Ricci measure for some singular Riemannian metrics

TL;DR

This paper introduces a measure-based Ricci curvature for specific singular Riemannian metrics, providing a new way to analyze curvature in singular geometric settings and proposing a weak Ricci flow concept.

Contribution

It defines Ricci curvature as a measure for singular torsion-free connections and introduces a weak Ricci flow framework for such manifolds.

Findings

Ricci measure can be explicitly computed in key examples

A new integral formula for Ricci curvature in singular contexts

Proposes a weak Ricci flow solution concept

Abstract

We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the Ricci measure can be computed. In the time dependent setting, we give a weak notion of a Ricci flow solution on a manifold.