Diffeomorphic theorems for open Riemannian manifolds with curvature decay

Abstract

In this paper, we study the topology of complete non-compact Riemannian manifolds with curvature decay to a non-positive constant. We show that such a complete open manifold M is diffeomorphic to a Euclidean n-space Rn if it contains enough rays starting from the base point. As applications, we also show that this kind of manifolds with Ricci curvature bounded from below by a non-positive constant are diffeomorphic to Rn if the volumes of geodesic balls in M grow properly. Our results generalize the main theorems of Wang�Xia for manifolds with quadratic curvature decay to zero.