Isometric rigidity of Wasserstein tori and spheres

TL;DR

This paper establishes that the geometric structure of Wasserstein spaces over tori and spheres is uniquely determined by their metrics, using a unified method based on Wasserstein potentials.

Contribution

It proves isometric rigidity for Wasserstein spaces over tori and spheres for all p, introducing a unified approach based on measure recovery from Wasserstein potentials.

Findings

Rigidity holds for Wasserstein spaces over tori and spheres.

A unified method for proving isometric rigidity is developed.

The approach applies to all p in Wasserstein spaces.

Abstract

We prove isometric rigidity for $p$-Wasserstein spaces over finite-dimensional tori and spheres for all $p$. We present a unified approach to proving rigidity that relies on the robust method of recovering measures from their Wasserstein potentials.