COMPLETE MAXIMAL SPACELIKE SUBMANIFOLDS

Qing-Ming Cheng* & Susumu Ishikawa

Abstract. We generalize Simons’ method to spacelike submanifolds of M_{q}^{n+p}(c) (1 ≤ q ≤ p) and we characterize the totally geodesic submanifolds of S_{q}^{n+p}(c) (1 ≤ q ≤ p) under the pinching conditions on scalar curvature, Ricci curvature and sectional curvature, respectively.

1. Introduction

Let M_{q}^{n+p}(c) be an (n+p)-dimensional connected indefinite Riemannian manifold of index q (1 ≤ q ≤ p) and of constant curvature c, which is called an indefinite space form of index q. According to c > 0, c = 0 and c < 0, it is denoted by S_{q}^{n+p}(c), R_{q}^{n+p} or H_{q}^{n+p}(c). A submanifold M^{n} of an indefinite space form M_{q}^{n+p}(c) is said to be spacelike if the induced metric on M^{n} from that of the ambient space is positive definite. R^{n} can be embedded in S_{1}^{n+1}(c) as a complete totally umbilical hypersurface. But it can not be embedded in the unit sphere S^{m}(c) as a totally umbilical submanifold. Hence it is very interesting to investigate complete spacelike submanifolds in M_{q}^{n+p}(c).

When p = q, we know that complete maximal spacelike submanifolds in S_{p}^{n+p}(c) or R_{p}^{n+p} are totally geodesic (cf. [3]). Hence the class of all such submanifolds are very small. But if q < p we shall see that the class of complete maximal spacelike submanifolds is very large. In fact, if M^{n} is a complete minimal submanifold in sphere S^{m}(c) (m > n) of constant curvature c, which is embeded in S_{q}^{m+q}(c) as a totally geodesic spacelike submanifold such that m − n + q = p, then M^{n} is a complete maximal spacelike submanifold in S_{q}^{n+p}(c). In [1], Alias and Romero studied compact maximal spacelike submanifolds in S_{q}^{n+p}(c). They proved that if M^{n} is a compact maximal spacelike submanifold in S_{q}^{n+p}(c) with Ricci curvature Ric(M^{n}) ≥ (n − 1)c, then M^{n} is totally geodesic. And they indicated that to get a Bernstein type result, the bound on the Ricci curvature is best possible. But their statement can not guarantee this fact. In fact, according to the theory of minimal submanifolds in sphere, we know that there are no n-dimensional compact minimal submanifolds in S^{m}(c) such that the Ricci curvature satisfies (n − 2)c < Ric(M^{n}) < (n − 1)c. Hence the set of examples which they supposed is empty if (n − 2)c < Ric(M^{n}) < (n − 1)c.

Typeset by AMS-TEX

Key words and phrases. spacelike maximal submanifold, totally geodesic submanifold, scalar curature, Ricci curvature and sectional curvature.

*Research partially Supported by NNSFC, FECC and CPSF

1991 Mathematics Subject Classification 53C20, 53C42

The purpose of this paper is to generalize the Simons’ method to complete spacelike submanifolds in M_{q}^{n+p}(c) and we get the following theorems. In particular, we obtain the best possible bound on the Ricci curvature of a complete maximal spacelike submanifold in the de Sitter space S_{1}^{n+2}(c).

Theorem 1. Let M^{n} be an n-dimensional compact maximal spacelike submanifold in the de Sitter space S_{q}^{n+p}(c) (1 ≤ q ≤ p). If

S ≤ max\{\frac{nc}{2 − \frac{1}{p−q}} , \frac{2nc}{3}\},

then (1) M^{n} is totally geodesic, or

(2) p − q = 1, M^{n} lies in the totally godesic spacelike submanifold S^{n+1}(c) of S_{q}^{n+q+1}(c) and is isometric to the Clifford torus S^{k}(\frac{n}{k}c) × S^{n−k}(\frac{n}{n−k} c) or

(3) n = 2 and p − q = 2, M^{2} lies in the totally godesic spacelike submanifold

S^{4}(c) of S_{q}^{4+q}(c) and is isometric to the Veronese surface, where S is the squared norm of the second fundamental form of M^{n}.

Remark 1. When M^{n} is complete and

sup S < max\{\frac{nc}{2 − \frac{1}{p−q}} , \frac{2nc}{3}\},

we can prove that M^{n} is totally geodesic.

Theorem 2. Let M^{n} be an n-dimensional compact maximal spacelike submanifold in the de Sitter space S_{q}^{n+q+1}(c). If the sectional curvature K of M^{n} is positive, then M^{n} is totally geodesic.

Remark 2. The Clifford torus S^{n−k}(\frac{n}{n−k} c)×S^{k}(\frac{n}{k}c) in S^{n+1}(c) can be embedded in S_{q}^{n+q+1}(c) as a compact maximal spacelike submanifold with nonnegative curvature and it is not totally geodesic. Hence, the bound on the sectional curvature is best possible.

Theorem 3. Let M^{n} be an n-dimensional complete maximal spacelike submanifold in the de Sitter space S_{1}^{n+2}(c). If Ric(M^{n}) ≥ (n − 2)c, then M^{n} is totally geodesic or M^{n} is a maximal spacelike Einstein submanifold with Ric(M^{n}) = (n − 2)c and its second fundamental form is parallel.

Remark 3. Let n = 2k. The Clifford torus S^{k}(2c) × S^{k}(2c) of S^{n+1}(c) can be embedded in S_{1}^{n+2}(c) as a compact spacelike maximal submanifold with Ric(M^{n}) = (n − 2)c and its second fundamental form is parallel. It is open for authors whether there exist the other compact maximal spacelike submanifolds in S_{1}^{n+2}(c) with Ric(M^{n}) = (n − 2)c and its second fundamental form is parallel except the Clifford torus S^{k}(2c) × S^{k}(2c). On the other hand, in [1], Alias and Romero proved that if M^{n} is a compact maximal spacelike submanifolds in S_{q}^{n+p}(c) with Ricci curvature Ric(M^{n}) ≥ (n − 1)c, then M^{n} is totally geodesic. And they indicated that to get a Bernstein type result the bound (n − 1)c on the Ricci curvature is best possible. But their statement can not guarantee this fact because there are no n-dimensional compact minimal submanifolds in S^{m}(c) such that the Ricci curvature satisfies (n − 2)c < Ric(M^{n}) < (n − 1)c.

2. Preliminaries

Let M_{q}^{n+p}(c) be an (n + p)-dimensional connected indefinite space form of constant curvature c whose index is q (1 ≤ q ≤ p) and M^{n} an n-dimensional connected Riemannian manifold immersed in M_{q}^{n+p}(c). We choose a local field of orthonormal frames {e_{1}, ⋯, e_{n+p}} adapted to the indefinite Riemannian metric of M_{q}^{n+p}(c) and the dual coframes {ω_{1}, ⋯, ω_{n+p}} in such a way that, restricted to the submanifold M^{n}, {e_{1}, ⋯, e_{n}} are tangent to M^{n}. Then the connection forms {ω_{AB}} of M_{q}^{n+p}(c) are characterized by the structure equations

(2.1) \{ \begin{matrix}dω_{A} = − ∑_{B=1}^{n+p} ϵ_{B}ω_{AB} ∧ ω_{B}, ω_{AB} + ω_{BA} = 0 \\ dω_{AB} = − ∑_{C=1}^{n+p} ϵ_{C}ω_{AC} ∧ ω_{CB} − \frac{1}{2} ∑_{C,D=1}^{n+p} ϵ_{C}ϵ_{D}K_{ABCD}ω_{C} ∧ ω_{D}, \end{matrix}

K_{ABCD} = cϵ_{A}ϵ_{B}(δ_{AD}δ_{BC} − δ_{AC}δ_{BD}),

where ϵ_{A} = 1 for 1 ≤ A ≤ n + p − q, ϵ_{A} = −1 for n + p − q + 1 ≤ A ≤ n + p and K_{ABCD} denotes the components of indefinite Riemannian curvature tensor of M_{q}^{n+p}(c).

The canonic forms {ω_{A}} and connection forms {ω_{AB}} restricted to M^{n} are also denoted by the same symbols. We then have

(2.2)  ω_{α} = 0, α = n + 1, ⋯, n + p.

We see that e_{1}, ⋯, e_{n} is a local field of orthonormal frames adapted to the induced Riemannian metric on M^{n} and ω_{1}, ⋯, ω_{n} is a local field of its dual coframes on M^{n}. It follows from (2.1), (2.2) and Cartan’s Lemma that

(2.3)  ω_{αi} = ∑_{j=1}^{n} h_{ij}^{α}ω_{j}, h_{ij}^{α} = h_{ji}^{α}.

The second fundamental form II and the mean curvature vector h of M^{n} are defined by

(2.4)  II = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n} ϵ_{α}h_{ij}^{α}ω_{i}ω_{j}e_{α},

(2.5)  h = \frac{1}{n} ∑_{α=n+1}^{n+p} ϵ_{α}(∑_{i=1}^{n} h_{ii}^{α})e_{α}.

The mean curvature H of M^{n} is defined by

(2.7)  H = \frac{1}{n}\sqrt{∑_{α=n+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α})^{2}}.

If H = 0, we recall that M^{n} is maximal. Let S = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} denote the squared norm of the second fundamental form of M^{n}. The connection form of M^{n} are characterized by the structure equations

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

(2.9)  dω_{ij} = − ∑_{k=1}^{n} ω_{ik} ∧ ω_{kj} − \frac{1}{2} ∑_{k,l=1}^{n} R_{ijkl}ω_{k} ∧ ω_{l},

(2.10)

R_{ijkl} = c(δ_{il}δ_{jk} − δ_{ik}δ_{jl}) + ∑_{α=n+1}^{n+p−q}(h_{il}^{α}h_{jk}^{α} − h_{ik}^{α}h_{jl}^{α}) − ∑_{α=n+p−q+1}^{n+p}(h_{il}^{α}h_{jk}^{α} − h_{ik}^{α}h_{jl}^{α}).

where R_{ijkl} is the components of the curvature tensor of M^{n}. Letting R_{ij} and r denote the components of the Ricci curvature and the scalar curvature of M^{n}, respectively, we have from (2.10):

R_{jk} = (n − 1)cδ_{jk} + ∑_{α=n+1}^{n+p−q}(∑_{i=1}^{n} h_{ii}^{α}h_{jk}^{α} − ∑_{i=1}^{n} h_{ik}^{α}h_{ji}^{α}) (2.11)

− ∑_{α=n+p−q+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α}h_{jk}^{α} − ∑_{i=1}^{n} h_{ik}^{α}h_{ji}^{α}),

r = n(n − 1)c + ∑_{α=n+1}^{n+p−q}(∑_{i=1}^{n} h_{ii}^{α})^{2} − ∑_{α=n+1}^{n+p−q} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} (2.12)

− ∑_{α=n+p−q+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α})^{2} + ∑_{α=n+p−q+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}.

We also have

(2.13)  dω_{αβ} = − ∑_{γ=n+1}^{n+p} ϵ_{γ}ω_{αγ} ∧ ω_{γβ} − \frac{1}{2} ∑_{i,j=1}^{n} R_{αβij}ω_{i} ∧ ω_{j},

(2.14)  R_{αβij} = − ∑_{l=1}^{n}(h_{il}^{α}h_{lj}^{β} − h_{jl}^{α}h_{li}^{β}).

By taking the exterior differentiation of (2.3) and defining h_{ijk}^{α} by

(2.15)  ∑_{k=1}^{n} h_{ijk}^{α}ω_{k} = dh_{ij}^{α} − ∑_{k=1}^{n} h_{ik}^{α}ω_{kj} − ∑_{k=1}^{n} h_{jk}^{α}ω_{ki} − ∑_{β=n+1}^{n+p} ϵ_{β}h_{ij}^{β}ω_{βα},

we get the Codazzi equation by straightforward computations

(2.16)  h_{ijk}^{α} = h_{ikj}^{α} = h_{jik}^{α}.

We take the exterior differentiation of (2.15) and define h_{ijkl}^{α} by

(2.17) ∑_{l=1}^{n} h_{ijkl}^{α}ω_{l} = dh_{ijk}^{α}−∑_{l=1}^{n} h_{ljk}^{α}ω_{li}−∑_{l=1}^{n} h_{ilk}^{α}ω_{lj}−∑_{l=1}^{n} h_{ijl}^{α}ω_{lk}− ∑_{β=n+1}^{n+p} ϵ_{β}h_{ijk}^{β}ω_{βα}.

Then the Ricci formula for the second fundamental form is given by

(2.18) h_{ijkl}^{α} − h_{ijlk}^{α} = − ∑_{m=1}^{n} h_{mj}^{α}R_{mikl} − ∑_{m=1}^{n} h_{im}^{α}R_{mjkl} − ∑_{β=n+1}^{n+p} ϵ_{β}h_{ij}^{β}R_{βαkl}.

The Laplacian Δh_{ij}^{α} of h_{ij}^{α} is defined by

Δh_{ij}^{α} = ∑_{k=1}^{n} h_{ijkk}^{α}.

From the Codazzi equation (2.16) and the Ricci formula (2.18) we get, for the maximal submanifold M^{n} in M_{q}^{n+p}(c),

(2.19) Δh_{ij}^{α} = ∑_{k=1}^{n} h_{kijk}^{α} = ∑_{k=1}^{n} h_{kikj}^{α}

− ∑_{k,m=1}^{n} h_{km}^{α}R_{mijk} − ∑_{k,m=1}^{n} h_{mi}^{α}R_{mkjk} − ∑_{k=1}^{n} ∑_{β=n+1}^{n+p} ϵ_{β}h_{ki}^{β}R_{βαjk}

= − ∑_{k,m=1}^{n} h_{km}^{α}R_{mijk} − ∑_{k,m=1}^{n} h_{mi}^{α}R_{mkjk} − ∑_{k=1}^{n} ∑_{β=n+1}^{n+p} ϵ_{β}h_{ki}^{β}R_{βαjk}.

Thus we get, for the maximal submanifold M^{n} in M_{q}^{n+p}(c),

Lemma. Let S denote the squared norm of the second fundamental form, i.e., S = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}.

\frac{1}{2}ΔS = ∑_{α=n+1}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n} h_{ij}^{α}Δh_{ij}^{α}

= ∑_{α=n+1}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} − ∑_{α=n+1}^{n+p} ∑_{i,j,k,m=1}^{n} h_{ij}^{α}h_{km}^{α}R_{mijk}

− ∑_{α=n+1}^{n+p} ∑_{i,j,k,m=1}^{n} h_{ij}^{α}h_{mi}^{α}R_{mkjk} − ∑_{α,β=n+1}^{n+p} ∑_{i,j,k=1}^{n} ϵ_{β}h_{ij}^{α}h_{ki}^{β}R_{βαjk}.

3. Proofs of Theorems

We define S_{1} and S_{2} by

S_{1} := ∑_{α=n+1}^{n+p−q} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}, S_{2} := ∑_{α=n+p−q+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2},

respectively. Then

S = S_{1} + S_{2}.

Proof of Theorem 1. Since

∑_{α=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{kl}^{α}R_{lijk} = −c ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}

+ ∑_{α,β=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} ϵ_{β}h_{ij}^{α}h_{kl}^{α}(h_{lk}^{β}h_{ij}^{β} − h_{lj}^{β}h_{ik}^{β}),

∑_{α=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{li}^{α}R_{lkjk} = c ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} − nc ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}

+ ∑_{α,β=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} ϵ_{β}h_{ij}^{α}h_{li}^{α}(h_{lk}^{β}h_{kj}^{β} − h_{lj}^{β}h_{kk}^{β})

and

∑_{α,β=n+1}^{n+p} ∑_{i,j,k=1}^{n} ϵ_{β}h_{ij}^{α}h_{ki}^{β}R_{βαjk}

= − ∑_{α,β=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} ϵ_{β}h_{ij}^{α}h_{ki}^{β}(h_{lk}^{α}h_{lj}^{β} − h_{lj}^{α}h_{lk}^{β}),

we conclude from the Lemma in the section 2

\frac{1}{2}ΔS (3.1)

= ∑_{α=n+1}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + nc ∑_{α=n+1}^{n+p} ∑_{i,j,k,l=1}^{n}(h_{ij}^{α})^{2}

− ∑_{α,β=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} ϵ_{β}h_{ij}^{α}h_{kl}^{α}(h_{lk}^{β}h_{ij}^{β} − h_{lj}^{β}h_{ik}^{β})

− ∑_{α,β=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} ϵ_{β}h_{ij}^{α}h_{li}^{α}(h_{lk}^{β}h_{kj}^{β} − h_{lj}^{β}h_{kk}^{β})

+ ∑_{α,β=n+1}^{n+p} ∑_{i,j,k,l=1}^{n} ϵ_{β}h_{ij}^{α}h_{ki}^{β}(h_{lk}^{α}h_{lj}^{β} − h_{lj}^{α}h_{lk}^{β})

= ∑_{α=n+1}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + nc ∑_{α=n+1}^{n+p} ∑_{i,j,k,l=1}^{n}(h_{ij}^{α})^{2}

− ∑_{α,β=n+1}^{n+p} ϵ_{β}[trace(H_{α}H_{β})]^{2} − ∑_{α,β=n+1}^{n+p} ϵ_{β}N(H_{α}H_{β} − H_{β}H_{α}),

where N(H_{α}) = trace(H_{α}^{t}H_{α}) and H_{α} = (h_{ij}^{α}) is an n×n-matrix. For any α and β, we know N(H_{α}H_{β} −H_{β}H_{α}) ≥ 0. Put S_{αβ} = ∑_{i,j=1}^{n} h_{ij}^{α}h_{ij}^{β}. Then the (p×p)-matrix (S_{αβ}) is symmetric. We can choose {e_{n+1}, ⋯, e_{n+p}} such that (S_{αβ}) is diagonal. Next we divide the proof of Theorem 1 into two cases.

Case 1: p − q = 1. From (3.1), we have

\frac{1}{2}ΔS = ∑_{α=n+1}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + nc ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}

+ ∑_{α,β=n+2}^{n+p} N(H_{α}H_{β} − H_{β}H_{α}) + ∑_{α,β=n+2}^{n+p}[trace(H_{α}H_{β})]^{2} − N(H_{n+1})^{2}

≥ nc ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} − [N(H_{n+1})]^{2} + ∑_{α,=n+2}^{n+p}[N(H_{α})]^{2}

≥ ncS − S^{2}.

From the assumptions of Theorem 1 and the Stokes formula, we get S = 0 or S = nc. If S = 0, then M^{n} is totally geodesic. If S = nc, from the above proof, we know S_{2} = 0 on M^{n}, i.e., h_{ij}^{α} = 0 on M^{n} for α = n + 2, ⋯, n + p. Hence M^{n} lies in the totally geodesic spacelike submanifold S^{n+1}(c) of S_{q}^{n+q+1}(c). Thus M^{n} become a compact minimal hypersurface in S^{n+1}(c) such that the squared norm S of the second fundamental form is equal to nc. From the result due to Chern-do Carmo and Kobayashi [2], we know that M^{n} is isometric to the Clifford torus. We complete the proof of Theorem 1 in this case.

Case 2: p − q > 1. In this case, we have

\frac{1}{2}ΔS = ∑_{α=n+1}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + nc ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}

+ ∑_{α,β=n+p−q+1}^{n+p} N(H_{α}H_{β} − H_{β}H_{α}) + ∑_{α,β=n+p−q+1}^{n+p}[trace(H_{α}H_{β})]^{2}

− ∑_{α,β=n+1}^{n+p−q} N(H_{α}H_{β} − H_{β}H_{α}) − ∑_{α,β=n+1}^{n+p−q}[trace(H_{α}H_{β})]^{2}

From a Lemma due to Li-Li in [4], we get

− ∑_{α,β=n+1}^{n+p−q} N(H_{α}H_{β} − H_{β}H_{α}) − ∑_{α,β=n+1}^{n+p−q}[trace(H_{α}H_{β})]^{2} ≥ −\frac{3}{2}[ ∑_{α=n+1}^{n+p−q} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}]^{2}

Hence, we get

(3.2)  \frac{1}{2}ΔS ≥ (ncS − \frac{3}{2}S^{2}) + ∑_{α=n+p−q+1}^{n+p}[N(H_{α})]^{2}.

From the Stokes formula, the assumptions in Theorem 1 and (3.2), we get S = \frac{2}{3}nc or S = 0. If S = 0, then M^{n} is totally geodesic. If S = \frac{2}{3}nc, we know h_{ij}^{α} = 0 on M^{n} for α = n + p − q, ⋯, n + p. Hence M^{n} lies in the totally geodesic spacelike submanifold S^{n+p−q}(c) of S_{q}^{n+p}(c). Thus M^{n} becomes a compact minimal submanifold in S^{n+p−q}(c) such that the squared norm S of the second fundamental form is equal to \frac{2}{3}nc. From the result due to Li-Li [4], we know that n = p − q = 2 and M^{n} is isometric to Veronese surface. Theorem 1 holds in this case. We complete the proof of Theorem 1.

Proposition. Let M^{n} be an n-dimensional compact spacelike submanifold in the de Sitter space S_{q}^{n+q+1}(c). If the sectional curvature K of M^{n} is nonnegative, then M^{n} is totally geodesic or M^{n} is a compact maximal spacelike submanifold with parallel second fundamental form.

Proof of Proposition. For any fixed α, we can choose e_{1}, ⋯, e_{n} such that h_{ij}^{α} = λ_{i}^{α}δ_{ij}. Then

− ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{kl}^{α}R_{lijk} − ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{li}^{α}R_{lkjk}

= [− ∑_{i,k=1}^{n} λ_{i}^{α}λ_{k}^{α}R_{kiik} − ∑_{i,k=1}^{n}(λ_{i}^{α})^{2}R_{ikik}]

= \frac{1}{2} ∑_{i,k=1}^{n}(λ_{i}^{α} − λ_{k}^{α})^{2}R_{kiik}

≥ \frac{1}{2} ∑_{i,k=1}^{n}(λ_{i}^{α} − λ_{k}^{α})^{2}K_{0} = nK_{0} ∑_{i,k=1}^{n}(h_{ik}^{α})^{2},

where K_{0} denotes the infimum of the sectional curvature of M^{n}. Since the both sides of the above inequality do not depend on the choice of the orthonormal frames {e_{1}, ⋯, e_{n}}, we have

− ∑_{α=n+1}^{n+q+1} ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{kl}^{α}R_{lijk} − ∑_{α=n+1}^{n+q+1} ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{li}^{α}R_{lkjk} (3.3)

≥ ∑_{α=n+1}^{n+q+1} nK_{0} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2}.

From Lemma and (3.3), we get

\frac{1}{2}ΔS = ∑_{α=n+1}^{n+q+1} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} − ∑_{α=n+1}^{n+q+1} ∑_{i,j,k,m=1}^{n} h_{ij}^{α}h_{km}^{α}R_{mijk}

− ∑_{α=n+1}^{n+q+1} ∑_{i,j,k,m=1}^{n} h_{ij}^{α}h_{mi}^{α}R_{mkjk} − ∑_{α,β=n+1}^{n+q+1} ∑_{i,j,k=1}^{n} ϵ_{β}h_{ij}^{α}h_{ki}^{β}R_{βαjk}

= ∑_{α=n+1}^{n+q+1} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} − ∑_{α=n+1}^{n+q+1} ∑_{i,j,k,m=1}^{n} h_{ij}^{α}h_{km}^{α}R_{mijk}

− ∑_{α=n+1}^{n+q+1} ∑_{i,j,k,m=1}^{n} h_{ij}^{α}h_{mi}^{α}R_{mkjk} − \frac{1}{2} ∑_{α,β=n+1}^{n+q+1} ϵ_{β}N(H_{α}H_{β} − H_{β}H_{α})

≥ nK_{0}S + \frac{1}{2} ∑_{α,β=n+2}^{n+q+1} N(H_{α}H_{β} − H_{β}H_{α})

≥ nK_{0}S.

Since the sectional curvature of M^{n} is nonnegative, we have K_{0} ≥ 0. From the Stokes formulas and the above inequality, we obtain S = 0, i.e., M^{n} is totally geodesic or S is constant and h_{ijk}^{α} = 0. We complete the proof of Proposition.

Proof of Theorem 2. From Proposition and its proof, it is obvious that Theorem 2 holds.

Proof of Theorem 3. From the assumptions of Theorem 3 and the Myers Theorem, we know that M^{n} is compact. According to (3.1), we get

\frac{1}{2}ΔS = ∑_{α=n+1}^{n+2} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + nc ∑_{α=n+1}^{n+2} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} (3.4)

− ∑_{α,β=n+1}^{n+2} ϵ_{β}N(H_{α}H_{β} − H_{β}H_{α}) − ∑_{α,β=n+1}^{n+2} ϵ_{β}[trace(H_{α}H_{β})]^{2}

≥ nc ∑_{α=n+1}^{n+2} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} − [N(H_{n+1})]^{2} + [N(H_{n+2})]^{2}

= (nc − S_{1} + S_{2})S,

where S_{1} = N(H_{n+1}), S_{2} = N(H_{n+2}) and S = S_{1} + S_{2}. From (2.11) and the assumptions of Theorem 3, we have

c − ∑_{i=1}^{n}(h_{ij}^{n+1})^{2} + ∑_{i=1}^{n}(h_{ij}^{n+2})^{2} ≥ 0.

Thus

(3.5)  nc − S_{1} + S_{2} ≥ 0.

From (3.4) and (3.5), we conclude

∑_{α=n+1}^{n+2} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} = 0

and S = 0 or nc − S_{1} + S_{2} = 0 and S is constant. If S = 0, then M^{n} is totally geodesic. If S ≠ 0, then all of the above inequalities become equalities. Hence, the Ricci curvature is equal to (n − 2)c. We complete the proof of Theorem 3.

References

1. L. J. Alias and A. Romero, Integral formulas for compact spacelike n-submanifolds in de Sitter spaces Applications to the papallel mean curvature vector case, Manuscripta Math. 87 (1995), 405-416.

2. S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length,, Functional analysis and related fields (1970), pp. 59-75.

3. T. Ishihara, Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature,, Michigan Math. J. 35 (1988), 345-352.

4. A. M. Li and J. M. Li, An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. 58 (1992), 582-594.

5. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62-105.

Department of Mathematics, Faculty of science, Josai University, Sakado, Saitama 350-02, Japan

Department of Mathematics, Saga University, Saga 840, Japan