【作者】 张德燕；

【导师】 刘建成；

【作者基本信息】 西北师范大学 ， 基础数学， 2007， 硕士

【摘要】 本论文主要研究de Sitter空间中具有平行平均曲率向量、常数量曲率或第二基本形式模长平方是常数的三类类空子流形,并通过分别估计三种情形下子流形的第二基本形式模长平方的Laplace,利用Yau的极大值原理和Stokes定理,获得了这些子流形的一些拼挤定理和刚性定理。论文共分为四节:第一节是准备工作,介绍de Sitter空间中类空子流形的基本概念和基本公式。第二节中,我们研究了de Sitter空间中具有平行平均曲率向量的类空子流形M~n,并证明了当M~n的第二基本形式模长平方||h||~2满足适当的上界时,M~n是全脐子流形(见定理2.1),同时获得了当M~n具有平坦法丛和正截面曲率时的一个刚性定理(详见定理2.2)。第三节中,我们主要讨论了de Sitter空间中具有常数量曲率的类空子流形M~n,获得了关于M~n的第二基本形式模长平方的两个拼挤定理,具体见定理3.1和定理3.2。第四节中,我们考虑了de Sitter空间中第二基本形式模长平方是常数的类空子流形M~n,并得到了M~n是全脐的三个充分条件,见定理4.1。更多还原

【Abstract】 In this article,we mainly study the space-like submanifolds with parallel mean curva-ture vector,constant scalar curvature or constant square length of the second fundamental form,respectively.We obtain some pinching theorems and rigidity results by estimating Laplace of the square length of the second fundamental form of the such submanifolds and using yau’s maximum principle or Stokes Theorem.The article divided into four sections.In section one,some basic concepts and associate formulas on space-like submanifolds in de Sitter space are presented.In section two,we consider the space-like submanifold Mⁿ with parallel mean curva-ture vector in de Sitter space S_p^n+P（c）,and shows that Mⁿ is totally umbilical if its square length of the second fundamental form ||h||² bounded from above（see Theorem 2.1）.At the same time,we give a rigidity theorem of Mⁿ with flat normal bundle and positive sectional curvature（see Theorem 2.2）.In section three,we study the space-like submanifold Mⁿ with constant scalar curva-ture in de Sitter space S_p^n+P（c）,and get two pinching theorems for Mⁿ about the square length of the second fundamental form（see Theorem 3.1 and 3.2）.In section four,we discuss the space-like submanifold Mⁿ with constant square norm of the second fundamental form in de Sitter space S_p^n+P（c）,and obtain three sufficient conditions for Mⁿ to be totally umbilical（see Theorem 4.1）.更多还原