具非负曲率完备非紧黎曼流形的闭测地线

Kingenberg证明了任意紧致黎曼流形上都存在闭测地线,Yau提出是否能够证明紧致黎曼流形上有无穷多条闭测地线.由著名的Cheeger-Gromoll的核心结构的思想,任意的具非负曲率完备非紧的黎曼流形与它的核心是同伦等价的.因此可以考虑具非负曲率完备非紧的黎曼流形闭测地线存在性和分布性问题.本文证明了当核心的余维数是奇数且具非负曲率的完备非紧的黎曼流形上存在有无穷多条闭测地线;并由此讨论了紧致的非单连通黎曼流形上无穷多的闭测地线存在性问题.

The Closed Geodesics in Complete Riemannian Manifolds with Nonnegative Curvature

Kingenberg obtained that there is a closed geodesic in a compact Riemannian manifold,Yau quoted whether there are infinite closed geodesics in a compact Riemannian manifold.By the well-known Cheeger-Gromoll soul's idea,a noncompact complete Riemannian manifold M with nonnegative curvature is homotopic to its soul S ,one can consider whether there are infinite closed geodesics in a noncompact Riemannian manifold.The paper proves that there are infinite closed geodesics in a complete noncompact Riemannian manifold with nonnegative curvature provided that codim S=2k+1 ,where S is the soul.Also,the paper discuss the existence of the infinite closed geodesics of a compact no-simply connected Riemannian manifold.