Soul, non-negative curvature, ray and coverings

完备非紧的黎曼流形\(M\)上总存在射线.(cf.do Carmo's book,P153)

假设\(M\)完备非紧, 且具有非负曲率, 则存在一个闭全凸子流形\(S\), 且\(S\)的法丛微分同胚于\(M\). (cf. Petersen's book. P462)
实际上, \(S\)可以是全测地的. (cf. Cheeger and Ebin's book, P128)

若\(M\)是紧的, 且具有非负Ricci曲率, 那么万有覆盖空间\(\tilde{M}\)可以分裂为一个平坦欧式空间和一个和\(M\)性质类似的流形的乘积. (cf.On the Structure of Complete Manifolds of Nonnegative Curvature, Jeff Cheeger and Detlef Gromoll)