黎曼流形中紧子集的两种灵魂

Two Kinds of Souls of Compact Subsets in Riemannian Manifolds

文章主要研究了黎曼流形中紧子集的λ-凸包的灵魂和紧子集的外蕴灵魂.作者首先列出了紧子集的外蕴灵魂唯一的一些充分条件(比如当流形为Hadamard流形时);接着证明了,对于Hadamard流形中给定的紧子集来说,这两种灵魂是重合的,并探究了在一般流形中这两种灵魂之间的距离;最后给出了黎曼流形中的一个子流形为全测地的由这两种灵魂所确定的充分必要条件.由于外蕴灵魂的定义仅涉及距离,所以本文的研究内容和思路容易推广到较黎曼流形更为一般的距离空间上,比如说Alexandrov空间.

In this paper,both the souls of theλ-convex hull of a compact subset and the extrinsic soul of a compact subset in a Riemannian manifold are studied.First of all,the authors list some conditions guaranteeing that the extrinsic soul of a compact subset is unique(e.g.,in Hadamard manifolds).Then,the authors show that,in Hadamard manifolds,such two kinds of souls of a compact subset are the same.Finally,using such two kinds of souls the authors can provide criteria to judge whether a submanifold in a Riemannian manifold is totally geodesic.Since the definition of extrinsic soul involves only the distance,the methods of this paper are hopefully adapted to more generalized metric spaces such as Alexandrov spaces.