A warped product version of the Cheeger-Gromoll splitting theorem

Abstract

We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form C D ( 0 , 1 ) CD(0,1) . Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is C D ( 0 , 1 ) CD(0,1) , we show that the fundamental group of M M is the fundamental group of a compact manifold with non-negative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is C D ( 0 , 1 ) CD(0,1) .