Pseudo-Umbilical Spacelike Submanifolds in the Indefinite Space Form

Xi-Fang Cao

Abstract

We prove an integral inequality for compact pseudo-umbilical spacelike submanifolds in the indefinite space form. As an application of the inequality, we give a necessary and sufficient condition for such submanifolds to be totally geodesic.

Mathematics Subject Classification: 53C40, 53C42, 53C50 Key words: pseudo-umbilical spacelike submanifold, indefinite space form

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, which is called as indefinite space form of index p. Let M^{n} be an n-dimensional Riemannian 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 spacelike submanifold. Let h be the second fundamental form of the immersion, and ξ the mean curvature vector. Denote by ⟨·, ·⟩ the scalar product of M_{p}^{n+p}(c). If there exists a function λ on M^{n} such that

(1.1)  ⟨h(X, Y), ξ⟩ = −λ⟨X, Y ⟩

for any tangent vector X, Y on M^{n}, then M^{n} is called a pseudo-umbilical spacelike submanifold of M_{p}^{n+p}(c). It is clear that λ ≥ 0. If the mean curvature vector ξ vanishes identically, then M^{n} is called a maximal spacelike submanifold of M_{p}^{n+p}(c). Every maximal spacelike submanifold of M_{p}^{n+p}(c) is itself a pseudo-umbilical spacelike submanifold of M_{p}^{n+p}(c).

Maximal and pseudo-umbilical spacelike submanifolds have been studied by many researches. For example in 1988 Ishihara [1] proved that if M^{n} is a complete and maximal spacelike submanifold of M_{p}^{n+p}(c), then either M^{n} is totally geodesic (when c ≥ 0) or 0 ≤ S ≤ −npc(when c < 0), where S is the square length of the second fundamental form of M^{n}. In 1995 Sun [2] first proved that the mean curvature H of the pseudo-umbilical submanifolds M^{n} in M_{p}^{n+p}(c) is constant, then he generalized Ishihara’s result to pseudo-umbilical submanifolds, obtaining the inequality

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 117-121 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

nH^{2} ≤ S ≤ \frac{1}{2}np[H^{2} − c − \sqrt{(H^{2} − c)^{2}} + 4H^{2}c/p].

In this paper, we prove an integral inequality for compact pseudo-umbilical spacelike submanifolds in the indefinite space form and as an application of the inequality, we give a necessary and sufficient condition for such submanifolds to be totally geodesic. We will prove the following Theorem 1. Let M^{n} be an n-dimensional compact pseudo-umbilical spacelike submanifold in M_{p}^{n+p}(c), then

(1.2)  ∫_{M^{n}} \{\frac{1}{2} ∑ R_{mijk}^{2} + ∑ R_{mj}^{2} − ncR + nH^{2}R\} ∗ 1 ≤ 0.

Theorem 2. Let M^{n} be an n-dimensional compact pseudo-umbilical spacelike submanifold in M_{p}^{n+p}(c), then (1.3)

∫_{M^{n}} \{\frac{1}{2} ∑ R_{mijk}^{2} + (n − 2)cS − n(n − 1)c^{2} + n^{2}(n − 1)cH^{2} + nH^{2}S\} ∗ 1 ≤ 0,

and equality holds if and only if M^{n} is totally geodesic.

In the above Theorem, ∑ R_{mijk}^{2} is the square length of the Riemannian curvature tensor of M^{n}, ∑ R_{mj}^{2} the square length of the Ricci curvature tensor and R the scalar curvature. We shall make use of the following convention on the ranges of indices:

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

and we shall agree that repeated indices are summed over the respective ranges.

2 Local formulas

We choose a local field of semi-Riemannian orthonormal frames e_{1}, ⋯, e_{n+p} in M_{p}^{n+p}(c) such that, restricted to M^{n}, e_{1}, ⋯, e_{n} are tangent to M^{n}. Let ω_{1}, ⋯, ω_{n+p} be its dual frame field such that the semi-Riemannian metric of M_{p}^{n+p}(c) is given by

ds^{2} = ∑(ω_{i})^{2} − ∑(ω_{α})^{2} = ∑ 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

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

(2.2)  dω_{AB} = − ∑ ε_{C}ω_{AC} ∧ ω_{CB} − \frac{1}{2} ∑ K_{ABCD}ω_{C} ∧ ω_{D},

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

We restrict these forms to M^{n}, then

(2.4)  ω_{α} = 0, ω_{αi} = ∑ h_{ij}^{α}ω_{j}, h_{ij}^{α} = h_{ji}^{α}

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

(2.6)  R_{ijkl} = c(δ_{il}δ_{jk} − δ_{ik}δ_{jl}) − ∑(h_{il}^{α}h_{jk}^{α} − h_{ik}^{α}h_{jl}^{α}),
(2.7)  R_{jk} = ∑ R_{ljkl} = c(n − 1)δ_{jk} + ∑ h_{lk}^{α}h_{jl}^{α},
(2.8)  R = ∑ R_{jj} = n(n − 1)c + S,

(2.9)  dω_{αβ} = − ∑ ω_{αγ} ∧ ω_{γα} − \frac{1}{2} ∑ R_{αβij}ω_{i} ∧ ω_{j},

(2.10)  R_{αβij} = − ∑(h_{il}^{α}h_{jl}^{β} − h_{jl}^{α}h_{il}^{β}).

For indefinite Riemannian manifolds, refer to O’Neill [3].

We call h = ∑ h_{ij}^{α}ω_{i}ω_{j}e_{α} the second fundamental form of the immersed manifold

M^{n}. Denote by S = ∑(h_{ij}^{α})^{2} the square length of h , ξ = \frac{1}{n} ∑ trH_{α}e_{α} the mean curvature vector and

H = \frac{1}{n}\sqrt{∑(trH_{α})^{2}}

the mean curvature of M^{n} respectively. Here tr is the trace of the matrix H_{α} = (h_{ij}^{α}). Now let e_{n+1} be parallel to ξ. Then we have

(2.11)  trH_{n+1} = nH, trH_{α} = 0, α ≠ n + 1.

Let h_{ijk}^{α} and h_{ijkl}^{α} denote the covariant derivative and the second covariant derivative of h_{ij}^{α} respectively, defined by

(2.12)  ∑ h_{ijk}^{α}ω_{k} = dh_{ij}^{α} − ∑ h_{ik}^{α}ω_{kj} − ∑ h_{jk}^{α}ω_{ki} − ∑ h_{ij}^{β}ω_{βα},
(2.13) ∑ h_{ijkl}^{α}ω_{l} = dh_{ijk}^{α} − ∑ h_{ijl}^{α}ω_{lk} − ∑ h_{ilk}^{α}ω_{lj} − ∑ h_{ljk}^{α}ω_{li} − ∑ h_{ijk}^{β}ω_{βα},

then we have

(2.14)  h_{ijk}^{α} − h_{ikj}^{α} = 0,
(2.15)  h_{ijkl}^{α} − h_{ijlk}^{α} = − ∑ h_{im}^{α}R_{mjkl} − ∑ h_{jm}^{α}R_{mikl} − ∑ h_{ij}^{β}R_{αβkl}.

The Laplacian △h_{ij}^{α} of h_{ij}^{α} is defined by △h_{ij}^{α} = ∑ h_{ijkk}^{α}. By a direct calculation we have (cf. [4])

(2.16)  \begin{aligned}∑ h_{ij}^{α} △ h_{ij}^{α} & = & ∑ h_{ij}^{α}h_{kkij}^{α} − ∑ h_{ij}^{α}h_{km}^{α}R_{mijk}− \\ & − & ∑ h_{ij}^{α}h_{mi}^{α}R_{mkjk} − ∑ h_{ij}^{α}h_{ki}^{β}R_{αβjk}. \end{aligned}

120 Xi-Fang Cao

3 Proof of Theorem

Proof of Theorem 1. From (1.1) and (2.11)

(3.1)  ⟨h(e_{i}, e_{j}), He_{n+1}⟩ = −H^{2}δ_{ij} i.e. h_{ij}^{n+1} = −Hδ_{ij},

therefore

(3.2)  ∑ h_{ij}^{α}h_{kkij}^{α} = −nH △ H.

Since H is constant (see [2]),

(3.3)  ∑ h_{ij}^{α}h_{kkij}^{α} = 0.

On the other hand, from (2.6)

(3.4)  \begin{matrix}− ∑ h_{ij}^{α}h_{km}^{α}R_{mijk} = −\frac{1}{2} ∑(h_{ij}^{α}h_{km}^{α} − h_{mj}^{α}h_{ik}^{α})R_{mijk} \\ = \frac{1}{2} ∑\{R_{imkj} − c(δ_{ij}δ_{mk} − δ_{mj}δ_{ik})\}R_{mijk} = \frac{1}{2} ∑ R_{mijk}^{2} − cR, \end{matrix}

(3.5)  \begin{matrix}− ∑ h_{ij}^{α}h_{mi}^{α}R_{mkjk} = ∑(h_{ij}^{α}h_{mi}^{α} − h_{ii}^{α}h_{mj}^{α} + nH^{2}δ_{mj})R_{mj} = \\ = ∑\{c(δ_{ij}δ_{mi} − δ_{ii}δ_{mj}) − R_{imij} + nH^{2}δ_{mj}\}R_{mj} = \\ = ∑ R_{mj}^{2} − (n − 1)cR + nH^{2}R. \end{matrix}

From (2.10)

(3.6)  − ∑ h_{ij}^{α}h_{ki}^{β}R_{αβjk} = ∑ h_{ij}^{α}h_{ki}^{β}h_{jl}^{α}h_{kl}^{β} − ∑ h_{ij}^{α}h_{ki}^{β}h_{kl}^{α}h_{jl}^{β}.

Since

(3.7)  \begin{aligned}∑ h_{ij}^{α}h_{jl}^{β}h_{lk}^{α}h_{ki}^{β} & = & ∑ tr(H^{α}H^{β})^{2} ≤ \\ & ≤ & ∑ tr((H^{α})^{2}(H^{β})^{2}) = ∑ h_{ij}^{α}h_{jl}^{α}h_{lk}^{β}h_{ki}^{β}, \end{aligned}

therefore

(3.8)  − ∑ h_{ij}^{α}h_{ki}^{β}R_{αβjk} ≥ 0.

From (2.14), (3.3-3.8)

(3.9)  ∑ h_{ij}^{α} △ h_{ij}^{α} ≥ \frac{1}{2} ∑ R_{mijk}^{2} + ∑ R_{mj}^{2} − ncR + nH^{2}R.

Since ∫_{M^{n}} {∑ h_{ij}^{α} △ h_{ij}^{α}} ∗ 1 ≤ 0 (see [4]), we have

(3.10)  ∫_{M^{n}} \{\frac{1}{2} ∑ R_{mijk}^{2} + ∑ R_{mj}^{2} − ncR + nH^{2}R\} ∗ 1 ≤ 0.

Theorem 1 is proved. Proof of Theorem 2. From (2.7)

(3.11)  ∑ R_{mj}^{2} = n(n − 1)^{2}c^{2} + 2(n − 1)cS + ∑_{m,j} (∑_{i,α} h_{im}^{α}h_{ij}^{α})^{2} ,

(3.12)  ∑_{m,j} (∑_{i,α} h_{im}^{α}h_{ij}^{α})^{2} ≥ ∑_{j} (∑_{i,α}(h_{ij}^{α})^{2})^{2} ≥ \frac{1}{n} (∑(h_{ij}^{α})^{2})^{2} = \frac{1}{n}S^{2},

therefore from (3.10), (3.13)

∫_{M^{n}} \{\frac{1}{2} ∑ R_{mijk}^{2} + (n − 2)cS − n(n − 1)c^{2} + n^{2}(n − 1)cH^{2} + nH^{2}S\} ∗ 1 ≤ 0.

If M^{n} is totally geodesic, i.e., S = 0, h_{ij}^{α} = 0, then from (2.6)

(3.14)  \frac{1}{2} ∑ R_{mijk}^{2} = n(n − 1)c^{2},

in this case, (3.13) becomes an equality; Inversely, if (3.13) becomes an equality, then from (3.12) S = 0, i.e. M^{n} is totally geodesic. Theorem 2 is proved.

References

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

[2] H. Sun, On spacelike submanifolds of a pseudo-Riemannian space form, Note Mat., 15(1995), 215-224.

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

[4] S.S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Springer, 1970, 59-75.

Department of Mathematics

Concordia University

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Department of Mathematics, Yangzhou University, Yangzhou 225002, People’s Republic of China