Spacelike Hypersurface In Local Symmetric Spaces And Locally Symmetric Lorentz Space Hypersurface

The study on the Riemannian manifold is an important field of the Riemannian geometry since Riemannian geometry had been born. When the ambient space has good symmetric, the submanifold of Riemannian manifold, especially submanifold in a sphere had been studied and obtained many rigidity theorems. While the ambient space is extended to locally symmetric space or locally symmetric Lorentz space, many researchers had obtained many important results.In this paper, by making use of the exterior differentiation, moving frames method and integral methods, we study the geometric properties of hypersurfaces or space-like hyper-surfaces in a locally symmetric space or a locally symmetric Lorentz space and obtain some rigidity theorems under its intrinsic invariant satisfying some intrinsic conditions, in detail, the main results in this paper may be stated as follows:1. The hypersurfaces with constant scalar curvature in locally symmetric space are studied by making use of the self operator of Yau S T and some rigidity theorems are obtained if the squared norm of the second fundamental form satisfying some pinching condition, the result of Hai Zhong Li is generalized.2. The definition of the locally symmetric Lorentz space is introduced, complete hypersurfaces with constant mean curvature and the compact space-like hypersurfaces with constant scalar curvature are studied by making use of the exterior differentiation, moving frames method. Two rigidity theorems are obtained and the main results of Chong Zhen OU-YANG, Zhen Qi Li and Xin Min Liu are generalized.3. First introduce the new define of the linear weingarten space-like hypersurfaces in de Sitter space and by making use of the self-adjiont differential operator of S.Y.Cheng and S.T.Yau study the linear weingarten space-like hypersurfaces we get two important rigidity theorems, which generalized the two main results of zheng[19] and Li[20] on the space-like hypersurfaces with constant mean curvature and constant scalar curvature in de sitter space.