| Geometry of submanifolds has caught much attention from mathematicians and physicians due to its significance in theoretical physics and differential geometry.The concept of pseudo-Riemannian manifold is proposed when the positive definiteness of inner product become non-degenerated.When the sectional curvature is positive in the outer space Npn+p?c?,the maximal space-like submanifold should be totally geodesic while the maximal time-like submanfold may not.As the special case of submani-fold in pseudo-Riemannian manifold,locally symmetry of the outer space has caught much attention.This paper focuses on the compact time-like submanifold of pseudo-Riemannian manifold in the locally symmetic outer space.Using moving frame method and assuming an outer space with constant sectional curvature or a maximum 2-har-monic space and together with some lemmas,the Pinching theorem on the length square of the second fundamental form of this kind of submanifold was deduced and the rela-tive rigidity theorems were also derived.The main contents include:In Chapter 1,we introduce some background of the rigidity theorems of locally symmetric pseudo-Riemannian submanifold;In Chapter2,we focuse on the fundamental knowledge,including relative concepts on pseudo-Riemannian manifold and the properties of pseudoumbilical and totally umbilic sub-manifolds.In addition,the fundamental formula on the pseudo-Riemannian subman-ifold is also deduced based on the properties of Riemannian submanifold.In Chapter3,we obtain the Laplace estimation and basic calculation on the length square of the second fundamental form based on the relative lemmas and the fundamental formula in Chapter 2.In Chapter 4,we give the integral inequality on the time-like submanifold of locally symmetric pseudo-Riemannian manifold and derive the Pinching theorem and the rigidity theorem and the expanded Simons'intergral inequality.In Chapter 5,we put forward several relevant questions for further possible research. |
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