Coarse Ricci curvature as a function on $M\times M$

TL;DR

This paper introduces a new notion of coarse Ricci curvature on smooth metric measure spaces, inspired by Bakry-Emery's framework, which can recover the Ricci tensor on Riemannian manifolds.

Contribution

It proposes an alternative definition of coarse Ricci curvature based on Bakry-Emery's approach, linking it to the Ricci tensor on smooth Riemannian manifolds.

Findings

The new curvature function can recover the Ricci tensor on Riemannian manifolds.

The approach provides a different perspective from Ollivier's notion of coarse Ricci curvature.

It connects curvature notions with logarithmic Sobolev inequalities.

Abstract

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula $$ \mathrm{Ric}(\gamma^{\prime}\left( 0\right) ,\gamma^{\prime}\left( 0\right) )=\frac{1}{2}\frac{d^{2}}{ds^{2}}\mathrm{Ric}_{\Delta_g}(x,\gamma\left( s\right) )$$ for any curve $\gamma(s).$