Soul, non-negative curvature, ray and coverings

完備非緊的黎曼流形\(M\)上總存在射線.(cf.do Carmo's book,P153)

假設\(M\)完備非緊, 且具有非負曲率, 則存在一個閉全凸子流形\(S\), 且\(S\)的法叢微分同胚於\(M\). (cf. Petersen's book. P462)
實際上, \(S\)可以是全測地的. (cf. Cheeger and Ebin's book, P128)

若\(M\)是緊的, 且具有非負Ricci曲率, 那麼萬有覆蓋空間\(\tilde{M}\)可以分裂為一個平坦歐式空間和一個和\(M\)性質類似的流形的乘積. (cf.On the Structure of Complete Manifolds of Nonnegative Curvature, Jeff Cheeger and Detlef Gromoll)