Space-like Submanifolds in the de Sitter Spaces^{∗}

Liu Ximin Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany E-mail: xmliu@dlut.edu.cn

Abstract

In this paper, by using Cheng-Yau’s self-adjoint operator □, we study the space-like submanifolds in the de Sitter spaces and obtain some general rigidity results. Key words and phrases: space-like submanifold, de Sitter space, totally umbilical submanifold. 1991 Mathmatics Subject Classification: 53C40, 53C42, 53C50.

1 Introduction

Let M_{p}^{n+p}(c) be an (n + p)-dimensional connected semi-Riemannian manifold of constant curvature c whose index is p. It is called an indefinite 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, denote it by S_{p}^{n+p}(c). The study of space-like hypersurfaces in de Sitter space has been recently of substantial interest from both physics and mathematical points of view. Akutagawa [2] and Ramanathan [16] investigated space-like hypersurfaces in a de Sitter space and proved independently that a complete space-like hypersurface in a de Sitter space with constant mean curvature is totally umbilical if the mean curvature H satisfies H^{2} ≤ c when n = 2 and n^{2}H^{2} < 4(n − 1)c when n ≥ 3. Later, Cheng [5] generalized this result to general submanifolds in a de Sitter space.

On the other hand, Cheng and Ishikawa [6] have recently shown that the totally umbilical round spheres are the only compact space-like hypersurfaces in S_{1}^{n+1}(1) with constant scalar curvature S < n(n − 1). Some other authors, such as Liu [12] and Zheng [18, 19], have also obtained interesting results related to the characterization of the totally umbilical round spheres as the only compact space-like hypersurfaces in the de Sitter space with constant scalar curvature.

In [11], H. Li first studied the Cheng-Yau’s self-adjoint operator □ for a given codazzi tensor field ϕ = ∑_{i,j} ϕ_{ij}ω_{i}ω_{j} on an n-dimensional compact Riemannian manifold and obtain a general rigidity theorem which generalized Cheng-Yau’s work [8]. By using this result he studied the hypersurfaces in real space form and obtain general rigidity results which unify some existing results. Also he studied the space-like hypersurfaces in Lorentzian space form and obtained some rigidity theorems which naturally generalize the existing results of Akutagawa [2], Ramanathan [16] and Montiel [13] about Goddard’s conjecture [9].

∗This work is supported in part by the National Natural Science Foundation of China

In the present paper, we would like to use Cheng-Yau’s self-adjoint operator □ to study the space-like submanifolds in de Sitter space and obtain some general rigidity results, also in the last section of this paper we discuss the space-like hypersurfaces in the de Sitter spaces and get some very interesting rigidity results.

2 Cheng-Yau’s self-adjoint operator □

In this section, we review the fundamental results about the Cheng-Yau’s self-adjoint operator □, for details see [11].

Let M^{n} be an n-dimensional Riemannian manifold, e_{1}, …, e_{n} a local orthonormal frame field on M^{n}, and let ω_{1}, …, ω_{n} be its dual coframe field. Then the structure equations of M^{n} are given by

dω_{i} = ∑_{j} ω_{ij} ∧ ω_{j}, ω_{ij} + ω_{ji} = 0,  (1)

dω_{ij} = ∑_{k} ω_{ik} ∧ ω_{kj} + Ω_{ij},  (2)

where

Ω_{ij} = −\frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l},  (3)

and

R_{ijkl} + R_{ijlk} = 0,

where ω_{ij} is the Levi-Civita connnection form and R_{ijkl} are the components of the curvature tensor of M^{n}.

For any C^{2}-function f defined on M^{n}, we defined its gradient and Hessian by the following formulas

df = ∑_{i} f_{i}ω_{i},  (4)
∑_{j} f_{ij}ω_{j} = df_{i} + ∑_{j} f_{j}ω_{ji}.  (5)

We know that f_{ij} = f_{ji} by exterior differentiation of (4).

Let ϕ = ∑_{ij} ϕ_{ij}ω_{i} ⊗ω_{j} be a symmetric tensor defined on M^{n}. The covariant derivative of ϕ_{ij} is defined by

∑_{k} ϕ_{ijk}ω_{k} = dϕ_{ij} + ∑_{k} ϕ_{kj}ω_{ki} + ∑_{k} ϕ_{ik}ω_{kj}.  (6)

We call the symmetric tensor ϕ = ∑_{i,j} ϕ_{ij}ω_{i} ⊗ ω_{j} a Codazzi tensor, if

ϕ_{ijk} = ϕ_{ikj}.  (7)

The second covariant derivative of ϕ_{ij} is defined by

∑_{l} ϕ_{ijkl}ω_{l} = dϕ_{ijk} + ∑_{m} ϕ_{mjk}ω_{mi} + ∑_{m} ϕ_{imk}ω_{mj} + ∑_{m} ϕ_{ijm}ω_{mk}.  (8)

By exterior differentiation of (6), we have

∑_{l,k} ϕ_{ijkl}ω_{j} ∧ ω_{k} = ∑_{m} ϕ_{mj}Ω_{mi} + ∑_{m} ϕ_{im}Ω_{mj}.  (9)

Then we have the following Ricci identities

ϕ_{ijkl} − ϕ_{ijlk} = ∑_{m} ϕ_{mj}R_{mikl} + ∑_{m} ϕ_{im}R_{mjkl}.  (10)

The definition of the following self-adjoint operator □ was first introduced by Cheng-Yau in [8].

Definition 2.1. Let ϕ = ∑_{i,j} ω_{i} ⊗ ω_{j} be a Codazzi tensor field on a Riemanian manifold M^{n}. We define the opertator □ associated to ϕ by

□f = ∑_{i,j}((∑_{k} ϕ_{kk})δ_{ij} − ϕ_{ij})f_{ij},

for any C^{2}-function f defined on M^{n}.

The Laplacian of the tensor ϕ_{ij} is defined to be ∑_{k} ϕ_{ijkk}, and therefore

Δϕ_{ij} = ∑_{k} ϕ_{ijkk}
= ∑_{k}(ϕ_{ijkk} − ϕ_{ikjk}) + ∑_{k}(ϕ_{ikjk} − ϕ_{ikkj}) + ∑_{k}(ϕ_{ikkj} − ϕ_{kkij}) + ∑_{k} ϕ_{kkij}
= ∑_{m,k} ϕ_{mk}R_{mijk} + ∑_{m,k} ϕ_{im}R_{mkjk} + ∑_{k}(ϕ_{ijkk} − ϕ_{ikjk})
+ ∑_{k}(ϕ_{ikkj} − ϕ_{kkij}) + (∑_{k} ϕ_{kk})_{ij}.  (11)

By use of (7), we have

Δϕ_{ij} = (∑_{k} ϕ_{kk})_{ij} + ∑_{m,k} ϕ_{mk}R_{mijk} + ∑_{m,k} ϕ_{im}R_{mkjk}  (12)

Let |ϕ|^{2} = ∑_{i,j} ϕ_{ij}^{2}, |∇ϕ|^{2} = ∑_{i,j,k} ϕ_{ijk}^{2} and tr ϕ = ∑_{k} ϕ_{kk}. Then from (12) we have

\frac{1}{2}Δ|ϕ|^{2} = |∇ϕ|^{2} + ∑_{i,j} ϕ_{ij}(tr ϕ)_{ij} + ∑_{i,j,m,k} ϕ_{ij}ϕ_{mk}R_{mijk} + ∑_{i,j,m,k} ϕ_{ij}ϕ_{im}R_{mkjk}.  (13)

Near a given point p ∈ M, we choose a local orthonormal frame field {e_{1}, …, e_{n}} and its dual frame field {ω_{1}, …, ω_{n}} such that ϕ = ∑_{i,j} ϕ_{ij}ω_{i} ⊗ ω_{j}, ϕ_{ij} = λ_{i}δ_{ij} at p. Then (13) is simplified to

\frac{1}{2}Δ|ϕ|^{2} = |∇ϕ|^{2} + ∑_{i} λ_{i}(tr ϕ)_{ii} + \frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i} − λ_{j})^{2}.  (14)

Denoting the second symmetric function of ϕ_{ij} by m, we have

m = ∑_{i≠j} λ_{i}λ_{j} = (tr ϕ)^{2} − |ϕ|^{2}.  (15)

From (14) and (15) we have

\frac{1}{2}Δ(tr ϕ)^{2} = \frac{1}{2}Δm + |∇ϕ|^{2} + ∑_{i} λ_{i}(tr ϕ)_{ii} + \frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i} − λ_{j})^{2}.  (16)

From Definition 2.1 of □, from (16) we have

□(tr ϕ) = \frac{1}{2}Δm + |∇ϕ|^{2} − |∇(tr ϕ)|^{2} + \frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i} − λ_{j})^{2}.  (17)

Since □ is self-adjoint and M^{n} is compact, we obtain by integration of (17)

∫_{M^{n}}[|∇ϕ|^{2} − |∇(tr ϕ)|^{2}] + ∫_{M^{n}} \frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i} − λ_{j})^{2} = 0.  (18)

So we have the following theorem.

Theorem 2.1 [11]. Let ϕ = ∑_{i,j} ω_{i} ⊗ ω_{j} be a Codazzi tensor field on a Riemanian manifold M^{n}. We assume the following condition

|∇ϕ|^{2} ≥ |∇(tr ϕ)|^{2}.  (19)

(1) If M^{n} has positive sectional curvature, then all the eigenvalues of ϕ_{ij} are the same on M^{n}.

(2) If M^{n} has nonnegative sectional curvature, then we have |∇ϕ|^{2} = |∇(tr ϕ)|^{2} and R_{ijij} = 0, when λ_{i} ≠ λ_{j} on M^{n}.

Proposition 2.1 [11]. If

tr ϕ = constant,  (20)

then (19) holds.

Proposition 2.2 [11]. If the second symmetric function of ϕ_{ij} is a nonnegative constant, i.e.

m = ∑_{i≠j} λ_{i}λ_{j} = (tr ϕ)^{2} − |ϕ|^{2} = constant ≥ 0,

then (19) holds.

M. Okumura [15] established the following lemma (see also [3]).

Lemma 2.1. Let μ_{i}, i = 1, …, n, be real numbers such that ∑_{i} μ_{i} = 0 and ∑_{i} μ_{i}^{2} = β^{2}, where β = constant ≥ 0. then

−\frac{n − 2}{\sqrt{n(n − 1)}}β^{3} ≤ ∑_{i} μ_{i}^{3} ≤ \frac{n − 2}{\sqrt{n(n − 1)}}β^{3},  (21)

and the equality holds in (21) if and only if at least (n − 1) of the μ_{i} are equal.

3 Space-like submanifolds in de Sitter space

Let S_{p}^{n+p}(c) be an (n + p)-dimensional de Sitter of constant curvature c whose index is p. Let M^{n} be an n-dimensional Riemannian manifold immersed in S_{p}^{n+p}(c). As the semi-Riemannian metric of S_{p}^{n+p}(c) induces the Riemannian metric of M^{n}, M^{n} is called a space-like submanifold. We choose a local field of semi-Riemannian orthonormal frames e_{1}, …, e_{n+p} in S_{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 field so that the semi-Riemannian metric of S_{p}^{n+p}(c) is given by ds̄^{2} = ∑_{i} ω_{i}^{2} −∑_{α} ω_{α}^{2} = ∑_{A} ϵ_{A}ω_{A}^{2}, where ϵ_{i} = 1 and ϵ_{α} = −1. Then the structure equations of S_{p}^{n+p}(c) are given by

dω_{A} = ∑_{B} ϵ_{B}ω_{AB} ∧ ω_{B}, ω_{AB} + ω_{BA} = 0,  (22)

dω_{AB} = ∑_{C} ϵ_{C}ω_{AC} ∧ ω_{CB} − \frac{1}{2} ∑_{C,D} K_{ABCD}ω_{C} ∧ ω_{D},  (23)
K_{ABCD} = c ϵ_{A}ϵ_{B}(δ_{AC}δ_{BD} − δ_{AD}δ_{BC}).  (24)

Restrict these form to M^{n}, we have

ω_{α} = 0, n + 1 ≤ α ≤ n + p,  (25)

the Riemannian metric of M^{n} is written as ds^{2} = ∑_{i} ω_{i}^{2}. From Cartan’s lemma we can write

ω_{αi} = ∑_{j} h_{ij}^{α}ω_{j}, h_{ij}^{α} = h_{ji}^{α}.  (26)

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

dω_{i} = ∑_{j} ω_{ij} ∧ ω_{j}, ω_{ij} + ω_{ji} = 0,  (27)

dω_{ij} = ∑_{k} ω_{ik} ∧ ω_{kj} − \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l},  (28)
R_{ijkl} = c(δ_{ik}δ_{jl} − δ_{il}δ_{jk}) − ∑_{α}(h_{ik}^{α}h_{jl}^{α} − h_{il}^{α}h_{jk}^{α}),  (29)

where R_{ijkl} are the components of the curvature tensor of M^{n} and

h = ∑_{α} h_{α}e_{α} = ∑_{i,j,α} h_{ij}^{α}ω_{i} ⊗ ω_{j} ⊗ e_{α}  (30)

is the second fundamental form of M^{n}.

For indefinite Riemannian manifolds in detail, refer to O’Neill [14].

Denote L_{α} = (h_{ij}^{α})_{n×n} and H_{α} = (1/n) ∑_{i} h_{ii}^{α} for α = n + 1, ⋯, n + p. Then the mean curvature vector field 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_{ij}^{α})^{2},  (31)

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_{km}^{α}h_{ml}^{β} − h_{lm}^{α}h_{mk}^{β}),
R_{ik} = (n − 1) c δ_{ik} + n ∑_{α}(H_{α})h_{ik}^{α} − ∑_{α,j} h_{ij}^{α}h_{jk}^{α},
R = c + \frac{1}{n(n − 1)}(S − n^{2}H^{2}).  (32)

If there exist p functions ρ_{α} such that h_{ij}^{α} = ρ_{α}δ_{ij} at each point of M^{n}, we call M^{n} a totally umbilical submanifold. For a totally umbilical submanifold, we have

ρ_{α} = \frac{1}{n}tr h_{α} = \frac{1}{n} ∑_{i} h_{ii}^{α}.  (33)

If R_{αβkl} = 0 at point x of M^{n} we say that the normal connection of M^{n} is flat at x and it is well known [4] that R_{αβkl} = 0 at x if and only if h_{α} are simultaneously diagonalizable at x.

Define the first and the second covariant derivatives of {h_{ij}^{α}}, say {h_{ijk}^{α}} and {h_{ijkl}^{α}} by

∑_{k} h_{ijk}^{α}ω_{k} = dh_{ij}^{α} + ∑_{k} h_{kj}^{α}ω_{ki} + ∑_{k} h_{ik}^{α}ω_{kj} + ∑_{β} h_{ij}^{β}ω_{βα},  (34)

∑_{l} h_{ijkl}^{α}ω_{l} = dh_{ijk}^{α} + ∑_{m} h_{mjk}^{α}ω_{mi} + ∑_{m} h_{imk}^{α}ω_{mj} + ∑_{m} h_{ijm}^{α}ω_{mk} + ∑_{β} h_{ijk}^{β}ω_{βα}. (35)

Then, by exterior differentiation of (26), we obtain the Codazzi equation

h_{ijk}^{α} = h_{ikj}^{α}.  (36)

It follows from Ricci’s identity that

h_{ijkl}^{α} − h_{ijlk}^{α} = ∑_{m} h_{mj}^{α}R_{mikl} + ∑_{m} h_{im}^{α}R_{mjkl} + ∑_{β} h_{ij}^{β}R_{βαkl}.  (37)

Suppose H > 0 on M^{n} and choose e_{n+1} = ξ/H. Then it follows that

H_{n+1} = H; H_{α} = 0,  α > n + 1.  (38)

We set

S_{α} = ∑_{i,j}(h_{ij}^{α})^{2}, S_{I} = ∑_{β>n+1} S_{β}.  (39)

Suppose that the normal bundle of M^{n} is flat, then we can choose e_{1}, …, e_{n} such that

h_{ij}^{α} = λ_{i}^{α}δ_{ij}, α = n + 1, ⋯, n + p.  (40)

Let ϕ_{ij} = h_{ij}^{α}. From (18) we have

∫_{M^{n}}[|∇h|^{2} − n^{2}|∇H|^{2}] + ∫_{M^{n}} \frac{1}{2} ∑_{i,j,α} R_{ijij}(λ_{i}^{α} − λ_{j}^{α})^{2} = 0.  (41)

By use of Gauss equation, we have

\frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i}^{n+1} − λ_{j}^{n+1})^{2} = ncS_{n+1} − n^{2}H^{2}c + S_{n+1}^{2} − nH ∑_{i}(λ_{i}^{n+1})^{3}.  (42)

Let μ_{i}^{n+1} = λ_{i}^{n+1} − H and |Z|^{2} = ∑_{i}(μ_{i}^{n+1})^{2}. We have

∑_{i} μ_{i}^{n+1} = 0, |Z|^{2} = S_{n+1} − nH^{2},  (43)

∑_{i}(λ_{i}^{n+1})^{3} = ∑_{i}(μ_{i}^{n+1})^{3} + 3H|Z|^{2} − nH^{3},  (44)

Putting (43) and (44) into (42), we get

\frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i}^{n+1} − λ_{j}^{n+1})^{2} = |Z|^{2}(nc − nH^{2} + |Z|^{2}) − nH ∑_{i}(μ_{i}^{n+1})^{3}.  (45)

By Using of Lemma 2.1, we have

\frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i}^{α} − λ_{j}^{α})^{2}

≥ (S_{n+1} − nH^{2})(nc − 2nH^{2} + S_{n+1} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H\sqrt{S_{n+1} − nH^{2}})  (46)

Putting (46) into (41), we hve

∫_{M^{n}}[(|∇h|^{2} − n^{2}|∇H|^{2})
+(S_{n+1} − nH^{2})(nc − 2nH^{2} + S_{n+1} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H\sqrt{S − nH^{2}})] ≤ 0. (47)

Note that

nc − 2nH^{2} + S_{n+1} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H\sqrt{S_{n+1} − nH^{2}}

= (\sqrt{S_{n+1} − nH^{2}} + \frac{1}{2}(n − 2)H\sqrt{\frac{n}{n − 1}})^{2} + n(c − \frac{n^{2}}{4(n − 1)}H^{2}).  (48)

From (47) and (48), we have the following result.

Theorem 3.1. Let M^{n} be an n-dimentional compact space-like submanifold in an (n + p)-dimensional de Sitter space S_{p}^{n+p}(c). Suppose that M^{n} has flat normal bundle, if

|∇h|^{2} ≥ n^{2}|∇H|^{2}  (49)

and

H^{2} < 4\frac{(n − 1)c}{n^{2}},  (50)

then S_{n+1} ≡ nH^{2} and M^{n} is a pseudo-umbilical submanifold.

Corollary 3.1 Let M^{n} be an n-dimentional compact space-like submanifold with constant scalar curvature R in an (n + p)-dimensional de Sitter space S_{p}^{n+p}(c). Suppose that M^{n} has flat normal bundle, if R − c ≤ 0 and

H^{2} < 4\frac{(n − 1)c}{n^{2}},  (51)

then S_{n+1} ≡ nH^{2} and M^{n} is a pseudo-umbilical submanifold.

Proof. From (32), 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_{ij}^{α} h_{ijk}^{α}, k = 1, ⋯, n.

For every k, it follows from Cauchy-Schwarz’s inequality that

n^{4}H^{2}H_{k}^{2} = (∑_{i,j,α} h_{ij}^{α} h_{ijk}^{α})^{2} ≤ S ∑_{i,j,α}(h_{ijk}^{α})^{2},  (52)

Taking sum on both sides of (52) with respect to k, we have

n^{4}H^{2}|∇H|^{2} = n^{4}H^{2} ∑_{k} H_{k}^{2} ≤ S ∑_{(i,j,k,α)}(h_{ijk}^{α})^{2} ≤ n^{2}H^{2} ∑_{(i,j,k,α)}(h_{ijk}^{α})^{2},  (53)

i.e. |∇h|^{2} ≥ n^{2}|∇H|^{2}, then Corollary 3.1 follows from Theorem 3.1.

Now consider the quadratic form Q(u, t) = u^{2} − \frac{n−2}{\sqrt{n−1}}ut − t^{2}. By the orthogonal transformation

\{ \begin{matrix}ū = \frac{1}{\sqrt{2n}}\{(1 + \sqrt{n − 1})u + (1 − \sqrt{n − 1})t\} \\ t̄= \frac{1}{\sqrt{2n}}\{(\sqrt{n − 1} − 1)u + (\sqrt{n − 1} + 1)t\} \end{matrix}

Q(u, t) turns into Q(u, t) = \frac{n}{2\sqrt{n−1}}(ū^{2} − t̄^{2}), where ū^{2} + t̄^{2} = u^{2} + t^{2}.

Take u = \sqrt{S̄_{n+1}}, t = \sqrt{n}H, then

nc − nH^{2} − n(n − 2)H\sqrt{\frac{S̄_{n+1}}{n(n − 1)}} + S̄_{n+1}

= nc + Q(u, t) = nc + \frac{n(ū^{2} − t̄^{2})}{2\sqrt{n − 1}}

= nc + \frac{n(−ū^{2} − t̄^{2})}{2\sqrt{n − 1}} + \frac{nū^{2}}{\sqrt{n − 1}}

≥ nc − \frac{nS̄_{n+1}}{2\sqrt{n − 1}}  (54)

From (54) and (47) we have

0 ≥ ∫_{M^{n}}(|∇h|^{2} − n^{2}|∇H|^{2})
+(S_{n+1} − nH^{2})\{nc − \frac{n(S_{n+1} − nH^{2}}{2\sqrt{n − 1}} \}  (55)

Therefore we have the following result

Theorem 3.2. Let M^{n} (n ≥ 3) be a compact space-like submanifold in the de Sitter space S_{p}^{n+p}(c). Suppose that |∇h|^{2} ≥ n^{2}|∇H|^{2}. If M^{n} has flat normal bundle and

S_{n+1} < nH^{2} + 2\sqrt{n − 1}c,  (56)

then S_{n+1} = nH^{2} and M^{n} is a pseudo-umbilical submanifold.

Theorem 3.3. Let M^{n} be an n-dimensional compact orientable space-like submanifold with constant scalar curvature and with nonnegative sectional curvature immersed in a de Sitter space S_{n}^{n+p}(c). Suppose that M^{n} has flat normal bundle, if the normalized scalar curvature R of M^{n} satisfies R < c, then M^{n} is totally umbilical.

Proof. We have from (32)

n^{2}H^{2} − ∥h∥^{2} = n(n − 1)(c − R),  (57)

where R is the normalized scalar curvature of M^{n}. Taking the covariant derivative of (57) and using the fact that R =const, we obtain

n^{2}HH_{k} = ∑_{i,j,α} h_{ij}^{α} · h_{ijk}^{α}

and hence, by Cauchy-Schwarz inequality, we have

∑_{k} n^{4}H^{2}(H_{k})^{2} = ∑_{k}(∑_{i,j,α} h_{ij}^{α} · h_{ijk}^{α})^{2} ≤ ∑_{i,j,α}(h_{ij}^{α})^{2} · ∑_{i,j,k,α}(h_{ijk}^{α})^{2},

that is

n^{4}H^{2}∥∇H∥^{2} ≤ ∥h∥^{2} · ∥∇h∥^{2}.  (58)

From (41) and (58), we have

0 ≥ ∫_{M^{n}}\{∥h∥^{−2}n^{3}(n − 1)(c − R)∥∇H∥^{2} + \frac{1}{2} ∑_{i,j,α} R_{ijij}(λ_{i}^{α} − λ_{j}^{α})^{2}\}.  (59)

Thus, by hypothesis, ∥∇H∥^{2} = 0, so H is constant on M^{n}. Therefore Theorem 3.3 follows immediately from a result of Aiyama [1] and this completes the proof of the Theorem 3.3.

4 Space-like hypersurfaces in de Sitter space

In this section, we will study the Space-like hypersurfaces in de Sitter space and prove some interesting rigidity results.

Let S_{1}^{n+1} be an (n + 1)-dimensional de Sitter of constant curvature 1 whose index is 1. Let M be an n-dimensional compact space-like hypersurface in S_{1}^{n+1}. We choose a local field of semi-Riemannian orthonormal frames e_{1}, …, e_{n+1} in S_{1}^{n+1} such that at each point of M, e_{1}, …, e_{n} span the tangent space of M and form an orthonormal frame there. We use the following convention on the range of indices:

1 ≤ A, B, C, … ≤ n + 1; 1 ≤ i, j, k, … ≤ n.

Let ω_{1}, …, ω_{n+1} be its dual frame field so that the semi-Riemannian metric of S_{1}^{n+1} is given by ds̄^{2} = ∑_{i} ω_{i}^{2}−ω_{n+1}^{2} = ∑_{A} ϵ_{A}ω_{A}^{2}, where ϵ_{i} = 1 and ϵ_{n+1} = −1. Then the structure equations of S_{1}^{n+1} are given by

dω_{A} = ∑_{B} ϵ_{B}ω_{AB} ∧ ω_{B}, ω_{AB} + ω_{BA} = 0,  (60)

dω_{AB} = ∑_{C} ϵ_{C}ω_{AC} ∧ ω_{CB} − \frac{1}{2} ∑_{C,D} K_{ABCD}ω_{C} ∧ ω_{D},  (61)
K_{ABCD} = ϵ_{A}ϵ_{B}(δ_{AC}δ_{BD} − δ_{AD}δ_{BC}).  (62)

Restrict these form to M^{n}, we have

ω_{n+1} = 0,  (63)

the Riemannian metric of M^{n} is written as ds^{2} = ∑_{i} ω_{i}^{2}. From Cartan’s lemma we can write

ω_{n+1i} = ∑_{j} h_{ij}ω_{j}, h_{ij} = h_{ji}.  (64)

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

dω_{i} = ∑_{j} ω_{ij} ∧ ω_{j}, ω_{ij} + ω_{ji} = 0,  (65)

dω_{ij} = ∑_{k} ω_{ik} ∧ ω_{kj} − \frac{1}{2} ∑_{k,l} R_{ijkl}ω_{k} ∧ ω_{l},  (66)
R_{ijkl} = (δ_{ik}δ_{jl} − δ_{il}δ_{jk}) − (h_{ik}h_{jl} − h_{il}h_{jk}),  (67)

where R_{ijkl} are the components of the curvature tensor of M and

h = ∑_{i,j} h_{ij}ω_{i} ⊗ ω_{j}  (68)

is the second fundamental form of M. We also have

R_{ij} = (n − 1)δ_{ij} − nHh_{ij} + ∑_{k} h_{ik}h_{kj},  (69)

n(n − 1)(R − 1) = −n^{2}H^{2} + |h|^{2},  (70)

where R is the normalized scalar curvature, and H the mean curvature.

Define the first and the second covariant derivatives of h_{ij}, say h_{ijk} and h_{ijkl} by

∑_{k} h_{ijk}ω_{k} = dh_{ij} + ∑_{k} h_{kj}ω_{ki} + ∑_{k} h_{ik}ω_{kj},  (71)

∑_{l} h_{ijkl}ω_{l} = dh_{ijk} + ∑_{m} h_{mjk}ω_{mi} + ∑_{m} h_{imk}ω_{mj} + ∑_{m} h_{ijm}ω_{mk}.  (72)

Then we have the Codazzi equation

h_{ijk} = h_{ikj},  (73)

and the Ricci’s identity

h_{ijkl} − h_{ijlk} = ∑_{m} h_{mj}R_{mikl} + ∑_{m} h_{im}R_{mjkl}.  (74)

We choose a local frame field e_{1}, ….e_{n} at any point p ∈ M, suth that h_{ij} = λ_{i}δ_{ij}. Now let ϕ_{ij} = h_{ij} = λ_{i}δ_{ij}, from (18) we have

∫_{M^{n}}[|∇h|^{2} − n^{2}|∇H|^{2} + \frac{1}{2} ∑_{i,j} R_{ijij}(λ_{i} − λ_{j})^{2}] = 0.  (75)

From (67), we have R_{ijij} = 1 − λ_{i}λ_{j}, i ≠ j, and by putting this into (75), we obtain

∫_{M^{n}}[|∇h|^{2} − n^{2}|∇H|^{2} + n|h|^{2} − n^{2}H^{2} + |h|^{4} − nH ∑_{i} λ_{i}^{3}] = 0.  (76)

Let μ_{i} = λ_{i} − H and |Z|^{2} = ∑_{i} μ_{i}^{2}, we have

∑_{i} μ_{i} = 0, |Z|^{2} = |h|^{2} − nH^{2},  (77)

∑_{i} λ_{i}^{3} = ∑_{i} μ_{i}^{3} + 3H|Z|^{2} − nH^{3},  (78)

From (76) -(78), we get

∫_{M^{n}}[|∇h|^{2} − n^{2}|∇H|^{2} + |Z|^{2}(n − nH^{2} + |Z|^{2}) − nH ∑_{i} μ_{i}^{3}] = 0.  (79)

By Using of Lemma 2.1, we have

∫_{M^{n}}[|∇h|^{2} − n^{2}|∇H|^{2} + (|h|^{2} − nH^{2})
×(n − 2nH^{2} + |h|^{2} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H\sqrt{|h|^{2} − nH^{2}})] = 0.  (80)

Now consider the quadratic form Q(u, t) = u^{2} − \frac{n−2}{\sqrt{n−1}}ut − t^{2}. By the orthogonal transformation

\{ \begin{matrix}ū = \frac{1}{\sqrt{2n}}\{(1 + \sqrt{n − 1})u + (1 − \sqrt{n − 1})t\} \\ t̄= \frac{1}{\sqrt{2n}}\{(\sqrt{n − 1} − 1)u + (\sqrt{n − 1} + 1)t\} \end{matrix}

Q(u, t) turns into Q(u, t) = \frac{n}{2\sqrt{n−1}}(ū^{2} − t̄^{2}), where ū^{2} + t̄^{2} = u^{2} + t^{2}. Take u = |Z|, t = \sqrt{n}H, then

n − nH^{2} − \frac{n(n − 2)}{\sqrt{n(n − 1)}}H|Z| + |Z|^{2}

= n + Q(u, t) = nc + \frac{n(ū^{2} − t̄^{2})}{2\sqrt{n − 1}}

= n + \frac{n(−ū^{2} − t̄^{2})}{2\sqrt{n − 1}} + \frac{nū^{2}}{\sqrt{n − 1}}

≥ n − \frac{n}{2\sqrt{n − 1}}|h|^{2}  (81)

From (81) and (80) we have

0 ≥ ∫_{M^{n}}\{(|∇h|^{2} − n^{2}|∇H|^{2}) + |Z|^{2}[n − \frac{n}{2\sqrt{n − 1}}|h|^{2}]\}.  (82)

Therefore we have the following result.

Theorem 4.1. Let M be an n-dimensional (n ≥ 3) compact space-like hypersurface in an (n + 1)-dimensional de Sitter space S_{1}^{n+1}. If

|∇h|^{2} ≥ n^{2}|∇H|^{2},  (83)

and

0 ≤ |h|^{2} ≤ 2\sqrt{n − 1},  (84)

then M is a totally umbilical hypersurface.

Proof. By the assumption of theorem, the right hand side is non-negative. Thus, either |Z|^{2} ≡ 0, that is , M is totally umbilical; or

|h|^{2} = 2\sqrt{n − 1}.  (85)

In the latter case, equality holds in Lemma 2.1, and it follows that M has two distinct constant principle curvatures. We conclude that M is totally umbilical from the compactness of M. This completes the proof of Theorem 4.1.

Corollary 4.1. Let M be an n-dimensional (n ≥ 2) compact space-like hypersurface with constant normalized scalar curcature R in an (n + 1)-dimensional de Sitter space S_{1}^{n+1}. If R ≤ 1 and

0 ≤ |h|^{2} ≤ 2\sqrt{n − 1},  (86)

then M is a totally umbilical hypersurface.

Proof. From (70),

−n^{2}H^{2} + ∑_{i,j} h_{ij}^{2} = n(n − 1)(R − 1).

Taking the covariant derivative of the above expression, and using the fact R = constant, we get

n^{2}HH_{k} = ∑_{i,j} h_{ij}h_{ijk}.

By Cauchy-Schwarz inequality, we have

∑_{k} n^{4}H^{2}(H_{k})^{2} = ∑_{k}(∑_{i,j} h_{ij}h_{ijk})^{2} ≤ (∑_{i,j} h_{ij}^{2}) ∑_{i,j,k} h_{ijk}^{2},  (87)

that is

n^{4}H^{2}∥∇H∥^{2} ≤ |h|^{2}|∇h|^{2}.  (88)

On the other hand, from R ≤ 1, we have n^{2}H^{2} − |h|^{2} ≥ 0. Thus

|∇h|^{2} ≥ n^{2}|∇H|^{2},

so from Theorem 4.1 the Corollary 4.1 follows.

Lemma 4.1. Let M be an n-dimensional compact space-like hypersurface in an (n + 1)-dimensional de Sitter space S_{1}^{n+1}. Suppose that the normalized scalar curvature R is proportional to the mean curvature H of M, that is

R = aH,  (89)

where a is any constant. Then |∇h|^{2} ≥ n^{2}|∇H|^{2}.

Proof. By use of Gauss equation −n^{2}H^{2} + |h|^{2} = n(n − 1)(R − 1) and the assumption (89), we have

|h|^{2} = n^{2}H^{2} − n(n − 1)(1 − aH).  (90)

Taking the covariant derivative of (90), we have for every k

2 ∑_{i,j} h_{ij}h_{ijk} = (2n^{2}H + n(n − 1)a)H_{k}.

It follows that

4|h|^{2}|∇h|^{2} ≥ 4 ∑_{k}(∑_{i,j} h_{ij}h_{ijk})^{2} = (2n^{2}H + n(n − 1)a)^{2}|∇H|^{2}.  (91)

By (89) and (90), we have

(2n^{2}H + n(n − 1)a)^{2} − 4n^{2}|h|^{2}

= (4n^{4}H^{2} + n^{2}(n − 1)^{2}a^{2} + 4n^{3}(n − 1)Ha) − n^{2}(4n^{2}H^{2} − 4n(n − 1)(1 − aH))

= n^{2}(n − 1)((n − 1)a^{2} + 4n) > 0.  (92)

Combining (91) with (92), we have |∇h|^{2} ≥ n^{2}|∇H|^{2}. This completes the proof of Lemma 4.1.

From Theorem 4.1 and Lemma 4.1, we have

Corollary 4.1. Let M be an n-dimensional (n ≥ 2) compact space-like hypersurface in an (n+1)-dimensional de Sitter space S_{1}^{n+1}. Suppose that the normalized scalar curvature R is proportional to the mean curvature H of M, that is, there exists a constant a such that

R = aH.  (93)

If

0 ≤ |h|^{2} ≤ 2\sqrt{n − 1},  (94)

then M is a totally umbilical hypersurface.

Acknowledgements

This work was carried out during the author’s visit to Max-Planck-Institut für Mathematik in Bonn. The author would like to exprss his thanks to Professor Yuri Manin for the invitation and the staffs of the MPIM for very warm hospitality.

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