黎曼流形在正交联络下的全脐点子流形

摘要

本文研究了正交联络下子流形基本方程以及在全脐点子流形中的应用.利用Cartan的方法将挠率张量分解成三个部分,计算得到正交联络下的三个基本方程,并考虑一个特殊的正交联络,证明了其黎曼曲率会有类似于Levi-Civita联络下的性质.利用基本方程得到常曲率空间中的全脐点子流形的性质,推广了Levi-Civita联络下的相应结果.%In this paper, we investigate the fundamental equations of submanifolds under or-thogonal connections and apply the results in totally umbilical submanifolds. By using the method of Cartan to split the torsion tensor into three components, we calculate and attain the fundamental equations. We consider a special orthogonal connection with which the Riemannian curvature has the same properties as the Levi-Civita connection. We use the fundamental equations to argue to-tally umbilical submanifolds on spaces with constant curvature, which generalizes the results under the Levi-Civita connection.