Ricci curvature and the topology of open manifolds

Abstract

In this paper, we prove that an open Riemannian n-manifold with Ricci curvature Ric _M ≥ 0 and $K_p^{\rm min} \geq K_0 >- \infty$ for some p ∈ M is diffeomorphic to a Euclidean n-space R ⁿ if the volume growth of geodesic balls around p is not too far from that of the balls in R ⁿ. We also prove that a complete n-manifold M with $K_p^{\rm min} \geq 0$ is diffeomorphic to R ⁿ if $lim_{r\to \infty} \frac{{\rm Vol} [B(p,r)]}{\omega_n r^n} \geq \frac{1}{2}$,where ω _n is the volume of unit ball in R ⁿ

Mathematics Subject Classification (1991): 53C20, 53C42

Supported by CNPq, Brasil