The Splitting Theorem and Topology of Noncompact Spaces with Nonnegative N-Bakry Émery Ricci Curvature

Abstract: In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative $N$-Bakry \'Emery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative $N$-Bakry \'Emery Ricci Curvature. In addition, we show that if $M<sup>n$ is a complete, noncompact Riemannian manifold with nonnegative $N$-Bakry \'Emery Ricci curvature where $N&gt;n$, then $H_{n-1}(M,\mathbb{Z})$ is $0$.