Ricci curvature bounds for warped products

TL;DR

This paper establishes generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds, extending known results to a broader class of metric measure spaces with curvature-dimension conditions.

Contribution

It extends lower Ricci curvature bounds to warped products over Finsler manifolds, unifying approaches from Finsler geometry, Alexandrov spaces, and curvature-dimension conditions.

Findings

Covers Bacher and Sturm's theorem on Euclidean and spherical cones.

Analogous to Bishop and Alexander's results in Alexandrov spaces.

Warped products satisfy curvature-dimension conditions despite metric degeneracies.

Abstract

We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below. For the proof we combine techniques developed in these papers. Because the Finsler product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again but a space satisfying a curvature-dimension condition in the sense of Lott-Villani/Sturm.