Ricci curvature and the topology of open manifolds

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