单位球面中紧致超曲面的曲率结构与拓扑性质

The Curvature Structure and Topological Property of Compact Hypersurfaces in a Unit Sphere

设M是单位球面S^(n+1)(1)中的n维(n■3)紧致连通定向超曲面,本文研究这种超曲面的曲率结构与拓扑性质,利用Lawson和Simons关于稳定k维流的不存在性与同调群消失定理,得到了曲率与拓扑的一个关系定理,从而对Cheng Q.M.所提出的一个分类问题从拓扑角度给出了一个肯定回答,并且部分肯定回答了Cheng的另一个问题.

Let M be n-dimensional compact connected oriented hypersurfaces in a unit sphere S^（n＋1）（1）. This paper studies the curvature structure and topological property of these hypersurfaces. Bying the Lawson-Simons formula for the nonexistence of stable k-currents, which enables us to eliminate the homology groups, we obtain some thorems on curvature and the topology, which give a topological answer to the Cheng Q. M.＇s problems.