TOTALLY REAL SUBMANIFOLDS

TOTALLY REAL SUBMANIFOLDS

OF

A COMPLEX SPACE FORM

U-HANG KIandYOUNG HO KIM

DepartmentofMathematics KyungpookNationalUniversity

Teacher’sCollege

Taegu

702-701

KOREA

(Received January^11, 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. We}then 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 curvature vector

H

is then given by

H

¹ Trh. An n-dimensional submanifold M in a Kaehler

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 the}decomposition

TM ^{J(TpM) Np(M).}

^{It followsthat thespace}

^Np(M)

is invariantunder theactionofJ.

Wenowconsider an m-dimensionaltotallyreal submanifoldMof2m-dimensionalKaehlermanifold M. Thenwemay^set

JX

O(X), (.1)

J U,

(1.2)

where

X

is avectorfieldtangentto

M, 0(X)

^{anormalvectorvalued}1-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 the}covariant derivative on

T(M)T+/-(M)

defined by

(-xh)(Y,Z)= ch(Y,Z)

-h(

,x

Y, Z)- h(Y,

_x

Z),R

^and

R

^+/- are the Riemann curvature tensor of

M

and that in the normal bundle respectively and

[A, An]

2. FUNDAMENTAL LEMMAS.

Inthissection,weassume thatM is an m-dimensionaltotallyreal submanifoldofacomplexspace form

M(c)

^{of real}dimension 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 no}simple 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 It}follows 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)}

where

# g(O(AUa), ).

^Because

A,

is a

setfic

k operator d h is asyetfic bilinfo,

#

^is

setfic

^{th respectto}Ml 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)}

for allvector fields

X

and

Y

onM.

(2.5)

togetherwith

(2.3)

yields

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"

i,3,k

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. Thisimplies}that each eigenvalue

,

^of

An

^isconstanton

^M.

^{Let#1,/2, ",/obe}

mutuallydistincteigenvalues of

AH

^andhi,r,

, _no

theirmultiplicities. Sothe smooth distributions

Ta

consisting of alleigenvectorscorrespondingto#a^{aredefinedand}orthogonal 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

^{isatotallyreal}submanifoldembeddedinsomeC

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^’e2x xC

w,

where

Me

^{is atotallyreal}submanifoldembeddedin 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 Sim;e n

+

rv2

+ +

^{r, m, r, must be 1. Hence}

^M,

^S

^1,

a circle in a complex space C.

(Q.E.D.)

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

M(c)

^withnonvanishing parallelmeancurvature vectorH If

An

^{hasmutuallydistinct}eigenvalues,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

^{hasmutuallydistinct}cigenvalues. (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.

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