摘要
对于S^(n_1)×S^(n_2)中的紧致子流形M,本文运用子流形M的形算子,在切空间T_xM的p维子空间上构造一个自伴线性算子Q^A,证明了当Q^A满足一定条件时M中没有p维稳定流,因而同调群H_p(M,Z)消没。在此基础上证明了,当子流形M的第二基本形式的长度的平方与平均曲率满足一定关系时,M与球面同胚。
For a compact submanifold Mimmersed in S^(n_1)×S^(n_2),by using the shape operators of M we construct a selfadjoint linear operator Q^A on a p-subspace of T_xM,and prove that when Q^A satisfies some conditions there is no stable p-current in M and thus H_p(M,Z)=H_(m-p)(M,Z)=0.From this,we prove that M is homeomorphic to a sphere when the square length of the second fundamental form and the mean curvature of M meet a specific relation.
基金
冶金部教育司基金资助项目
关键词
几何
拓扑
形算子
同调群
稳定流
homology group
homeomorphism/submanifold
stable current
shape operator
selfadjoint linear operator
