Ricci curvature, entropy and optimal transport

Shin-Ichi Ohta (1)

1 Deparment of mathematics

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.

Mots clés: Ricci curvature; entropy; optimal transport; curvature-dimension condition <br> 53C21; 53C23; 53C60; 28A33; 28D20

Domaines: Analyse classique [math.CA]; Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Shin-Ichi Ohta. Ricci curvature, entropy and optimal transport. 2009. ⟨hal-00518974⟩