【作者】 陈琦；

【导师】 李光汉；

【作者基本信息】 湖北大学 ， 基础数学， 2017， 硕士

【摘要】 子流形几何的研究一直受到数学家和物理学家的关注,所研究的内容与理论物理、微分几何等密切相关,具有重要的理论意义.在爱因斯坦的广义相对论的影响和推动下,内积的正定性减弱为非退化的情形,就有了伪黎曼流形的概念和对伪黎曼子流形的研究.当外围空间N_p^n+p（c）的截面曲率为正常数时,极大类空子流形一定是全测地的,但其极大类时子流形未必是全测地.本文对局部对称伪黎曼流形中的紧致类时子流形进行研究,通过对子流形的条件加以限制（具有常平均曲率或者是极大或者2-调和）,利用活动标架法和一些引理得到这类子流形的Pinching定理以及相关的刚性定理.本文的结构安排如下:第一章简要介绍关于局部对称伪黎曼流形中子流形的刚性定理的研究背景及目前相关的一些研究成果;第二章为基础知识部分,介绍伪黎曼流形的相关概念和伪脐子流形及全测地子流形的一些性质,同时参考黎曼子流形的性质,计算得到伪黎曼子流形的基本公式;第三章利用伪黎曼流形中类时子流形的基本公式以及相关的引理,对第二基本形式模长平方的Laplacian进行估计和初步运算,为本文的主要结论的证明做好准备;第四章给出局部对称伪黎曼流形中类时子流形在相应条件下的的积分不等式,得到对应的Pinching定理及推广的Simons型积分不等式和刚性定理;第五章指出本文的不足以及对伪黎曼子流形未来研究的展望.更多还原

【Abstract】 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 N_p^n+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.更多还原