Application of an integral formula to CR submanifolds of complex hyperbolic space

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CR-SUBMANIFOLDS OF COMPLEX HYPERBOLIC SPACE

JIN SUK PAK AND HYANG SOOK KIM

Received 24 May 2004 and in revised form 2 March 2005

The purpose of this paper is to studyn-dimensional compact CR-submanifolds of com-plex hyperbolic space CH(n+p)/2, and especially to characterize geodesic hypersphere in CH(n+1)/2by an integral formula.

1. Introduction

Let ¯M be a complex space form of constant holomorphic sectional curvaturecand let Mbe ann-dimensional CR-submanifold of (n−1) CR-dimension in ¯M. ThenMhas an almost contact metric structure (F,U,u,g) (seeSection 2) induced from the canonical complex structure of ¯M. Hence on ann-dimensional CR-submanifold of (n−1) CR-dimension, we can consider two structures, namely, almost contact structureF and a submanifold structure represented by second fundamental formA. In this point of view, many differential geometers have classified M under the conditions concerning those structures (cf. [3,5,8,9,10,11,12,14,15,16]). In particular, Montiel and Romero [12] have classified real hypersurfacesMof complex hyperbolic space CH(n+1)/2which satisfy the commutativity condition

(C)

FA=AF (1.1)

by using the S1-fibration π:Hn+2

1 CH(n+1)/2 of the anti-de Sitter space H1n+2 over

CH(n+1)/2, and obtainedTheorem 4.1stated inSection 2. We notice that among the model spaces inTheorem 4.1, the geodesic hypersphere is only compact.

In this paper, we will investigaten-dimensional compact CR-submanifold of (n−1) CR-dimension in complex hyperbolic space and provide a characterization of the geo-desic hypersphere, which is equivalent to condition (C), by using the following integral formula established by Yano [17,18]:

Mdiv

∇XX−(divX)X∗1=

M

Ric(X,X)+1 2ᏸXg

2

−∇X2−(divX)21=0,

(1.2)

Copyright©2005 Hindawi Publishing Corporation

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whereX is an arbitrary vector field tangent toM. Our results of the paper are complex hyperbolic versions of those in [6,15].

2. Preliminaries

LetMbe ann-dimensional CR-submanifold of (n−1) CR-dimension isometrically im-mersed in a complex space form ¯M(n+p)/2(c). Denoting by (J, ¯g) the K¨ahler structure of

¯

M(n+p)/2(c), it follows by definition (cf. [5,6,8,9,13,16]) that the maximalJ-invariant

subspace

Ᏸx:=TxM∩JTxM (2.1)

of the tangent spaceTxMofM at each pointxinMhas constant dimension (n−1). So there exists a unit vector fieldU1tangent toMsuch that

x =Span

U1

, ∀x∈M, (2.2)

whereᏰx denotes the subspace ofTxMcomplementary orthogonal toᏰx. Moreover, the vector fieldξ1defined by

ξ1:=JU1 (2.3)

is normal toMand satisfies

JTM⊂TM⊕Spanξ1

. (2.4)

Hence we have, for any tangent vector field X and for a local orthonormal basis 1,

ξα}α=2,...,pof normal vectors toM, the following decomposition in tangential and normal components:

JX=FX+u1(X)ξ

1, (2.5)

Jξα= −Uα+Pξα, α=1,. . .,p. (2.6)

Since the structure (J, ¯g) is Hermitian andJ2= −I, we can easily see from (2.5) and (2.6)

thatFandPare skew-symmetric linear endomorphisms acting onTxMandTxM⊥, re-spectively, and that

gFUα,X= −u1(X) ¯gξ

1,Pξα, (2.7)

gUα,Uβ=δαβ−g¯Pξα,Pξβ, (2.8)

whereTxM⊥denotes the normal space ofMatxandgthe metric onMinduced from ¯g. Furthermore, we also have

gUα,X=u1(X)δ1α, (2.9)

and consequently,

gU1,X

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Next, applyingJto (2.5) and using (2.6) and (2.10), we have

F2X= −X+u1(X)U1, u1(X)1= −u1(FX)ξ1, (2.11)

from which, taking account of the skew-symmetry ofPand (2.7),

u1(FX)=0, FU1=0, 1=0. (2.12)

Thus (2.6) may be written in the form

1= −U1, Jξα=Pξα, α=2,. . .,p. (2.13)

These equations tell us that (F,g,U1,u1) defines an almost contact metric structure on

M(cf. [5,6,8,9,16]), and consequently,n=2m+ 1 for some integerm.

We denote by ¯andthe Levi-Civita connection on ¯M(n+p)/2(c) andM, respectively.

Then the Gauss and Weingarten formulas are given by ¯

∇XY = ∇XY+h(X,Y), (2.14)

¯

∇Xξα= −AαX+∇⊥Xξα, α=1,. . .,p, (2.15)

for any vector fieldsX,Ytangent toM. Here∇⊥denotes the normal connection induced from ¯in the normal bundleTM⊥ofM, andhandthe second fundamental form and the shape operator corresponding toξα, respectively. It is clear thathandare related by

h(X,Y)= p

α=1

gAαX,Yξα. (2.16)

We put

∇⊥

Xξα= p

β=1

sαβ(X)ξβ. (2.17)

Then (sαβ) is the skew-symmetric matrix of connection forms of∇⊥.

Now, using (2.14), (2.15), and (2.17), and taking account of the K¨ahler condition ¯∇J= 0, we differentiate (2.5) and (2.6) covariantly and compare the tangential and normal parts. Then we can easily find that

∇XFY=u1(Y)A

1X−gA1Y,XU1, (2.18)

∇Xu1

(Y)=gFA1X,Y

, (2.19)

∇XU1=FA1X, (2.20)

gAαU1,X= −

p

β=2

s1β(X) ¯gPξβ,ξα, α=2,. . .,p, (2.21)

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In the rest of this paper,we suppose that the distinguished normal vector fieldξ1is parallel

with respect to the normal connection∇⊥.Hence (2.17) gives

s1α=0, α=2,. . .,p, (2.22)

which, together with (2.21), yields

AαU1=0, α=2,. . .,p. (2.23)

On the other hand, the ambient manifold ¯M(n+p)/2(c) is of constant holomorphic

sec-tional curvaturecand consequently, its Riemannian curvature tensor ¯Rsatisfies

¯

RX¯Y¯Z¯= c

4

¯

g( ¯Y, ¯Z) ¯X−g¯( ¯X, ¯Z) ¯Y+ ¯g(JY¯, ¯Z)JX¯−g¯(JX¯, ¯Z)JY¯2 ¯g(JX¯, ¯Y)JZ¯ (2.24)

for any ¯X, ¯Y, ¯Ztangent to ¯M(n+p)/2(c) (cf. [1,2,4,19]). So, the equations of Gauss and

Codazzi imply that

RXYZ=c 4

g(Y,Z)X−g(X,Z)Y+g(FY,Z)FX−g(FX,Z)FY−2g(FX,Y)FZ

+ α

gAαY,ZAαX−gAαX,ZAαY, (2.25)

∇XA1

Y−∇YA1

X= c 4

gX,U1

FY−gY,U1

FX−2g(FX,Y)U1

, (2.26)

for anyX,Y,Z tangent toM with the aid of (2.22), whereRdenotes the Riemannian curvature tensor ofM. Moreover, (2.11) and (2.25) yield

Ric(X,Y)= c 4

(n+ 2)g(X,Y)3u1(X)u1(Y)+ α

trAαgAαX,Y−gA2αX,Y

, (2.27) ρ=c

4(n+ 3)(n−1) +n

2µ2

α

tr2, (2.28)

where Ric andρdenote the Ricci tensor and the scalar curvature, respectively, and

µ=1 n α

trAαξα (2.29)

is the mean curvature vector (cf. [1,2,4,19]).

3. Codimension reduction ofCR-submanifolds ofCH(n+p)/2

LetMbe ann-dimensional CR-submanifold of (n−1) CR-dimension in a complex hy-perbolic space CH(n+p)/2with constant holomorphic sectional curvaturec= −4.

Applying the integral formula (1.2) to the vector fieldU1, we have

M

RicU1,U1

+1

2ᏸU1g

2

−∇U1 2

divU1

2

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Now we take an orthonormal basis{U1,ea,ea∗}a=1,...,(n1)/2of tangent vectors toMsuch that

ea∗:=Fea, a=1,. . .,n−1

2 . (3.2)

Then it follows from (2.11) and (2.20) that

divU1=trFA1= (n−1)/2

a=1

gFA1ea,ea+gFA1ea∗,ea∗=0. (3.3)

Also, using (2.20), we have

U12

=gFA1U1,FA1U1+ (n−1)/2

a=1

gFA1ea,FA1ea+gFA1ea∗,FA1ea∗, (3.4)

from which, together with (2.11) and (2.12), we can easily obtain

U12

=trA2

1−A1U12. (3.5)

Furthermore, (2.20) yields

ᏸU1g

(X,Y)=g∇XU1,Y

+g∇YU1,X

=gFA1−A1F

X,Y, (3.6)

and consequently,

ᏸU

1g

2

=FA1−A1F2. (3.7)

On the other hand, (2.27) and (2.28) withc= −4 yield

RicU1,U1= −(n−1) +u1A1U1trA1−A1U12, (3.8)

trA21

= −ρ−(n+ 3)(n−1) +n2µ2

p

α=2

tr2. (3.9)

Substituting (3.3), (3.5), (3.7), (3.8), and (3.9) into (3.1), we have

M

1

2FA1−A1F

2

+ RicU1,U1

+ρ−n2µ2

+A1U12+ (n+ 3)(n−1) +

p

α=2

tr2

1=0,

(3.10)

or equivalently,

M

1

2FA1−A1F

2

+u1A 1U1

trA1

trA2

1(n−1)

1=0. (3.11)

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Lemma3.1. LetMbe ann-dimensional compact orientableCR-submanifold of(n−1) CR-dimension in a complex hyperbolic spaceCH(n+p)/2. If the distinguished normal vector field ξ1is parallel with respect to the normal connection and if the inequality

RicU1,U1

+ρ−n2µ2+A 1U1

2

+ (n+ 3)(n−1)0 (3.12)

holds onM, then

A1F=FA1 (3.13)

andAα=0forα=2,. . .,p.

Corollary3.2. LetM be a compact orientable real hypersurface ofCH(n+1)/2over which the inequality

RicU1,U1

+ρ−n2µ2+A

1U12+ (n+ 3)(n−1)0 (3.14)

holds. ThenMsatisfies the commutativity condition (C).

CombiningLemma 3.1and the codimension reduction theorem proved in [7, Theo-rem 3.2, page 126], we have the following theoTheo-rem.

Theorem3.3. LetMbe ann-dimensional compact orientableCR-submanifold of(n−1) CR-dimension in a complex hyperbolic spaceCH(n+p)/2. If the distinguished normal vector fieldξ1is parallel with respect to the normal connection and if the inequality

RicU1,U1+ρ−n2µ2+A1U12+ (n+ 3)(n−1)0 (3.15)

holds onM, then there exists a totally geodesic complex hyperbolic spaceCH(n+1)/2immersed inCH(n+p)/2 such that M⊂CH(n+1)/2. MoreoverM satisfies the commutativity condition (C) as a real hypersurface ofCH(n+1)/2.

Proof. Let

N0(x) :=η∈TxM⊥|Aη=0 (3.16)

and letH0(x) be the maximal holomorphic subspace ofN0(x), that is,

H0(x)=N0(x)∩JN0(x). (3.17)

Then, by means ofLemma 3.1,

H0(x)=N0(x)=Spanξ2,. . .,ξp. (3.18)

Hence, the orthogonal complementH1(x) ofH0(x) inTM⊥is Span{ξ1}and so,H1(x)

is invariant under the parallel translation with respect to the normal connection and dimH1(x)=1 at any pointx∈M. Thus, applying the codimension reduction theorem

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be regarded as a real hypersurface of CH(n+1)/2which is totally geodesic in CH(n+p)/2. Ten-tatively, we denote CH(n+1)/2 byM, and byi1 we denote the immersion ofM intoM,

and byi2the totally geodesic immersion ofMinto CH(n+p)/2. Then it is clear from (2.14)

that

i1Xi1Y=i1∇XY+h

(X,Y)=i

1∇XY+g(AX,Y)ξ, (3.19) where is the induced connection onM from that of CH(n+p)/2,h the second

fun-damental form ofMinM, andAthe corresponding shape operator to a unit normal vector fieldξtoMinM. Sincei=i2◦i1andMis totally geodesic in CH(n+p)/2, we can

easily see that (2.15) and (3.19) imply that

ξ1=i2ξ, A1=A. (3.20)

SinceMis a holomorphic submanifold of CH(n+p)/2, for anyXinTM,

Ji2X=i2JX (3.21)

is valid, whereJis the induced K¨ahler structure onM. Thus it follows from (2.5) that JiX=Ji2◦i1X=i2Ji1X=i2

i1FX+u(X)ξ

=iFX+u(X)i2ξ=iFX+u(X)ξ1

(3.22)

for any vector fieldXtangent toM. Comparing this equation with (2.5), we haveF=F andu1=u, which, together withLemma 3.1, implies that

AF=FA. (3.23)

4. An integral formula on the model spaceMh2p+1,2q+1(r)

We first explain the model hypersurfaces of complex hyperbolic space due to Montiel and Romero for later use (for the details, see [12]).

Consider the complex (n+ 3)/2-space C1(n+3)/2 endowed with the pseudo-Euclidean

metricg0given by

g0= −dz0dz¯0+

m

j=1

dzjdz¯j,

m+ 1 :=n+ 3 2

, (4.1)

where ¯zkdenotes the complex conjugate ofzk. OnC(1n+3)/2, we define

F(z,w)= −z0w¯0+

m

k=1

zkwk.¯ (4.2)

Put

Hn+2

1 =

z=z0,z1,. . .,zm

∈C1(n+3)/2:z,z = −1

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where·,·denotes the inner product onC1(n+3)/2 induced from g0. Then it is known

thatHn+2

1 , together with the induced metric, is a pseudo-Riemannian manifold of

con-stant sectional curvature−1, which is known as an anti-de Sitter space. Moreover,Hn+2 1

is a principalS1-bundle over CH(n+1)/2 with projectionπ:Hn+2

1 CH(n+1)/2which is a

Riemannian submersion with fundamental tensorJand time-like totally geodesic fibers. Given p, q integers with 2p+ 2q=n−1 and r∈R with 0< r <1, we denote by M2p+1,2q+1(r) the Lorentz hypersurface ofH1n+2defined by the equations

−z0 2

+ m

k=1

zk2

= −1, r

−z0 2

+ p

k=1

zk2

= − m

k=p+1

zk2

, (4.4)

wherez=(z0,z1,. . .,zm)∈C1(n+3)/2. In fact,M2p+1,2q+1(r) is isometric to the product

H12p+1

1

r−1

×S2q+1

r

1−r

, (4.5)

where 1/(r−1) andr/(1−r) denote the squares of the radii and each factor is embedded in Hn+2

1 in a totally umbilical way. SinceM2p+1,2q+1(r) is S1-invariant,M2hp+1,2q+1(r) :=

π(M2p+1,2q+1(r)) is a real hypersurface of CH(n+1)/2 which is complete and satisfies the

condition (C).

As already mentioned inSection 1, Montiel and Romero [12] have classified real hy-persurfacesM of CH(n+1)/2 which satisfy the condition (C) and obtained the following classification theorem.

Theorem4.1. LetMbe a complete real hypersurface ofCH(n+1)/2which satisfies the condi-tion (C). Then there exist the following possibilities.

(1)M has three constant principal curvaturestanhθ,cothθ,2 coth 2θwith multiplici-ties2p,2q, 1, respectively,2p+ 2q=n−1. Moreover,Mis congruent toM2hp+1,2q+1

(tanh2θ).

(2)Mhas two constant principal curvaturesλ12with multiplicitiesn−1and 1,

respec-tively. (i) Ifλ1>1, thenλ1=cothθ,λ2=2 coth 2θwithθ >0, andMis congruent to

a geodesic hypersphereMh

1,n(tanh2θ). (ii) Ifλ1<1, thenλ1=tanhθ,λ2=2 coth 2θ

withθ >0, andMis congruent toMnh,1(tanh2θ). (iii) Ifλ1=1, thenλ2=2andMis

congruent to a horosphere.

CombiningCorollary 3.2andTheorem 4.1, we have the following theorem.

Theorem4.2. LetMbe a compact orientable real hypersurface ofCH(n+1)/2over which the

inequality

RicU1,U1+ρ−n2µ2+A1U12+ (n+ 3)(n−1)0 (4.6)

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Theorem4.3. LetMbe ann-dimensional compact orientableCR-submanifold of(n−1) CR-dimension in a complex hyperbolic spaceCH(n+p)/2. If the distinguished normal vector fieldξ1is parallel with respect to the normal connection and if the inequality

RicU1,U1

+ρ−n2µ2+A 1U1

2

+ (n+ 3)(n−1)0 (4.7)

holds onM, thenMis congruent to a geodesic hypersphereMh

1,n(tanh2θ)inCH(n+1)/2. Remark 4.4. As already shown in (3.10) and (3.11), the equality

RicU1,U1

+ρ−n2µ2+A 1U1

2

+ (n+ 3)(n−1) =u1A1U1

trA1

trA21(n−1)

(4.8)

holds onM. On the other hand, the geodesic hypersphereM1,hn(tanh2θ) inTheorem 4.1 has constant principal curvatures cothθ and 2 coth 2θ with multiplicitiesn−1 and 1, respectively. Hence we can easily verify the equality

u1A 1U1

trA1

trA2

1(n−1)=0, (4.9)

and consequently,

RicU1,U1

+ρ−n2µ2+A 1U1

2

+ (n+ 3)(n−1)=0 (4.10)

onM1,hn(tanh2θ).

Remark 4.5. If we putV := ∇U1U1(divU1)U1, then it easily follows from (2.11) that V=FA1U1. Taking account of (3.3), (3.5), (3.7), and (3.8), we obtain

divV=1

2FA1−A1F

2

+u1A 1U1

trA1

trA2

1(n−1). (4.11)

Hence if the commutativity condition (C) holds onM, then the vector field V is zero sinceU1is a principal vector ofA1, and consequently,

u1A 1U1

trA1

trA2

1(n−1)=0. (4.12)

Thus, onn-dimensional CR-submanifoldMof (n−1) CR-dimension in a complex hy-perbolic space CH(n+p)/2over which the commutativity conditionCholds, the function u1(A

1U1) cannot be zero at any point ofM. A real hypersurface of a complex hyperbolic

space CH(n+p)/2satisfying the commutativity condition (C) cannot be minimal.

Acknowledgment

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Jin Suk Pak: Department of Mathematics, College of Natural Science, Kyungpook National Uni-versity, Daegu 702-701, Korea

E-mail address:jspak@knu.ac.kr

Hyang Sook Kim: Department of Computational Mathematics, School of Computer Aided Sci-ence, College of Natural SciSci-ence, Inje University, Kimhae, Kyungnan 621-749, Korea

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