Nonnegative Ricci Curvature And The Topological Finiteness Of Riemannian Manifolds

In this paper,we study the topology of complete open manifolds with nonnegative Ricci curvature. We use comparison theorems and the theory of critical points of distance functions on Riemannian manifolds to get some structrual results of these manifolds. Specifically,we prove the following theorems.Theoremâ… Let (M,g) be a complete noncompact n-manifolds with nonnegative Ricci curvature and large volume growth.Suppose thatfor some p∈M.Then M has finite topological type,provided that the sectional curvature K_M≥-C> -∞.Theoremâ…¡Let M be a complete open Riemannian n-manifold with Ric_M≥0,α_M >0.Assume that k_p(r)≥-C/(1+r)^αfor some p∈M and all r > 0,where C > 0 andα∈[0,2] are constants. If there is a constant (?) = (?)(n, C,α) > 0 such thatfor all r > 0.Then M is diffemorphic to Rⁿ.Theoremâ…¢Let M be a complete noncompact Riemannian manifold and discrete group of isometries G act properly and discontinuously on M,p∈M.Ï€: M→M/G is the natural projection.If the quotient manifold M := M/G is noncompact andthen G is finite.In particular,if M is a universal cover, thenÏ€₁(M) is finite.