Totally real submanifolds of a complex space form

TOTALLY REAL SUBMANIFOLDS

OF

A COMPLEX

SPACE FORM

U-HANG KIandYOUNG HO KIM

DepartmentofMathematics

KyungpookNationalUniversity Teacher’sCollege

Taegu

702-701

KOREA

(Received January11, 1994)

ABSTRACT. Totallyrealsubmanifoldsofacomplex space formare studied. Inparticular,totallyreal submanifolds ofacomplexnumberspacewithparallelmean curvature vectorareclassified.

KEY WORDS AND PHRASES. Totallyreal submanifolds,isoperimetric section andcomplex space

form.

1991AMSSUBJECT CLASSIFICATION CODE(S). Primary 53C40,Secondary53B25. 0. INTRODUCTION.

Totallyreal submanifoldsofaKaehler manifold arevery typicalsubmanifoldsofaKaehler manifold introducedbyChen andOgiue

[2]

andYau

[9].

In particular Chen,HouhandLue

1]

pointedoutthat it is interesting to study totally real submanifolds of the complex number space

U

with parallel isoperimetric section and theyclassified compact totally real submanifolds with nonnegative sectional curvature in

U".

In 1987, Urbano

[7]

studied compact totally real submanifold with non-vanishing parallelmean curvature vector.

Inthispaper,weshallstudym-dimensionalcomplete totallyreal submanifolds ofacomplex space

form

M

(c)

andobtain some classificationtheorems. 1. PRELIMINARIES.

Let

M

be aKaehler manifold of real dimension 2m withalmostcomplex structure and metric tensorg. Wethen havej2

I

and

g(JX,

JY)

g(X,

Y)

foranyvector fields.

X

and

Y

on/r,

whereIdenotes the identitytransformation on thetangent bundle. Let be theLevi-Civitaconnection ofM satisfying J O. LetMbe an n-dimensional Riemannian manifoldisometricallyimmersed in

M

bytheimmersioni:M

--

M. Wethen obtain theinduced metric onMwhich willberepresentedthe same notation g. Wealsoidentify

X

with

i.(X)

and

M

with

i(M).

Let be the induced Levi-Civita connectiononM. Then theequationsofGaussandWeingarten

are respectively givenby _xY _VxY

+

h(X, Y)

and

x

AX

+

V

,

where h isthe second fundamental form,

A

the Weingarten map associated to the normal vector field

(

satisfying

g(h(X,

Y),)

g(AX, Y)

and ₇

+/-the connection inthe normal bundle

T+/-M

of

M.

The mean

manifold//is

calledtotallyreal ifJ (TpM) c

T

MforeachPin

M,

whereTpMisthe tangent space of

M

atPand

T,

Mthe normalspace ofMatP.

SinceJhas themaximalrank,m

_>

n. Let Np(M)be theorthogonal complementof

J(TpM)

in

TM.

Thenweget thedecomposition

TM

J(TpM)

Np(M).

It followsthat thespace

Np(M)

is invariantunder theactionofJ.

Wenowconsider an m-dimensionaltotallyreal submanifoldMof2m-dimensionalKaehlermanifold

M. Thenwemayset

JX

O(X),

(.1)

J

U,

(1.2)

where

X

is avectorfieldtangentto

M,

0(X)

anormalvectorvalued1-form, anormalvectorfield and

U

avectorfield onMsatisfying

g(U,

X)

g(0(X), ).

Applying Jto

(1.

l)and

(1.2),

wehave

X

Uo(x)

and

O(U)

.

(1.3)

Differentiating (1 1) and (1.2) covariantly and making use of the equations of Gauss and Weingarten,we get

Uh(X,Y)

Ao(x)Y,

(1.4)

o(

v

xY)

v

i,o(x),

(.5)

7x

U

Uv

}

,

(1.6)

O(AX)

h(X,

U),

(1.7)

where

X

and

Y

are vectorfields tangentto

M

and a vectorfieldnormalto

M.

We now assumethat theambientmanifold

M

isofconstantholomorphic sectional curvature4c, which iscalledacomplex spaceform andit isdenotedby

M(c).

Then theRiemann Christoffel curvature tensor

R

of

M(c)

has the form

gCt(x,

)z,

w)

((x, w)(, z)

(Y,

w)(x, z)

+

(sx, w)(JY, z)

g(JY, W)g(JX,

Z)

2g(JX, Y)g(JZ,

W)).

Since themanifoldM istotally real, itfollows from equations(1,l)-(1.7)that the equations ofGauss,

CodazziandRiccifor

M

arerespectivelyobtained

g(R(X,

Y)Z,

W)

c(g(X, W)g(Y,

Z)

g(Y,

W)g(X,

Z))

+

g(h(X,

W), h(Y, Z))

g(h(Y,

W), h(X, Z)),

(1.$)

(-xh)(Y, Z)

(yh)(X,

Z),

(1.9)

g(R

+/-(X,

Y),,

rt)

c(g(O(X), rl)g(O(Y),

)

g(O(Y), rt)g(O(X),

)

+

g([A,

An]X,

Y),

where

-

is thecovariant derivative on

T(M)T+/-(M)

defined by

(-xh)(Y,Z)=

ch(Y,Z)

2. FUNDAMENTAL LEMMAS.

Inthissection,weassume thatM is an m-dimensionaltotallyreal submanifoldofacomplexspace form

M(c)

of realdimension 2m Anormal vectorfield(is said tobeparallelif 7

-(

0 for any

vectorfieldXonMand(iscalledan moperlmemcsectionifTr

A

is non-zero constant

LEMMA

1. Let M be an m-dimensional totally real submanifold of

M(c)

with parallel

isoperimetricsection( If

A

has nosimple eigenvalues, thenM

(c)

isflat

PROOF. Since

A

is self-adjoint with respect to g, there exists an orthonormal basis

{et,

e2,

,

e,}

forTpM suchthat

g(Ae,,

e,) A6,3, where A1, )2,

,

Am

areeigenvalues of

A.

Since isparallel,we seethat

g([A,

Ao]e,, %)

(,

j)g(A,e,,

%)

((o(,),

)g(O(e), )

g(o(), v)(o(,), ))

for y nodal vector field because of (1.10). Since

A

has no simple eigenvues, for each

e

{

1, 2,

.,

m}

thereis j suchthat

(g(o(,),

)g(o(),

)-

(o(),,)(o(e,),

))

o.

Choosing qas

O(e,),

weget

cg(O(e,),()

0 By (1 1),wesee that

{O(e,)

1,2,

,m}

fos onhonofl basisfor

TM.

Itfollows that

M(c)

isflat.(Q.E.D.)

1. Let

M

be an m-dimensionaltotallyrealsubmanifoldof

M(c)(c

0).

If

M

has an isopefimetficsection

,

then

A

has simple eigenvalues

Let H be the me cuature vector field defined byH Trh. We now assumethat H is

nonvsng

pallel inthe nodal bundle. Wechooseanonhonoal

am

{,

2,

,}

inthe nofl bundleinsuchaway that

(1

H/

H Itfollows that

TrA,

0for 2,where

A,

A,

and

U,

U2,

U

fo onhonoflbasisforTpMbecause of

(1.2),

where

U,

U,.

Then

(1.3)

d

(1.4)

imply

AtU

Uh(u,,u,),

(2.1)

wMch shows that

A,U

AU,.

Tng

the

scM

productth d

mng

useof

(1.3), (1.7)

and(2.1),wemayset

A,

Ua

#Ua,

(2.2)

k

where

#

g(O(AUa), ).

Because

A,

is a

setfic

operator d h is asyetfic bilinfo,

#

is

setfic

th respecttoMl indices i,jd k.

On the other hd,

(2.2)

implies

h(u,,

u)

O(A,U)

.

k

Siny

veor

field

X

on

M

c beexpressed asX

ag(X,

U)U,

hcbe

tten

by

h(X, Y)

Pokg(O(X), (,)g(O(Y),

(a)k,

whichimplies

Trh

-’Pkk,

where

Pk

’Pik.

Since

1

isparallelinthe

norma

bundle, (1

0)

gives

(2.3)

(2.4)

g([A,,

A1]X,

Y)

c(g(O(Y), l)g/(0(X), (,)

g(0(X),

1

)g(O(Y),

(2.5)

and hence where

P

Pll

1.

Wenowprove

’PkP

(TrA1)Pk

c(m

1)6k

(2.6)

-’(PI)

(TrAI)P

+

c(m-

1),

(2.7)

LEMMA

2. Let

M

beanm-dimensionaltotallyreal submanifold ofacomplex spaceform

M(c)

with nonvanishingparallelmean curvature vectorH. Then

An

isparallel.

PROOF. Let

{el,e2,

,em,l,2,

,)

beanorthonormal frame of

M(c)

atapoint

P

of

M

such that el,e2,’. ",em are tangent to

M

and

,2,"" ",(m

are normal to

M,

where

n/

n

II.

Thenweget

ATrA

g(A’A,,A)

+

v

A,

(2.s)

where AistheLaplacianoperator and

A’A1

denotes the restrictedLaplacian

A’

of

A1

isgivenby

(A’A,)X

-][R(ei, X),A,]et

(se

[6]

fordetail). Makinguseof(1.8)ofGauss

ad

thefact that

M

istotally real,wehave

A’A1

c(m

1)A

c(TrA)(I

UI

(R)

U)

+

(TrA)’]PijP#Ua

(R)

Uk

t,j,k

withthehelp

of(2.3),

(2.4)

and

(2.5).

Ifweuse

(2.5)

and

(2.6),

weobtain

(2.9)

g(A’A1,A1)

=0.

(2.10)

Onthe otherhand,wecan put

AIX

EPijg(Ui,

X)Uj

(2.11)

t,j

because of

(2.3).

Wenow extend

1,2,

,m

to differentiableorthonormal normalvector fields defined on a normalneighborhoodOofPbyparalleltranslation withrespecttonormalconnectionalong

geodesicsinM. Thenweget

V yA1)X

E(

V

YPijl)g(Ut, X)Uj

at

P

(2.12)

becauseof

(1.6).

Therefore,

A’A1

isreducedto

A’A1

(

V

yP31)U,

(R)U

i,3

(2.13)

Ifwe

use

(2.9),

thenwehave

g((A’A)UI,

U1)

c(m

1)P

+

(TrA1)E(/ll

)2

E

Pt2kPtjl

J::k11"

Makinguseof(2.6),weobtain

9((A’A1)U1,U1)

=0.

Thus(2 3)implies

AP

0. (2.14)

Since

TrA21

,,g(AU,,AU,)=

-’,,o(/:’,s)

(TrA)P +c(m-

1),

weseethat

A(TrA1)

(TrA

)AP

O.

Combining(2.8), (2.10)and the lastequation,weget the result (Q.E.D)

3. MAIN THEOREMS.

Let

M

be an m-dimensional totally real submanifold of a complex space form

M(c)

with nonvanishing parallelmeancurvature vector. Bylemrna2,weknow that

AH

isparallel. Wenowdefine afunction

hn

forany integern

>

1by h,

Tr(A).

Thenh,is constant on

M

for any integernsince

AH

isparallel. Thisimpliesthat each eigenvalue

,

of

An

isconstanton

M.

Let#1,/2, ",/obe mutuallydistincteigenvalues of

AH

andhi,r,

,

no

theirmultiplicities. Sothe smooth distributions

Ta

consisting of alleigenvectorscorrespondingto_#aaredefinedandorthogonal eachother.

Since

AH

is parallel,

Ta

are parallel and completely integrable.

By

the de Rham decomposition

theorem [4], the submanifold M is a product manifold

M

Mg.

M,,

wherethe tangent bundle of

Ma

correspondsto

Ta.

Wenowassume thattheambient manifold isfiat, that is, acomplex

number space

C

and

M

is embeddedin C

’.

Then asin

[1]

wecan choose an orthonormal basis e, e, -,

e

for

TM

aseigenvectors of

AH

and

J,J,

.,

J

for

J(TpM)

insuchaway that

’

o(A4

e)

and

hi

hjk

ho,,

where

hs/

e,,

h,,/=

0for

eo

E

Lu3],

e,E

’r],/3

:/=

7, where

O]

is the eigenspace correspondingtothe eigenvalue

Let

7re(H

be the component ofHinthe subspaceC

re.

Then

7re(H

isaparallelnormal section of

Me

inC’’e andM

e

isumbilical with respectto

7re(H

).

Therefore, Me is a minimal submanifoldofa

hypersphere in C

"e.

Hence

M

is a product submanifold

M1

x

M2

x x

M,

embedded in

Cr,

C"1

C C

",

where

Me

isatotallyrealsubmanifoldembeddedinsomeC

we.

Thus we have

THEOREM 1. Let

M

be anm-dimensional complete totally real submanifold embedded ina

complex number space C

m.

If

M

hasparallel mean curvature vector

H,

then

M

is either a minimal submanifold or a product submanifold

MI

M2

x x

M

embedded in

Cn _C01 _xC’e2_x

xC

w,

where

Me

is atotallyrealsubmanifoldembeddedin someC

isalsoaminimal submanifold of ahypersphereofC

ve

THEOREM 2. Let

M

be an m-dimensional complete totally real submanifold embedded ina

complex number space C

r".

If

M

has the nonvanishing parallel mean curvature vector and

An

has mutuallydistincteigenvalues, then

M

is aproductsubmanifoldofcirclesS xS S

1.

PROOF.

By

a lemma ofMoore [5], M

M1

x

M2

Mm

is a product immersion embeddedin

C’’,

and

M,

isatotally real submanifold inC and containedin ahypersphere in

THEOREM 3. Let Mbe an m-dimensional totallyreal submanifold ofa complex space form

M(c)

withnonvanishing parallelmeancurvature vectorH If

An

hasmutuallydistincteigenvalues,then

Misflat.

PROOF. Letel,e2, "erabe eigenvectors of

An

correspondingtoeigenvaluesA1,

A.,

.,

Am

respectively. Since

An

isparallelbyLemma2,wehave

AH

R(X, Y)e,

A,R(X, Y)e,

foranyvectorfieldsX andYon

M,

thatis

R(X,

Y)e,is aneigenvector of

AH

correspondingtoA,.

Taking theinnerproductwith%,weobtain

(A,-

A)g(R(X, Y)e,,

%)

0

because

AH

isasymmetric operator. Thus

M

isflat if

AH

hasmutuallydistinctcigenvalues. (Q.E.D.) REMARK. Let Mbe atotallyreal submanifoldofcomplexspace form

M(c)

withnonvanishing

parallel mean curvature vector H. Considering Lemma 1, we see that

M(c)

is flat if the sectional curvaturesdefinedby principalvectorsofHare nonzero.

ACKNOWLEDGEMENT. Thisworkwaspartiallysupported by TGRC-KOSEF. REFERENCES

1.

CHEN, B.-Y., HOUH, C.-S.

and

LUE, H.-S.,

Totallyreal submanifolds,

J.

Diff.

Geom

12

(1977),

473-480.

2. CHEN, B.-Y.and OG1UE,

K.,

On totallyreal submanifolds,Trans.Amer.Math.Soc., 193

(1974),

257-266.

3.

KI, U.-H.

and

NAKAGAWA, H., Compact

totallyreal submanifoldswithparallelmean curvaturevector in acomplex space form,

J.

KoreanMath.

Soc.

23

(1986),

141 150. 4.

KOBAYASHI,

S. and

NOMIZU, K.,

Foundations

of

Differnetial

Geometry I

and

II,

Interscience Publishing,New York,1963and1969.

5. MOORE,

J.D.,

Isometric immersionsofRiemannianproducts,J.

Diff.

Geom.5(1971), 159-168.

6.

SMYTH,

B.,

Submanifoldsofconstantmean curvature, Math.

Ann.

205

(1973),

265-280. 7.

URBANO, F.,

Totallyreal submanifolds,

Geometry

andTopology

of

Submanifolds, World

Scientific, Singapore1989.

8.

YANO, K.

and

KON,

M.,

Anti-invariant

Submanifolds,

Marcel Dekker

Inc.,

1976.