具非负Ricci曲率的完备非紧黎曼流形

The Complete Open Riemannian Manifolds with Nonnegative Ricci Curvature

几何学研究的一个中心问题是曲率与拓朴性质之间的关系.本文讨论了具非负Ricci曲率的完备非紧黎曼流形的体积增长与其拓扑性质之间的关系.通过对测地球内的由球心点出发的最短测地线集合的测度与非最短测地线的测度的比较分析,根据距离函数临界点理论所隐含的拓扑性质,在大体积增长的情况下,得到流形拓扑的有限性.

The relationship of curvature and topology is important in geometry. For an open complete Riemannian manifold with nonnegative Ricci curvature, the present paper discusses the relation between the topology and the volume growth. On a given geodesic ball ,by comparing the measures of the shortest geodesics with the measure of the geodesics which are not shortest, if the manifold is with large volume growth,one gets its finite topologiacal type by crtical point theory of distance function.