be differentiab le and of C∞. Let

M bc covered by a systcm of coordinatc ncighbo rhoods 띠 ; l]. Then 、，ve have

(1.1) 1/ 1/= -δlh， l ltI ,sgts = glt ’

ð; being the Kroneckcr dclta.// and gji the components of 1 and if. respec­

tivcly. Here and in thc scquc l. thc following convention on thc rangc of

‘ ) This research was partially supported by JSPS and KOSEF.

100 U-Hallg Kz" ond Ht'sao Nakagawa

indices arc used. unless otherwÎse stated h, i , j , k. "' ::: 1, n, n+ l • ...• 2m.

a. b. c. d. ‘ = 1, n. X , y. Z, 1，ι … =π+ 1. …. 2m.

The summation convention will bc used with rcspect to thosc systems on

indices. Denoting by \1; the operator of covariant diffcrcntiation with respect

to g ji ' we get ( 1. 2) V;J/ = O.

Lct M bc an n-dimensional Riemannian manifoJd covered by a system of

coordinatc neighborhoods (U; x") and immcrscd isomctrically in M by thc

immersion rþ : M• M. Whcn thc argumcnt is locaJ. M nced not be distinguished

from rþ(M ). Wc rcpr않ent the immersion rþ Jocally by l =/'(x") and put B;' =

8bYh, 이=å/åi. then Bb=(Bbh) are ’,- lincarly independent JocaJ tangent vectors

of M. We choose 2m-n mutually orthogonal' unit normaJs Cx =(Cx) to thc sub

manifold. Sincc thc immersion ø is isometric, thc Ricmannian metric gcb induccd

on M is given by

(1. 3) g cb= gllBc1Bbl.

Thcreforc. denoting by '11, the operator of van der Waerdcn.Bortolotti covariant

differentiation 、vith respect to g'b’ the cquations of Gauss and Weingartcn for

M arc respectivcJy obtaincd

( 1. 4) V ,B:=""xC:. V,C:= - h: xS:. where hCb arc thc second fundamcntaI forms in thc dircction of Cx and hc

ax=

llcbxgbo= liefg%gyx, gyx= g”CylCxt belng the metrlC tensor of the nmo야Orma M‘ and (gba) = (gc6) ~l

An n-dimensional Ricmannian manifoJd M immcrscd isometrically into 껴 JS

callcd a totally real s뼈nanifoJd of M if JMxC M f for each point x and M f

denotes the normal space of M at x [2], [15). Jn this case. J X is a normaJ

vector to M. provided that X is a tangent vcctor. Thus it follows that thc

dimensions satisfy m르n. Let N x(M) be an orthogonaJ compJcment of JMx in M f. and then the foJJowing decomposition is obtai뼈 M상=JMx잉N/M).

Hencc it foJJows that the space Nx(M) is invariant under thc action of J.

AccordingJy we can put in each coordinatc ncighborhoods of M ,

(1. 5)

(1. 6)

IfBcl= I cXCxh,

Ifcj= -lfBr+frcj.

Totally real submallifolds witlJ !Jarm01t’c curνature 101

wherc wc havc put J" = g (JB,, C,) , J，，= -강(JC" B,) and l y"= g (JC, , C,). From thcsc dcfinitions wc casily sce that l y,+/ ,y=O and J cx = J " . By tak ing

account of (1. I) and (1. 3) , it follows from (1. 5) and ( 1. 6) that

( 1. 7) [! / J X" =δ Q’ J/자，y= O，

lf xff zy= -- t7xY+ l xeI ey,

、vhcrc J / = J ,ygYX, 1; =l y,g"'. Thcse show that 1 3 +/ = 0. 1 bcing of constant

rank, it defines the so-called l - slruclure in thc normal bundlc [13] . If wc apply the opcrator '11, of thc covariant differcnt iation to ( 1. 5) and ( 1. 6) and make

usc of (1. 1) , ( 1. 2) , (1. 4) and thcsc cquations, 、，\'C get

(1. 8)

(1. 9)

( 1. 10)

hcbXJ xa= hcaxJ :rb’

'VcJ / = hcbZf :/'.

\Icl / = h: x J /- heexJ ey.

lf VJ: vanishes iden tially, then the I -structure 1 in the normal bundle is said

to bc parallel [15] . ln this case, (1. 10) reduccs to m l (

l‘ceJIex= I;cexl ey-

Multiplying ,," y J; to ( 1. 8) and summing up for c, b and a, and using (1. JJ ) ,

wc find llcbYllcbxf yalxa= hC6YlZcajI axl xb’

or, taking account of (1 .7) , we havc

μltCbY(fy;fix+gyx) =hcbYhCC.

Therefore it follows that "야，Zfz x= O and hencc

(1. 12) Vcf,'=O,

becausc of (1. 9). We noticc from (1. 7) that 1; vanishes idcntically if 1Il=n. Thus an 1Il-dimcnsional totally real submanifold of a rcal 2m-dimensional Kach­

lerian manifold has always thc cquation (1. 12).

In thc scqucl , the ambicnt Kaehlerian manifold is assumcd to bc of constan t holomorphic scctional curvature 4c and of rcal dimcnsion 2m, which is denotcd

by 1\12m(c) . Then thc curvaturc tensor of 1\12m(c) is givcn by

R.jih = C(gkhgji- gkigjh + J khJ ji - J kiJjh - 2J kjJ ih).

Sincc thc submanifold 1\1 is totally rcal , wc scc, using ( 1. 5) , ( 1. 6) and (1. 7) , that thc cquatiom‘ of Gauss, Codazzi and Ricci for 1\1 arc rcspcctivcly givcn by

102 U-Hang Ki 011d /I isao Nakagawo

( 1. 13) Rd ,1Jo = c(gd.g,,- gdbK<a) + h. /h" , - hd, ' h,." ( 1. 14) \ldh,:- \l,hdb= O,

( 1. 15) Rd,y,= c(j d,J 'y - J dyJ ,,) + h/,h"y - h: ,hd" ,

‘vhcrc Rdcbo and Rdcyx arc lhc Ricmannian curvalurc lcnsor and that of l hc

conncction induced in thc normal bundlc of M , respcctivcly,

씨r c scc, f rom ( 1. 13) thal thc Ricci tcnsor R" and lhe scalar curvalure r of

M can be exprcsscd as f ollows

( 1. 16) R,,= c(n - 1)g, .. t h'h" ,- h,/h:"

(1 .17) r cn (n - l) + h'h,- h,:h" ,’

“ hcrc lf= g cblicbx.

A Ricmannian curvalurc tensor is said to bc Itarn’onic if thc Ricci tCIlsor

salisfies thc Codazzi cqualion , narncly, \I,RIJo is symmclric with rcspcct to all

indices c, b and a. Differentiating ( 1. 16) covariantly along M , wc find

V dRcb= ( V dh X) h'bx+ ltxV d"Cbx - (V d ltc f'X，Iz~x- h/xV dhbtx.

By mcans of (1. 4) , it f이 10、\' s that it is ncccssary and sufficicnt for M to be

of harmonic curvaturc that it satisfics

( 1. 18) ( \1 dh')h" ,- (\I,h')hd" - h;'V d/'싸f+ hdUV，h“x :...;.. o.

2. Totally rca l 5ubmanifolds with pa rallcl mcan curvature vecto r

Lcl 1\1' bc an n-dimcnsional total ly real submanifold with harmonic curvaturc

in M 2m ( c) such that thc j -5tructurc in thc normal bundlc is paralle l. This

scction is dcvotcd to thc in vcstigation of a totally rcal submanifold with parallcl

mcan curvalurc vcctor. Thc covariant deri vativc of thc Ricci tensor satisfics

(2. 1) \ldR，，= \I，R，μ·

T.ct .7 bc a mcan curvaturc vcctor ficld. Namcly , it is defincd by

‘..7 = g"h,: C,In h'C,In,

、\' hich is indepcndcnt of thc choicc of thc local ficld of orthonormal frames

[C,l . Thc [act that thc mcan curvaturc vcctor is parallcl is assnurncd , and we

may choosc a local fie ld [C,l in 5uch a way that / = a Cn +1, whcrc a =I!/ 1I

is constan l. Bccause of thc choicc of thc local field , the parallclism of /

yiclds

(2. 22) {kX=0 X르n +2

h’ +l= nll/II

Thus. (1 .18) lurns out to bc

Totally real subma’ufolds witll harmollic cιrvature 103

(2. 3) hcexV dhbex - h/~Vilher= O.

/ being a normal yectOr f icld on M , thc cur‘ aturc tcnsor R dcyx of the connec.

t ion in thc normal bundlc shows Rdon+ ,,= O for any index x. Thus ( 1. 15) yields

(2. 4) c( - 1 dl ,,+ l J d' ) + IId" lI / - 11", 11/ = 0,

“ here wc havc put hC6= hc~.q and J c= J / 1

On thc other hand, since l hc !-structurc in thc normal bundle is parallcl. il

\\'0 apply 1: to (1. 8) and sum for a, then wc obtain

hChy= hca%J yaJ xb’ “ hcrc \vc ha\"c used ( 1. 7) and ( l.l1) , from which , taking the ske\\'.symmctric

pa rt of this wilh respcct to indiccs c and b

(11,,' 1/)1.,- (h/><' 1/)1,,=0.

Thus wc scc, using ( 1. 7) and (1. 9) , l hat

(2. 5) ltcexlye= Pyzxf c;, x 1 c T h \\"here we ha\"c put Py/= hc/1 y" J / o Dcnoting P xy

“ = g zwP x/. wc sec that P xy: is

symmetric for a ll indices, bccause 01 (1. 11). It follo\Vs from (1. 7), (1. 9) and

0 . 12) that

(2. 6) P y" ! ,, Y= O

and hcncc h,: h" ,= P xy,p Y'. Thus thc square of thc lcnglh of II ，，'- Pμ" 1서'1/ lor

cach x vanishcs idcntically, which sh。、、 s

(2. 7) Itch:= PyZI cYj b:

lf 、、 c differentiate lh is cquation covariantly and takc account of ( 1. 12) . \\'e

oblain

(2. 8) 'ï1d h,;‘ = ('ï1dPyη1/1，‘， which togcthcr \\' ith thc Codazz i equation ( 1. 14) g ivcs

J / 'VdP y/ - J / V CP y."‘ = 0.

and hencc

(2. 9) V dPy?x = (J zeVcP y;/) J (/‘ ’ “ herc wc ha‘ c uscd (1.7) , (1. 10) , (1.11) and (2. 6)

By the way, thc equations ( 1. 7) , ( 1. 11) , (2. G) and (2. 7) give 11' = P / ' . Since

the mean curvaturc vcctor is parallel in t hc normal bundle, it follows that

'ï1dP/'=O. By contracting z and y in (2. 9) , wc easily sec that

J zeveP z!('x = O.

which toge ther with (2. 8) yields

(2. 10) 1.:'ï1 dh,: = O.

l여 U.Hallg Ki Qlld Hisao Nakagawa

Combining (2.8) and (2. 9) , \\'c vcrify that

(2 1l) f j?tllcbx- I xe?ehcby= 0.

because P xy, is symmct ric for all indiccs. By making usc of ( 1. 12) , thc co,'a

riant derivative of (2.~ ) gives

(2. 12) ('V까ct) hd“+h，，'\J‘:'= ('V ';'d,)h," 수 hd，'V아cIX‘ \\'hich implies

(2. 13) ('V ';',, )h:, h:x - (\'아d e) llce:il/% = ('9아ce:r)μ/h:'- ('V';'d") 1，낀lZd. By thc propertics of ( 1. 18), (2. 1) and (2.4) thc sccond tcrm in thc righl hand

sidc is dcformed as fol lo\\'s :

-('V아dtx) Il/11/X = - C\l아d: )h, ' h,/ = - ('V il'd: )hd'hec, .;-cUJ d,- J dJcx)자lladX. Hence lhc last t“ o cquations giyc

('V' I.'O) ('V싸")"d' ' h/ x_ ('V'h'O)('V아/)h"，":' cUJ d,- J dJ cx) ('V' h'O) ('\ ';'do ),

“’hich togclhcr wilh (2.10) and (2.11) yiclds .

(2. 11 ) ('V，"，，)('V，h")":'h，d，_('V싸ca) (?bhdf)kcexhadx= - c(Fchκ) (\" hba) ‘

\\'herc \\'e havc uscd ( 1. 7).

On the other hand , for any fixcd indiccs c and x, ('V''''')''d:- ('Vd",,) h,: can

bc rcgarded as a squarc matrix of dcgrcc n. By laking l he nOrm of this matrix

and by using thc cquation (2. 14) , the following cquation

(2. 15) ('V씨ce) hd ，x-('Vd 'zce)h/:，/'I'! 2cViZba 2 0

can be obtaincd

3. Main thcorcm

This section is dcvotcd to lhc sludy of a lOlally rca l submanifold on “ hich

thc Istructurc in thc norma l bundlc is parallc l. Fir"l of aIJ, thc f‘ ’1I0\\in)(

propcrty can bc 、 crificd

Lnl~l.\ 3. 1. Lel M be aκ n-dimellsional totally real 51tb1}“zni[old wilh harlllo

씨c cun:ature zOn .i\{ !m(c) , (c르0) 511C" I"al Ihe [slrucI"re in Ihc norlllal bundlc

i5 þara l1el. 1[ Ihe mean wrvalllre veclor is þaral1el. tllen h,," 1 i5 þara l1el

PROOF. It is sufficicnl to pro、 e thc abovc rcsult in thc casc ‘\'here thc

ambicn t space is Euclidcan. Thcn thc cquation (2.15) sho“ s

(3. 1) ('V A ,)hd: - ('V dh")h,,,' =0

fo r any ind iccs anù henccs it follows from (2.12) that

(3. 2) " " 'V d";" = hd, 'V "" " . In the casc whcrc x - n+ 1 in (3. 2) , IIC gct

T otally real sllbmamfolds toith harmollic Cl‘rvature 105

(3.3) hceV dlz/= hdeViz,/. By thc similar mcthod to the discussion on hypcrsurfaces with harmonic

curvaturc in a Euclidean space, the result is vcrified and the proof is thcrcforc

skctchcd briefl y. When a function il ", for any intcger 111르 1 is givcn by il…= h ha' /t ha‘ h~altl .

”’ “: “ l “ 1 ’

it satisfies thc fo l1owing eguation

V Jl ... =m lt~a ' ... ILa"'- IV JLa",

ν “.， ‘ ι1

By using ( 1. 14) and by combining (3. 3) togcthcr with thc above cquation, it

fo l1ows that thc funclion hm is constan t. for any integcr 111. Since thc matrix

A" 1= (11/'" 1) is diagonalizablc. thc local ficld (ea) on .1.1[ can be spccializcd so

that h~= ì.cδcG， 、이삐1 implies the fact that all functio따 서 h2• lz m' .. . arc

constant mcans that cach eigen\"aIue λa is constant on ]11. Then . bγ taking

account of thc calculation for the Laplacian of thc constant 1z2, t hc fo l1owing

rclation

(3.4)

can bc obtaincd

On thc othcr hand.

(3.5)

\7 dlz" \7d Iz" + R ,,,,,,(À , - À, )2= 0

the equation (3.2) givcs

(1., -?,) \7 dlz,{ 0

for any indiccs d and x at a point Þ in M and hcncc

(3. 6) \7 dlz,{ = 0

for any indcx x providcd that ?， .-"λ. By μJ. μ“ mutually distincl eigcn. values of A" 1 arc dcnotcd. Lct nl' ... 1'Zπ bc thcir multiplicitics. By using thc

notation [a) = (b λa = ^,) we scc from (3.6) that

which yicJds

(3.7)

V dhcb'': = O [or cδ [a). bc- [a) ,

\7dlz,,= O for cξ [aJ. bE [a)

Thcrc[orc, differentiating (3.3) covariantly, ‘vc havc

\1 a1lceV dhb t! + hceV a V dlzbc = V itdcV chbe + hdeV a \1 ch lle,

from which it fo l1ows by using thc Ricci formula

2(\7 "J'" \7 d" : • \7 a"d' \7 /':)

= lzacC Rab/hc'" - R ab/h/) - hceCRab/lld’ -Ra,jh/) .

Putting a = c and d =b. and diagonalizing thc matrix A h 1. ‘\'e get

2(\7, 11, '\7'h,' - \7,11,,\7' lt") = R""O,-μ)"， whcrc thc summation convcntion is not uscd with respect to c and b. ...\ccording

to (3.7) thc abovc cquation means

106 U-Hallg Ki alzd Hisao Nakagaωa

R,bbcU,-Àbi=o for cè; [aJ. bε [aj

Consequen tIy. by combining this cquation logcthcr with (3.4) it turns our thar

thc shapc opcrator A，， ~1 is parallc l. Sincc distinct eigen\'alucs μ， (r=l • .... a)

arc constant. thc smooth distributions $, \l'hich consists of all cigcnspaccs

associatcd 、.\'ith thc cigcJ1\'aluc can bc dcfincd. and thcy arc mutually orthogo.

na l. Furthcrmorc. A" 1 bcing parallcl. \l'C SCC that distribulions $, arc all

parallcl and hcnce complctcly intcgrablc. bccausc all cigcnvalues !Ø, are cons.

tant. Thus. by means of thc local dccomposition thcorcm [5j lhc abo\'e discu' ssions are summcrizcd in thc follo ,,'jng way

'1‘HEORE~I 3.2. Lc/ j\f2", (c) bc a real 2m-di’ucnsi01zal complcx space 107m 01

cor.s/a/!I ilolomorpltic cκrvature 4c>O, alld ,\1 a’‘ n-dilllensional /olally real

sκb’lIa"'fold of JI12’If(C) with harmonic curvaturc such that tlle [-struclure in thc

norlllal bltndle is paral/el. l f Iile mea/z Cltrvatll1'C ,'cc/or of M is paral/cl. tlzcn

M is local y a producl of Ricmannian lIla’ufolds

COROLLARY. Lcl M be an ’ll-dimcnsional totally real submanifold lOith Jzar'

’Ilonic ctlrvaturc in M2m(c) , C르O. lf tilc lIlealt cνrva/ure vector of M is parallc/.

t/zcn .11 is locally a produc/ of Rienraκnian man i.folds .

RE.\IλRK 1. Let S'’(c) bc an n-d imcnsional sphcrc of conslant curvalurc c. 2n...;...1 II'hcn S •• - ' is considcrcd as a hypcrsurfacc in an (n+ I)-dimcnsional com1'lc"

EucJ idean spacc C" J thc product SI(Cl) ... SI(C" ‘• 1) which is denotcd by

N(1t-1) is a lotally real submanifold of C". Then thc mcan cur‘ ature 、 cctOI

JS para lJcl and the normal conncction is flat

RE.\IARK 2. Thc product manifold N(η-'- 1) is conlaincd in a unit sphcre

S'!.II 1’ “;hcrc 1 "cl...;.... , .. + l /cn 1 1. For thc isomctric immcrsion ø’ of N(n +I)

inlO 5111 1 and a principal Sl -bund lc ovcr an n-dimensional comp1cx projcctivc

spacc P"C 、，vith thc projcction r. : S2n • P"C. 、\'0 assumc that therc exists a principal SI-bundlc N(n +l) O\'cr M with totally geodesic fibcrs and thc pr이cc­tion ~ such that ~ is compalible 、.\'ilh thc fibration " . Thcn M=(N(n수1)) is

an n-d imcnsional totally rcal submanifold of P"C and it is casily seen that thc

mcan curvature vector is para lIcl, thc normal conncction is fIa t and moreover

thc second fundamctal form is also parallc l. Of coursc. thc condition of lhe

cur‘ ature tensor is satificd. In thc case where c>O. under thc additional con­

di t ion that thc normal connection is f1at in Theorem 3.2. thcrc cχits a totally

Totally rea/ s“ómauifo/ds wit lJ harmollic c“rvotllre 107

geodesic subman ifold P"C in P ",C such that M satisfics thc abovc situation in

P"C.

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