Integral inequalities for maximal space-like submanifolds in the indefinite space form

Liu Ximin

Abstract In this note, we give two intrinsic integral inequalities for compact maximal space-like submanifolds in the indefinite space form and a sufficent and necessary condition for such submanifolds to be totally geodesic.

Mathematics Subject Classification: 53C40, 53C42, 53C50. Key words: maximal space-like submanifold, indefinite space form, flat normal bundle.

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. 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 space-like submanifold. A space-like submanifold with vanishing mean curvature is called a maximal space-like submanifold. Kobayashi [5] gave the Weierstrass-Enneper representation formulas for maximal space-like surfaces in 3-dimensional Minkowski space and exhibited various examples. In particular, he determined the maximal spacelike surfaces which are rotation surfaces or ruled surfaces. Montiel [6] give an integral inequality for compact space-like hypersurfaces in the de Sitter space and by use of this integral inequality, he studied the constant mean curvature space-like hypersurfaces. Also, Akutagawa [1] and Ramanathan [8] 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 [3] generalized this result to general submanifolds in a de Sitter space.

In this paper, we study compact maximal space-like submanifolds in the indefinite space form with flat normal bundle and obtain two intrinsic integral inequalities for such submanifolds. We also give a sufficent and necessary condition for such submanifolds to be totally geodesic. We will prove the following Theorem 1. Let M^{n} be an n-dimensional compact maximal space-like submanifold in M_{p}^{n+p}(c) with flat normal bundle, then

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 109-114 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

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

Theorem 2. Let M^{n} be an n-dimensional compact maximal space-like submanifold in M_{p}^{n+p}(c) with flat normal bundle, then

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

Theorem 3. Let M^{n} be an n-dimensional compact maximal space-like submanifold in M_{p}^{n+p}(c) with flat normal bundle, then M^{n} is totally geodesic if and only if

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

In the above theorems, ∑ R_{mijk}^{2} is the square length of the Riemannian curvature tensor, ∑ R_{mj}^{2} the square length of the Ricci curvature tensor, S the square length of the second fundamental form, R the scalar curvature. All these are intrinsic properties of M^{n}.

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 Riemannian manifold immersed in M_{p}^{n+p}(c). We choose a local field 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 field so that the semi-Riemannian metric of

M_{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 M_{p}^{n+p}(c) are given by

dω_{A} = ∑_{B} ϵ_{B}ω_{AB} ∧ ω_{B}, ω_{AB} + ω_{BA} = 0, (1)

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

K_{ABCD} = c ϵ_{A}ϵ_{B}(δ_{AC}δ_{BD} − δ_{AD}δ_{BC}). (3)

By restricting these forms to M^{n}, we have ω_{α} = 0, n + 1 ≤ α ≤ n + p, (4)

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}. (5)

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

dω_{i} = ∑_{j} ω_{ij} ∧ ω_{j}, ω_{ij} + ω_{ji} = 0, (6)

dω_{ij} = ∑_{k} ω_{ik} ∧ ω_{kj} − \frac{1}{2} ∑_{k,l} K_{ijkl}ω_{k} ∧ ω_{l}, (7)

R_{ijkl} = c(δ_{ik}δ_{jl} − δ_{il}δ_{jk}) − ∑_{α}(h_{ik}^{α}h_{jl}^{α} − h_{il}^{α}h_{jk}^{α}), (8)

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

For indefinite Riemannian manifolds in detail, refer to O’Neill [7].

We call

h = ∑_{α} h_{α}e_{α} = ∑_{i,j,α} h_{ij}^{α}ω_{i} ⊗ ω_{j} ⊗ e_{α} (9)

the second fundamental form of M^{n} and the square length of the second fundamental form is defined by

S = ∑_{α} tr(h_{α})^{2} = ∑_{α,i,j}(h_{ij}^{α})^{2}. (10)

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

N = \frac{1}{n} ∑_{α} tr(h_{α})e_{α} = \frac{1}{n} ∑_{α}(∑_{i} h_{ii}^{α})e_{α}, (11)

and it is well known that N is independent of the choice of unit normal vectors e_{n+1}, …, e_{n+p} to M^{n}. The length of the mean curvature vector is called the mean curvature of M^{n}, denoted by H.

If M^{n} is maximal, then

∑_{i} h_{ii}^{α} = 0, α = n + 1, …, n + p. (12)

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}, (13)

\begin{aligned}∑_{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}. \end{aligned} (14)

Then we have h_{ijk}^{α} = h_{ikj}^{α}, (15)

h_{ijkl}^{α} − h_{ijlk}^{α} = ∑_{m} h_{im}^{α}R_{mjkl} + ∑_{m} h_{jm}^{α}R_{mikl} + ∑_{β} h_{ij}^{β}R_{αβkl}, (16)

where R_{αβkl} are the components of the normal curvature tensor of M^{n}, that is

R_{αβkl} = ∑_{i}(h_{ik}^{α}h_{il}^{β} − h_{il}^{α}h_{ik}^{β}). (17)

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 [2] that R_{αβkl} = 0 at x if and only if h_{α} are simultaneously diagonalizable at x.

The Laplacian Δh_{ij}^{α} of the fundamental form h_{ij}^{α} is defined to be ∑_{k} h_{ijkk}^{α}, and hence, if M^{n} has flat normal bundle, from (15) and (16) we have

Δh_{ij}^{α}  = ∑_{k}(h_{ijkk}^{α} − h_{ikjk}^{α}) + ∑_{k}(h_{ikjk}^{α} − h_{ikkj}^{α}) + ∑_{k}(h_{ikkj}^{α} − h_{kkij}^{α})

= ∑_{m,k} h_{im}^{α}R_{mkjk} + ∑_{m,k} h_{mk}^{α}R_{mijk} (18)

3 Proofs of the Theorems

Proof of Theorem 1. From (8), (12) and (18), we have

∑ h_{ij}^{α}Δh_{ij}^{α} = ∑ h_{ij}^{α}h_{mk}^{α}R_{mijk} + ∑ h_{ij}^{α}h_{im}^{α}R_{mkjk}
= \frac{1}{2} ∑(h_{ij}^{α}h_{mk}^{α} − h_{mj}^{α}h_{ik}^{α})R_{mijk} + ∑(h_{ij}^{α}h_{im}^{α} − h_{ii}^{α}h_{jm}^{α})R_{mj}
(19)  = \frac{1}{2} ∑[c(δ_{ij}δ_{mk} − δ_{mj}δ_{ik}) − R_{imjk}]R_{mijk}
+ ∑[c(δ_{ij}δ_{im} − δ_{ii}δ_{jm}) + R_{ijim}]R_{mj}
= \frac{1}{2} ∑ R_{mijk}^{2} + ∑ R_{mj}^{2} − ncR.

Since ∫_{M^{n}}{∑ h_{ij}^{α}Δh_{ij}^{α}} ∗ 1 ≤ 0, we have

∫_{M^{n}} \{\frac{1}{2} ∑ R_{mijk}^{2} + ∑ R_{mj}^{2} − ncR\} ∗ 1 ≤ 0. (20)

Theorem 1 is proved.

In order to prove Theorem 2, we need the following algebraic lemma Lemma. Let a_{1}, …, a_{n} be real numbers, then

∑(a_{i})^{2} ≥ \frac{1}{n}(∑ a_{i})^{2}, (21)

and the equality holds if and only if a_{1} = ⋯ = a_{n}.

In fact, n ∑(a_{i})^{2} − (∑ a_{i})^{2} = ∑(a_{i} − a_{j})^{2}, (22)

then Lemma follows immediately from (22). Proof of Theorem 2. From (8), we have

R_{mj} = (n − 1)cδ_{mj} + ∑ h_{mi}^{α}h_{ij}^{α}, (23)

R = n(n − 1)c + S. (24)

Since M^{n} has flat normal bundle, so we can diagonalize the second fundamental form simultaneously so that h_{ij}^{α} = λ_{i}^{α}δ_{ij}, then from (21), we have

R_{mj} = (n − 1)cδ_{mj} + ∑(λ_{j}^{α})^{2}δ_{mj}, (25)

∑ R_{mj}^{2} = n(n − 1)^{2}c^{2} + 2(n − 1)cS + ∑(λ_{j}^{α})^{4}
(26)  ≥ n(n − 1)^{2}c^{2} + 2(n − 1)cS + \frac{1}{n}\{∑(λ_{j}^{α})^{2}\}^{2}
= n(n − 1)^{2}c^{2} + 2(n − 1)cS + \frac{1}{n}S^{2},

therefore from (20), we have

∫_{M^{n}} \{\frac{1}{2} ∑ R_{mijk}^{2} + \frac{1}{n}S^{2} + (n − 2)cS − n(n − 1)c^{2}\} ∗ 1 ≤ 0. (27)

Theorem 2 is proved. Proof of Theorem 3. From (27), we have

∫_{M^{n}} \{\frac{1}{2} ∑ R_{mijk}^{2} + (n − 2)cS − n(n − 1)c^{2}\} ∗ 1 ≤ 0. (28)

If M^{n} is totally geodesic, i.e., S = 0, h_{ij}^{α} = 0, then from (8), we have

R_{mijk} = c(δ_{mj}δ_{ik} − δ_{mk}δ_{ij}), ∑ R_{mijk}^{2} = 2n(n − 1)c^{2}, (29)

in this case, (28) becomes an equality.

Inversely, if (28) becomes an equality, then S = 0, M^{n} is totally geodesic. Acknowledgements. This note was written during the author’s stay at the Max-Planck-Institut für Mathematik in Bonn. The author would like to exprss his thanks to the institute for its hospitality and support.

This work is partially supported by the National Natural Science Foundation of China.

References

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

[2] B.Y. Chen, Geometry of submanifolds, Marcel Dekker, New York, 1973.

[3] Q.M. Cheng, Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector, Math. Z., 206(1991), 333-339.

[4] S. Y. Cheng and S. T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math., 104(1976), 407-419.

[5] O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space L^{3}, Tokyo J. Math., 6(1983), 297-309.

[6] S. Montiel, An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J., 37(1988), 909-917.

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

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

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China E-mail: xmliu@dlut.edu.cn