摘要
在给出伪Riemann流形中一般等距浸入子流形的基本公式后,我们证明了极大类空子流形的一个广义Bernstein定理,并研究这种子流形的稳定性.
Let Nnv be a pseudo-Riemannian n-manifold with index v, Mnμ an m(< n)-dimensional pseudo-Riemannian submanifold with index μ(≤v) in Nnv. If the mean curvature of Mmμ vanishes identically, then Mmμ is called an extremal submanifold. Particularly, an extremal submanifold in Nnv with μ=0 and m = n - v is called maximal spacelike submanifold. In this paper, main results are as follows.
Theorem 1. Let Snv(c) denote an n-dimensional complete simply-connected pseudo-Riemannian manifold with index v and constant sectional curvature c. Then, for c≥0, any complete maximal spacelike submanifold in Snv(c) must be totally geodesic.
Theorem 2. Let Nnv be a pseudo-Riemannian manifold with nonpositive sectional curvature. Then any maximal spacelike submanifold in Nnv is stable.
Theorem 3. Let Nnv be a pseudo-Riemannian manifold with sectional curvature bounded from below by c0 and M a maximal spacelike submanifold in Nnv. Assume that D=M is a simply-connected compact domain with piecewise smooth boundary. If the restriction of the scalar curvature of M to D is less than (n-v)2c0-λ(D), where λ1(D) denotes the first Dirichlet eigenvalue of the Laplacian on D, then D is unstable.
出处
《杭州大学学报（自然科学版）》
CSCD
1991年第4期371-376,共6页
Journal of Hangzhou University Natural Science Edition
基金
国家自然科学基金
关键词
伪黎曼流形
极大子流形
稳定性
pseudo-Riemannian manifold
maximal submanifold
generalized Bernstein theorem
stability
