In this article, we prove that an n-dimensional complete open Riemannian manifold M with Ricci curvature Ric M≥−( n−1) is diffeomorphic to a Euclidean n-space if M has positive conjugate radius and if the volume growth of geodesic balls in M is not far from that of the balls in an n-dimensional hyperbolic space H n(−1) of sectional curvature −1.