Ricci curvature , entropy and optimal transport – Summer School in Grenoble 2009 – ‘ Optimal Transportation : Theory and Applications

These notes are the planned contents of my lectures. Some parts could be only briefly explained or skipped due to the lack of time or possible overlap with other lectures. The aim of these lectures is to review the recent development on the relation between optimal transport theory and Riemannian geometry. Ricci curvature is the key ingredient. Optimal transport theory provides a good characterization of lower Ricci curvature bounds without using differentiable structure. Then it can be considered as the ‘definition’ of lower Ricci curvature bounds of metric measure spaces. In §1, we recall the definition of the Ricci curvature of a Riemannian manifold and the classical Bishop-Gromov volume comparison theorem. In §2, we start with BrunnMinkowski inequalities in (weighted) Euclidean spaces, and show that a lower weighted Ricci curvature bound for a weighted Riemannian manifold is equivalent to some convexity inequality of entropy, called the curvature-dimension condition. In §3, we give the precise definition of the curvature-dimension condition for metric measure spaces, and see that it is stable under the measured Gromov-Hausdorff convergence. §4 is devoted to some geometric applications of the curvature-dimension condition. The final lecture will be concerned with some of related topics summarized in §5. Although we concentrate on rather geometric aspects, these lectures will be far from exhaustive. Interested readers can find more references in Further Reading at the end of each section (except §5).