A warped product version of the Cheeger-Gromoll splitting theorem

TL;DR

This paper extends the Cheeger-Gromoll splitting theorem to warped products under a $CD(0,1)$ curvature condition, leading to new topological insights and obstructions for certain Riemannian metrics.

Contribution

It introduces a warped product splitting theorem under $CD(0,1)$ conditions, broadening the classical splitting theorem's scope and applications.

Findings

Warped product splitting under $CD(0,1)$ curvature condition.

Fundamental group of certain manifolds relates to nonnegative sectional curvature.

Obstructions to $CD(0,1)$ densities on locally homogeneous spaces.

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 $CD(0,1)$. 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 $CD(0,1)$, we show that the fundamental group of $M$ is the fundamental group of a compact manifold with nonnegative 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 $CD(0,1)$.