Ricci Bounds for Euclidean and Spherical Cones (revised/extended version)

TL;DR

This paper establishes lower Ricci curvature bounds for Euclidean and spherical cones over Riemannian manifolds, extending curvature-dimension conditions to spaces that are not necessarily manifolds or Alexandrov spaces.

Contribution

It proves new curvature-dimension conditions for Euclidean and spherical cones over manifolds, generalizing Ricci bounds to non-manifold metric measure spaces.

Findings

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

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

Equivalence of CD conditions for weighted spaces and their cones for N > 1

Abstract

We prove generalized lower Ricci bounds for Euclidean and spherical cones over complete Riemannian manifolds. These cones are regarded as complete metric measure spaces. In general, they will be neither manifolds nor Alexandrov 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). More generally, for each N > 1 we prove that the condition CD(N-1;N) for a weighted Riemannian space is equivalent to the condition CD(0;N +1) for its N-Euclidean cone as well as to the condition CD(N;N + 1) for its N-spherical cone.