【作者】 江桔丽；

【导师】 宋卫东；

【作者基本信息】 安徽师范大学 ， 基础数学， 2007， 硕士

【摘要】 本文通过计算de Sitter空间中子流形的第二基本形式模长平方的Laplace和引入一个自共轭的二阶微分算子,以及定义高阶平均曲率,并且利用Hopf定理,J.Simons技巧,得到了一些积分公式和刚性定理。主要结论为;1.得到了de Sitter空间中具有常平均曲率的超曲面,满足截面曲率非负时的刚性定理以及截面曲率与Ricci曲率之间存在的不等式。2.得到了de Sitter空间中具有常数量曲率超曲面的一个积分不等式,以及关于第二基本形式模长平方的一个拼挤定理。3.得到了de Sitter空间中的超曲面用高阶平均曲率表示的一些积分公式和刚性定理。4.得到了de Sitter空间中具有平行平均曲率的子流形关于截面曲率,Ricci曲率的拼挤定理,以及积分不等式和刚性定理。5.得到了de Sitter空间中具有平行第二基本形式的子流形的一个刚性定理。更多还原

【Abstract】 Firstly, we calculate the Laplace of the square of the length of the second fundamental form for submanifold and introduce a self-adjoint second order operator in de Sitter space. Secondly, we define the higher mean curvature and use Hopf Theorem, the method of J.Simons’ proof of his pinching theorem. Finally, we obtain some integral inequalities and rigidity theorems for hypersurfaces and submanifolds in de Sitter space. It mainly concludes the following:1. We obtain the rigidity theorem for hypersurfaces with constant mean curvature and nonnegative sectional curvature in de Sitter space. Also, the inequality for sectional curvature and Ricci curvature is got.2. A integral inequality of constant scalar curvature in de Sitter space is obtained. At the same time, we get the pinching theorem about the square of the length of the second fundamental form.3. We obtain some integral inequalities and rigidity theorems on condition that it uses the higher mean curvature in de Sitter space.4. The submanifolds get the pinching theorems, the integral inequality and some rigidity theorems with the sectional curvature and Ricci curvature, which have parallel mean curvature in de Sitter space.5. We obtain a rigidity theorem for submanifold with parallel the second fundamental form in de Sitter space.更多还原