Ricci Bounds for Euclidean and Spherical Cones

TL;DR

This paper establishes lower Ricci curvature bounds for Euclidean and spherical cones over compact Riemannian manifolds, demonstrating their satisfaction of specific curvature-dimension conditions based on the base manifold's Ricci curvature.

Contribution

It extends Ricci curvature bounds to Euclidean and spherical cones over manifolds, linking the base manifold's Ricci bounds to the cones' curvature-dimension conditions.

Findings

Euclidean cone over Ricci ≥ n-1 manifold satisfies CD(0,n+1)

Spherical cone over same manifold satisfies CD(n,n+1)

Cones are complete metric measure spaces with generalized Ricci bounds

Abstract

We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n,n+1).