Space-like Submanifolds With Uint Parallel Mean Curvature Vector In The De Sitter Space

This paper contains three chapters. We discuss the pinching problems on the length of the second fundamental form of compact space-like submanifolds Mn with unit parallel mean curvature vector in the de Sitter space Spn+p. In particular, a sufficient condition for Mn with constant scalar curvature to be totally umbilical is given. We obtain the following:Theorem 1 Let Mn be an n-dimensional compact space-like submanifold with unit parallel mean curvature in the de Sitter space Spn+p with p > 1. If either n = 2and H2 ≤ c or n≥ 3, H2 ≤ c and S ≤ ; then Mn lies in an (n + 1)-dimensional totally geodesic submanifold S1n+1 of Spn+p, where 5 and H are the length square of second fundamental form and the mean curvature of Mn , respectively.Theorem 2 Let Mn be an n-dimensional compact space-like submanifold with unit parallel mean curvature in the de Sitter space Spn+p with p > 1, n > 3. Ifand Mn has constant scalar curvature R < cn(n -1), then Mn is totally umbilical (and isometric to the standard sphere).