Manifolds with Bakry-Emery Ricci Curvature Bounded Below

Abstract

In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.