On Differential and Riemannian Calculus on Wasserstein Spaces

TL;DR

This paper develops a formalism to analyze the topology and geometry of Wasserstein spaces on Riemannian manifolds, providing new characterizations and proofs, especially for compact Lie groups.

Contribution

It introduces an intrinsic formalism for Wasserstein spaces that characterizes topology, smooth structure, and Riemannian geometry, with applications to convexity and Lie groups.

Findings

Wasserstein spaces of closed manifolds are geodesically convex.

New characterization of the Weak topology via convergent sequences.

Explicit example for Wasserstein spaces on compact Lie groups.

Abstract

In this paper we develop an intrinsic formalism to study the topology, smooth structure, and Riemannian geometry of the Wasserstein space of a closed Riemannian manifold. Our formalism allows for a new characterisation of the Weak topology via convergent sequences of the subjacent space. Applying it we also provide a new proof that Wasserstein spaces of closed manifolds are geodesically convex. Our framework is particularly handy to address the Wasserstein spaces of compact Lie groups, where we refine our formalism and present an explicit example.