Manifolds Of Large Volume Growth And Curvature And Topology

In this thesis,we mainly study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. We study the topology of these manifolds via comparisonal geometry methods.In chapter 1 ,we will make a general description on the recent researches in our field.In chapter 2,we study complete noncompact Riemannian manifolds with nonegative Ricci curvature.We prove that an n-dimensional complete open Riemannian manifold M with Ricci curvature Ric_M ≥ 0 is diffeomorphic to an Euclidean n-space if M satisfies some volume growth conditions.In chapter 3,we prove that a complete noncompact n-dimensional Riemannian manifold M whose Ricci curvature Ric_M ≥(n- 1) has finite topological type or diffeomorphic to R~n if it's Excess function has some upper bound at a point.