Isometric rigidity of Wasserstein spaces over Euclidean spheres

TL;DR

This paper proves that the Wasserstein space over Euclidean spheres has a rigid isometry group, contrasting with the non-rigidity of the Euclidean case, and introduces new techniques for establishing this rigidity.

Contribution

It establishes the isometric rigidity of Wasserstein spaces over spheres and develops methods applicable to other Wasserstein spaces.

Findings

Wasserstein space over spheres is isometrically rigid.

Contrast with non-rigidity of Euclidean Wasserstein spaces.

Rigidity results extend to certain p-Wasserstein spaces.

Abstract

We study the structure of isometries of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ over the sphere endowed with the distance inherited from the norm of $\mathbb{R}^{n+1}$. We prove that $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ is isometrically rigid, meaning that its isometry group is isomorphic to that of $\left(\mathbb{S}^n,\|\cdot\|\right)$. This is in striking contrast to the non-rigidity of its ambient space $\mathcal{W}_2\left(\mathbb{R}^n,\|\cdot\|\right)$ but in line with the rigidity of the geodesic space $\mathcal{W}_2\left(\mathbb{S}^n,\sphericalangle\right)$. One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$. A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of $\mathcal{W}_p\left(\mathbb{S}^1, \|\cdot\|\right)$ for $1 \leq p<2.$