On tangent cones in Wasserstein space

TL;DR

This paper computes the tangent cone of Wasserstein space at a measure supported on a submanifold of a smooth compact Riemannian manifold, providing insights into its geometric structure.

Contribution

It introduces a method to explicitly compute tangent cones in Wasserstein space at measures supported on submanifolds, advancing geometric understanding.

Findings

Explicit formula for tangent cone at measures on submanifolds

Enhanced understanding of Wasserstein space geometry

Potential applications in optimal transport and geometric analysis

Abstract

If M is a smooth compact Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. If S is an embedded submanifold of M, and $\mu$ is an absolutely continuous measure on S, then we compute the tangent cone of P(M) at $\mu$.