Ricci curvature, entropy and optimal transport

TL;DR

This paper explores how optimal transport theory relates to Riemannian geometry, showing that entropy convexity along transport paths characterizes Ricci curvature bounds and extends to metric measure spaces.

Contribution

It provides a comprehensive overview of the connection between entropy convexity, Ricci curvature, and optimal transport in both Riemannian and metric measure spaces.

Findings

Convexity of entropy characterizes Ricci curvature lower bounds.

Extension of geometric properties to general metric measure spaces.

Insights into the interplay between optimal transport and Riemannian geometry.

Abstract

This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure spaces satisfying this convexity condition.