关于外在刚性定理的两个结果

THE TWO RESULTS REGARDING EXTRINSIC RIGIDITY THEOREMS

在［１］中陈省身证出：如果Ｍ是Ｓｎ＋１中闭的可定向极小超曲面。则若Ｍ的高斯象位于Ｓｎ＋１的开半球内，那末Ｍ必是全测地。在［２］中李安民对ｎ＝２的情形给予了改进。本文进一步改进了［２］中的结果，并对Ｓ４中的二维极小子流形，也作了讨论。

In,Cherm Shiing-Shen had proved:Let M be a closed oriented minimal hypersurface in S n+1 .If the image of the Gauss map of M lies in an open hemisphere of S n+1 ,then M must be totally geodesic.In ,when n=2,Li An-Min improved the result of Chern's.This paper makes further improvement on the result in .In the meantime,it gives a discussion on the case of two-dimensional minimal submanifolds in S 4.