Space-like submanifolds with constant scalar curvature in the de Sitter spaces

SPACE-LIKE SUBMANIFOLDS WITH CONSTANT SCALAR CURVATURE IN THE DE SITTER SPACES

Liu Ximin

Abstract. Let M

be a space-like submanifold in a de Sitter space M

(c) with constant scalar curvature. We rstly extend Cheng- Yau’s technique to higher codimensional cases. Then we study the rigidity problem for M

with parallel normalized mean curvature vector eld.

1. Introduction

Let M _p ^n+p (c) be an (n+p)-dimensional connected semi-Riemannian man- ifold of constant curvature c whose index is p. It is called an indenite space form of index p and simply a space form when p = 0. If c > 0, we call it as a de Sitter space of index p. Akutagawa [3] and Ramanathan [11]

investigated space-like hypersurfaces in a de Sitter space and proved inde- pendently that a complete space-like hypersurface in a de Sitter space with constant mean curvature is totally umbilical if the mean curvature H sat- ises H ² ≤ c when n = 2 and n ² H ² < 4(n − 1)c when n ≥ 3. Later, Cheng [4] generalized this result to general submanifolds in a de Sitter space.

To our best knowledge, there are almost no intrinsic rigidity results for the space-like submanifolds with constant scalar curvature in a de Sitter space until Zheng [15] obtained the following result.

Theorem. Let M ⁿ be an n-dimensional compact space-like hypersur- face in M ₁ ⁿ⁺¹ (c) with constant scalar curvature. If M ⁿ satises

(1) K(M ) > 0,

(2) Ric(M ) ≤ (n − 1)c, (3) R < c,

where R is the normalized scalar curvature of M ⁿ , then M ⁿ is totally umbilical.

Received February 23, 2000.

2000 Mathematics Subject Classication: 53C40, 53C42, 53C50.

Key words and phrases: space-like submanifold, scalar curvature, normalized mean curvature vector eld.

This project is supported in part by the National Science Foundation of China.

In [5], Cheng-Yau rstly studied the rigidity problem for a hypersurface with constant scalar curvature in a space form by introducing a self-adjoint second order dierential operator (See Theorems 1 and 2 in [5]). They proved that, for an M ⁿ in M ⁿ⁺¹ (c), if R is constant and R ≥ c, then

|∇σ| ² ≥ n ² |∇H| ² where σ and H denote the second fundamental form and the length of the mean curvature vector eld of M ⁿ respectively. By using Cheng-Yau’s technique, Li [7] [8] studied the pinching problem and also proved some global rigidity theorems for hypersurfaces with constant scalar curvature.

In the present paper, we would like extend Cheng-Yau’s technique to higher codimensional cases and use this result to study the rigidity prob- lem for space-like submanifolds in a de Sitter space with constant scalar curvature.

2. Preliminaries

Let M p ^n+p (c) be an (n + p)-dimensional semi-Riemannian manifold of constant curvature c whose index is p. Let M ⁿ be an n-dimensional Rie- mannian manifold immersed in M p ^n+p (c). As the semi-Riemannian metric of M p ^n+p (c) induces the Riemannian metric of M ⁿ , M ⁿ is called a space- like submanifold. We choose a local eld of semi-Riemannian orthonormal frames e 1 , . . . , e n+p in M p ^n+p (c) such that at each point of M ⁿ , e 1 , . . . , e n

span the tangent space of M ⁿ and form an orthonormal frame there. We use the following convention on the range of indices:

1 ≤ A, B, C, . . . ≤ n + p; 1 ≤ i, j, k, . . . ≤ n; n + 1 ≤ α, β, γ ≤ n + p.

Let ω 1 , . . . , ω n+p be its dual frame eld so that the semi-Riemannian metric of M _p ^n+p (c) is given by ds ² = ∑

i ω ² _i − ∑

α ω ² _α = ∑

A ϵ _A ω _A ² , where ϵ _i = 1 and ϵ α = −1. Then the structure equations of M p ^n+p (c) are given by

(1) dω A = ∑

B

ϵ B ω AB ∧ ω B , ω AB + ω BA = 0,

(2) dω AB = ∑

C

ϵ C ω AC ∧ ω CB − 1 2

∑

C,D

K ABCD ω C ∧ ω D ,

(3) K ABCD = c ϵ A ϵ B (δ AC δ BD − δ AD δ BC ).

Restrict these form to M ⁿ , we have

(4) ω _α = 0, n + 1 ≤ α ≤ n + p,

the Riemannian metric of M ⁿ is written as ds ² = ∑

i ω _i ² . From Cartan’s lemma we can write

(5) ω _αi = ∑

j

h ^α _ij ω _j , h ^α _ij = h ^α _ji .

From these formulas, we obtain the structure equations of M ⁿ :

(6) dω i = ∑

j

ω ij ∧ ω j , ω ij + ω ji = 0,

(7) dω _ij = ∑

k

ω _ik ∧ ω kj − 1 2

∑

k,l

K _ijkl ω _k ∧ ω l ,

(8) R _ijkl = c(δ _ik δ _jl − δ il δ _jk ) − ∑

α

(h ^α _ik h ^α _jl − h ^α il h ^α _jk ), where R ijkl are the components of the curvature tensor of M ⁿ .

For indenite Riemannian manifolds in detail, refer to O’Neill [9].

Denote L α = (h ^α _ij ) n ×n and H α = (1/n) ∑

i h ^α _ii for α = n + 1, · · · , n + p.

Then the mean curvature vector eld ξ, the mean curvature H and the square of the length of the second fundamental form S are expressed as

ξ = ∑

α

H α e α , H = |ξ|, S = ∑

α,i,j

(h ^α _ij ) ² ,

respectively. Moreover, the normal curvature tensor {R αβkl }, the Ricci curvature tensor {R ik } and the normalized scalar curvature R are expressed as

R αβkl = ∑

m

(h ^α _km h ^β _ml − h ^α _lm h ^β _mk ), R ik = (n − 1) c δ ik − n ∑

α

(H α )h ^α _ik + ∑

α,j

h ^α _ij h ^α _jk ,

R = c + 1

n(n − 1) (S − n ² H ² ).

(9)

Dene the rst and the second covariant derivatives of {h ^α _ij }, say {h ^α _ijk } and {h ^α _ijkl } by

(10) ∑

k

h ^α _ijk ω _k = dh ^α _ij + ∑

k

h ^α _kj ω _ki + ∑

k

h ^α _ik ω _kj + ∑

β

h ^β _ij ω _βα ,

∑

l

h ^α _ijkl ω _l = dh ^α _ijk + ∑

m

h ^α _mjk ω _mi + ∑

m

h ^α _imk ω _mj (11)

+ ∑

m

h ^α _ijm ω mk + ∑

β

h ^β _ijk ω βα .

Then, by exterior dierentiation of (5), we obtain the Codazzi equation

(12) h ^α _ijk = h ^α _ikj .

It follows from Ricci’s identity that (13) h ^α _ijkl − h ^α _ijlk = ∑

m

h ^α _mj R mikl + ∑

m

h ^α _im R mjkl + ∑

β

h ^β _ij R βαkl . The Laplacian of h ^α _ij is dened by h ^α _ij = ∑

k h ^α _ijkk . From (13), we have h ^α _ij = nH α,ij + ∑

k,m

h ^α _km R _mijk + ∑

k,m

h ^α _im R _mkjk + ∑

k,β

h ^β _ik R _βαjk

= nH _α,ij + n c h ^α _ij − n c H α δ _ij − n ∑

β,m

H _β h ^α _im h ^β _mj + ∑

β

S _αβ h ^β _ij

−2 ∑

β,k,m

h ^β _ik h ^α _km h ^β _mj + ∑

m,k,β

h ^α _im h ^β _mk h ^β _kj + ∑

β,k,m

h ^β _ik h ^β _km h ^α _mj ,

where S αβ = ∑

i,j h ^α _ij h ^β _ij for all α and β. Dene N (A) = ∑

i,j a ² _ij for any real matrix A = (a ij ) n ×n . Then we have

∑

i,j

h ^α _ij h ^α _ij = n ∑

i,j

H α,ij h ^α _ij + n c S α − c n ² H _α ² − n ∑

β

H _β T r(L ² _α L _β )

+ ∑

β

S _αβ ² + ∑

β

N (L _α L _β − L β L _α ), (14)

where S _α = ∑

i,j (h ^α _ij ) ² , for every α.

Suppose H > 0 on M ⁿ and choose e _n+1 = ξ/H. Then it follows that (15) H _n+1 = H; H _α = 0, α > n + 1.

From (10) and (15) we can see

(16) H _n+1,k ω _k = dH, H _α,k ω _k = Hω n+1α α > n + 1.

From (11), (15) and (16) we have

(17) H _n+1,kl = H _kl − 1

H

∑

β>n+1

H _β,k H _β,l , where dH = ∑

i H _i ω _i and ∇H k = ∑

l H _kl ω _l ≡ dH k + H _l ω _lk for all k.

Using (14) and (17), we have

∑

i,j

h ⁿ⁺¹ _ij h ⁿ⁺¹ _ij = n ∑

i,j

H ij h ⁿ⁺¹ _ij − n H

∑

i,j

∑

β>n+1

H _β,i H _β,j h ⁿ⁺¹ _ij +n c S _n+1 − c n ² H ² − nHf n+1 + S _n+1 ² + ∑

β>n+1

S _n+1β ²

+ ∑

β>n+1

N (L n+1 L _β − L β L n+1 ), (18)

where f n+1 = T r(L n+1 ) ³ .

M. Okumura [10] established the following lemma (see also [2]).

Lemma 2.1. Let {a i } ⁿ _i=1 be a set of real numbers satisfying ∑

i a i = 0,

∑

i a ² _i = t ² , where t ≥ 0. Then we have

− √ n − 2

n(n − 1) t ³ ≤ ∑

i

a ³ _i ≤ √ n − 2 n(n − 1) t ³ ,

and the equalities hold if and only if at least (n − 1) of the a i are equal.

Denote the eigenvalues of L n+1 by {λ i } ⁿ i=1 . Then we have

(19) nH = ∑

i

λ i , S n+1 = ∑

i

λ ² _i , f n+1 = ∑

i

λ ³ _i .

Set L _n+1 = L _n+1 − H I n , f _n+1 = f _n+1 − 3HS n+1 + 2nH ³ , S _n+1 = S n+1 − nH ² , and λ i = λ i − H, where I n denotes the identity matrix of degree n. Then (19) changes into

(20) 0 = ∑

i

λ i , S n+1 = ∑

i

λ ² _i , f n+1 = ∑

i

λ ³ _i .

By applying Okumura’s Lemma to f n+1 , we have f n+1 ≤ √ n − 2

n(n − 1) S n+1

√

S n+1 ⇐⇒

f _n+1 ≤ 3HS n+1 − 2nH ³ + √ n − 2

n(n − 1) S _n+1

√ S _n+1 . So we have

n c S n+1 − c n ² H ² − nHf n+1 + S _n+1 ² (21)

≥ S n+1 {n c + S n+1 − nH ² − n(n − 2)H

√ S n+1

n(n − 1) }.

It follows from (15) that

(22) ∑

β>n+1

S _n+1β ² = ∑

β>n+1

{ ∑

i,j

(h ⁿ⁺¹ _ij − Hδ ij )h ^β _ij } ² . Denote S I = ∑

β>n+1 S β . From (22), we have

(23) ∑

β>n+1

S _n+1β ² ≤ S n+1 S _I . Let T = ∑

i,j T _ij ω _i ω _j be a symmetric tensor on M ⁿ dened by (24) T _ij = h ⁿ⁺¹ _ij − nHδ ij .

We introduce an operator  associated to T acting on f ∈ C ² (M ⁿ ) by

f = ∑

i,j

T ij f ij = ∑

i,j

h ⁿ⁺¹ _ij f ij − nH f,

where is the Laplacian. Since ( T ij ) is divergence-free, it follows from [5]

that the operator  is self-adjoint relative to the L ² -inner product of M ⁿ . Choosing f = H in above expression, we have

(25) ∑

i,j

h ⁿ⁺¹ _ij H _ij = H + nH H.

Denote S = S _n+1 + S _I . Substituting (21), (23) and (25) into (18), we get

∑

i,j

h ⁿ⁺¹ _ij h ⁿ⁺¹ _ij ≥ nH + 1

2 n ² ( H ² ) − n ² |∇H| ²

− n H

∑

β>n+1

∑

i,j

H _β,i H _β,j h ⁿ⁺¹ _ij

+ ∑

β>n+1

N (L n+1 L _β − L β L n+1 )

+S n+1

{

n c − nH ² + S n+1 − n(n − 2)H

√ S n+1

n(n − 1) }

. (26)

3. An extension of Cheng-Yau’s technique

Cheng-Yau [5] gave a lower estimation for |∇σ| ² , the square of the length

of the covariant derivative of σ, which plays an important role in their

discussion. They proved that, for a hypersurface in a space form of constant

scalar curvature c, if the normalized scalar curvature R is constant and

R ≥ c, then |∇σ| ² ≥ n ² |∇H| ² .

For the space-like submanifolds in a de Sitter space, we can prove the following

Theorem 3.1. Let M ⁿ be a connected submanifold in M p ^n+p (c) with nowhere zero mean curvature H. If R is constant and R < c, then

(27) |∇σ| ² = ∑

i,j,k,α

(h ^α _ijk ) ² ≥ n ² |∇H| ²

and the symmetric tensor T dened by (24) is negative semi-denite. More- over, if the equality in (27) holds on M ⁿ , then H is constant and T is negative denite.

Proof. From (9), we have n ² H ² − S = n(n − 1)(c − R) > 0. Taking the covariant derivative on both sides of this equality, we get

n ² H H _k = ∑

i,j,α

h ^α _ij h ^α _ijk , k = 1, · · · , n.

For every k, it follows from Cauchy-Schwarz’s inequality that (28) n ⁴ H ² H _k ² = ( ∑

i,j,α

h ^α _ij h ^α _ijk ) ² ≤ S ∑

i,j,α

(h ^α _ijk ) ² ,

where the equality holds if and only if there exits a real function c _k such that

(29) h ^α _ijk = c k h ^α _ij

for all i, j and α. Taking sum on both sides of (28) with respect to k, we have

(30) n ⁴ H ² |∇H| ² = n ⁴ H ² ∑

k

H _k ² ≤ S ∑

(i,j,k,α)

(h ^α _ijk ) ² ≤ n ² H ² ∑

(i,j,k,α)

(h ^α _ijk ) ² . Therefore (27) holds on M ⁿ .

Denote the eigenvalues of L _n+1 by {λ i } ⁿ _i=1 . Then (λ _i ) ² ≤ S n+1 ≤ S ≤ n ² H ² for all i. Hence |λ i | ≤ nH for all i. Therefore T = (T ij ) = L n+1 − nHI _n is negative semi-denite.

Suppose that ∑

i,j,k,α (h ^α _ijk ) ² = n ² |∇H| ² holds on M ⁿ . It follows from (30) that

(31) 0 ≤ n ³ (n − 1)(c − R)|∇H| ² ≤ S



 ∑

i,j,k,α

(h ^α _ijk ) ² − n ² |∇H| ²



 .

Hence (c −R)|∇H| ² = 0 on M ⁿ . Because R < c, |∇H| ² = 0 on M ⁿ . In this case, |λ i | ≤ (S n+1 ) ^1/2 ≤ S ^1/2 < nH for all i. Thus T is negative denite.

This completes the proof of Theorem 3.1.

4. Submanifolds with at normal bundle

In this section, we propose to use the extension of Cheng-Yau’s technique given in section 3 to study the rigidity problem for compact submanifolds in the de sitter space M p ^n+p (c). We continue use the same notations as in section 2. Let M ⁿ be a compact submanifold in M p ^n+p (c) with nowhere zero mean curvature H. Suppose that ξ/H is parallel and choose e _n+1 = ξ/H.

Then ω n+1α = 0 for all α. It follows from (11) and (16) that

(32) H _α,k = 0, H _α,kl = 0,

for all α > n + 1 and k, l = 1, · · · , n.

Suppose in addition that the normal bundle of M ⁿ is at. Then

(33) _αβ = − 1

2 R _αβkl ω _k ∧ ω l = 0,

for all α and β on M ⁿ . For all α and β we have L α L _β = L _β L α , which is equivalent to that {L α } ^n+p _α=n+1 can be diagonized simultaneously.

We denote the eigenvalues of L _α by {λ ^α ₁ , · · · , λ ^α _n } for every α. It follows from [13] that

(34) 1

2 S = ∑

i,j,k,α

(h ^α _ijk ) ² + n ∑

i,j,α

H α,ij h ^α _ij + ∑

α

∑

i<j

K ij (λ ^α _i − λ ^α j ) ² ,

where K _ij = c + ∑

β λ ^β _i λ ^β _j denotes the sectional curvature of M ⁿ corre- sponding to the plane section spanned by {e i , e _j } for every pair of i < j.

Assume that R is constant and R < c. From (25) and (32), we have

∑

i,j,k,α

(h ^α _ijk ) ² + n ∑

i,j,α

H _α,ij h ^α _ij = n H + 1

2 ( n ² H ² ) + ∑

i,j,k,α

(h ^α _ijk ) ² −n ² |∇H| ² . Note that S = ( n ² H ² ). Therefore (34) turns into

0 = n H + ∑

i,j,k,α

(h ^α _ijk ) ² − n ² |∇H| ² + ∑

i<j

∑

α

K ij (λ ^α _i − λ ^α j ) ² . Integrating the both sides of above equality on M ⁿ , we have

0 =

∫

M



 ∑

i,j,k,α

(h ^α _ijk ) ² − n ² |∇H| ²



 ∗ 1 + ∑

i<j

∑

α

∫

M

K ij (λ ^α _i − λ ^α _j ) ² ∗ 1.

If K ij ≥ 0 on M ⁿ , it follows from (27) and the above equality that

(35) ∑

(i,j,k,α)

(h ^α _ijk ) ² ≡ n ² |∇H| ² ; K _ij (λ ^α _i − λ ^α j ) ² ≡ 0,

for every α and i < j. Hence we can prove the following theorem

Theorem 4.1. Let M ⁿ be a compact submanifold with non-negative sectional curvature in M p ^n+p (c). Suppose that the normal bundle N (M ) is at and the normalized mean curvature vector is parallel. If R is constant and R < c, then M ⁿ is totally umbilical.

Proof. From the rst equality of (35) and Theorem 3.1, we have that H is constant on M ⁿ , then ξ is parallel. From Theorem 3 of [1] we know that M ⁿ is totally umbilical.

Remark 4.1. In Theorem 4.1, we have used the assumptions that are dierent from that in Theorem 3 [1] to obtain the same result.

Also, we need the following

Lemma 4.1 [12]. Let A and B be n × n-symmetric matrices satisfying T r A = 0, T r B = 0 and A B − B A = 0. Then

− √ n − 2

n(n − 1) (T r A ² )(T r B ² ) ^1/2 ≤ T r A ² B (36)

≤ √ n − 2

n(n − 1) (T r A ² )(T r B ² ) ^1/2 , and the equality holds on the right (resp. left) hand side if and only if n −1 of the eigenvalues x _i of A and the corresponding eigenvalues y _i of B satisfy

|x i | = ^{(T r A} √

⁾

n(n −1) , x i x j ≥ 0, y i = − ^{(T r B} √

⁾

n(n −1) (resp. y i = ^{(T r B} √

⁾

n(n −1) ).

Choose a suitable normal frame eld {e β } ^n+p _β=n+2 such that S _αβ = 0 for all α ̸= β. Then

(37) ∑

α,β>n+1

S _αβ ² = ∑

β>n+1

S _β ² ≤ S _I ² ,

where the equality holds if and only if at least p − 2 numbers of S α ’s are zero.

Taking sum with respect to α > n + 1 on both-sides of (14), we have

∑

i,j,α>n+1

h ^α _ij h ^α _ij = (n c − n H ² )S _I − nH ∑

α>n+1

T r(L ² _α L _n+1 )

+ ∑

α>n+1

S _n+1α ² + ∑

α>n+1

S _α ² . (38)

Using the left hand side of (36) to T r(L ² _α L n+1 ), we have T r(L ² _α L n+1 ) ≤ (n − 2)S α

√ S n+1

n(n − 1) .

Substituting this into (38) and using (23) and (37), we have (39)

∑

i,j,α>n+1

h ^α _ij h ^α _ij ≥ S I

{

(n c − n H ² ) − n(n − 2)H

√ S _n+1

n(n − 1) + S n+1

} . Substituting (32) into (26), we have

∑

i,j

h ⁿ⁺¹ _ij h ⁿ⁺¹ _ij ≥ nH + 1

2 ( n ² H ² ) − n ² |∇H| ²

+S _n+1 {

(n c − nH ² ) − n(n − 2)H

√ S _n+1

n(n − 1) + S _n+1 }

. (40)

Note that S = ( n ² H ² ) and (41) 1

2 S = ∑

i,j,k,α

( h ^α _ijk ) 2

+ ∑

i,j

h ⁿ⁺¹ _ij h ⁿ⁺¹ _ij + ∑

i,j,α>n+1

h ^α _ij h ^α _ij . From (39) and (40), we obtain

0 ≥ nH + ∑

(i,j,k,α)

( h ^α _ijk ) 2

− n ² |∇H| ²

+S {

(n c − nH ² ) − n(n − 2)H

√ S _n+1

n(n − 1) + S _n+1 }

. (42)

Consider the quadratic form Q(u, t) = u ² − ^{√ n} _n ⁻² ₋₁ ut − t ² . By the orthogonal transformation

{ u = √ ¹

2n {(1 + √

n − 1)u + (1 − √

n − 1)t}

t = ^{√ 1} _2n {( √

n − 1 − 1)u + ( √

n − 1 + 1)t}

Q(u, t) turns into Q(u, t) = ⁿ

2 √

n −1 (u ² − t ² ), where u ² + t ² = u ² + t ² . Take u = √

S _n+1 , t = √

nH, then nc − nH ² − n(n − 2)H

√ S n+1

n(n − 1) + S _n+1

= nc + Q(u, t) = nc + n(u ² − t ² ) 2 √

n − 1

= nc + n( −u ² − t ² ) 2 √

n − 1 + nu ²

√ n − 1

≥ nc − nS n+1

2 √

n − 1

(43)

Note that

(44) S _n+1 ≤ S n+1 + S _I = S.

From (43), (44) and (27) we have

(45) 0 ≥ nH + S

{

nc − nS 2 √

n − 1 }

. Integrating the both sides of (45) on M ⁿ , we have

(46) 0 ≥

∫

M

S {

nc − nS 2 √

n − 1 }

∗ 1.

Therefore we can prove the following

Theorem 4.2. Let M ⁿ (n ≥ 3) be a closed space-like submanifold with parallel normalized mean curvature vector eld immersed into M p ^n+p (c).

Suppose that R is constant and R = c − R > 0. If the normal bundle N (M ) is at and

(47) S < nH ² + 2 √

n − 1c,

then S = nH ² and M ⁿ is umbilical (hence isometric to a sphere).

Proof. Denote R = c − R. Then S = n(n − 1)(H ² − R) and S = nR + n ² (H ² − R). Since n ≥ 3, we have

(48) nc − nS 2 √

n − 1 = n(c − n(n − 1)(H ² − R 2 √

n − 1 ) = n(c − S − nH ² 2 √

n − 1 ).

It is clear that the condition (47) is equivalent to

(49) nc − S

2 √

n − 1 > 0.

From (46) and (49) we have S = 0 on M ⁿ , so H ² = R and S = nR, that is S = nH ² . Since H is constant on M ⁿ , hence ξ is parallel, from Theorem 3 of [1] we know that M ⁿ is totally umbilical.

References

[1] R. Aiyama, Compact space-like submanifolds in a pseudo-Riemannian sphere S

(c), Tokyo J. Math. 18 (1995), 81–90.

[2] H. Alencar and M. P. do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120 (1994), 1223–1229.

[3] K. Akutagawa, On space-like hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 196 (1987), 13–19.

[4] Q. M. Cheng, Complete space-like submanifolds in a de Sitter space with parallel

mean curvature vector, Math. Z. 206 (1991), 333–339.

[5] S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math.

Ann. 225 (1977), 195–204.

[6] Z. H. Hou, Hypersurfaces in sphere with constant mean curvature, Proc. Amer. Math.

Soc. 125 (1997), 1193–1196.

[7] H. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305 (1996), 665–672.

[8] , Global rigidity theorems of Hypersurfaces, Ark. Mat. 35 (1997), 327–351.

[9] B. O’Neill, Semi-Riemannian Geometry, New York, Academic Press, 1983.

[10] M. Okumuru, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213.

[11] J. Ramanathan, Complete space-like hypersurfaces of constant mean curvature in the de Sitter space, Indiana Univ. Math. J. 36 (1987), 349-35.

[12] W. Santos, Submanifolds with parallel mean curvature vector in spheres, Tˆ ohoku Math. J. 46 (1994), 403–415.

[13] K. Yano and S. Ishihara, Submanifolds with parallel mean curvature vector, J. Di.

Geom. 6 (1971), 95–118.

[14] S. T. Yau, Submanifolds with constant mean curvature, Amer. J. Math. 96 (1975), 346–366; 97 (1975), 76–100.

[15] Y. Zheng, Space-like hypersurfaces with constant scalar curvature in the de Sitter spaces, Di. Goem. and its Appl. 6 (1996), 51–54.

Department of Applied Mathematics Dalian University of Technology Dalian 116024, China

and

Institute of Mathematics

Academy of Mathematics and Systems Sciences Chinese Academy of Sciences

Beijing 100080, China

E-mail : [email protected]