Geometry on the Wasserstein space over a compact Riemannian manifold

TL;DR

This paper explores the intrinsic differential geometry of the Wasserstein space over a compact Riemannian manifold, building on foundational work by prominent researchers in the field.

Contribution

It revisits and synthesizes the geometric structure of Wasserstein space on Riemannian manifolds, clarifying and extending prior theoretical frameworks.

Findings

Detailed geometric characterization of Wasserstein space

Connections to optimal transport and Riemannian geometry

Potential implications for analysis on metric measure spaces

Abstract

We will revisit the intrinsic differential geometry of the Wasserstein space over a Riemannian manifold, due to a series of papers by Otto, Villani, Lott, Ambrosio, Gigli, Savar\'e and so on.