Space-like Submanifolds In Pseudo-Riemannian Space Forms

In this paper, we mainly investigated the totally umbilical of compact space-like submanifolds and hypersurfaces in pseudo-Riemannian space forms N_q^n+p（c）, and we got the following results.1. Let Mⁿ be a submanifolds of N_q^n+p（c） （1≤q≤p）, with nonzero mean curvature H and parallel mean curvature vector form. If the second fundamental form of Mⁿ is locally timelike, and there exist a positive constant C（n, H） such that ||S||_n/2<C（n,H）, then Mn is totally umbilical, where S is the squared norm of second fundamental form of Mn, and ||S||_n/2= (∫MS^n/2dvg)^2/n.2. Let Mⁿ be hypersurface of N₁ⁿ⁺¹（c） （c= 1,-1）. If the higher order mean curvature of Mⁿ satisfies the conditions that H_k/H_l is a constant, and H_l> 0, then Mⁿ is totally umbilical, where 1≤k≤l≤n-2.