Isometric Rigidity of Metric Constructions with respect to Wasserstein Spaces

TL;DR

This paper investigates the isometric rigidity of various metric spaces within Wasserstein spaces, revealing conditions under which certain spaces preserve or lose their rigidity properties.

Contribution

It establishes new results on the isometric rigidity of infinite rays, half-cylinders, and spherical suspensions in Wasserstein spaces, extending prior understanding.

Findings

Infinite rays are isometrically rigid for all p ≥ 1.

Half-cylinders over compact non-branching spaces preserve rigidity for p > 1.

Spherical suspensions with diameter less than π/2 are rigid for p > 1.

Abstract

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the $p$-Wasserstein space. We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to $\mathbb{P}_2$. We then prove that infinite rays are isometrically rigid with respect to $\mathbb{P}_p$ for any $p\geq 1$, whereas taking infinite half-cylinders (i.e.\ product spaces of the form $X\times [0,\infty)$) over compact non-branching geodesic spaces preserves isometric rigidity with respect to $\mathbb{P}_p$, for $p>1$. Finally, we prove that spherical suspensions over compact spaces with diameter less than $\pi/2$ are isometrically rigid with respect to $\mathbb{P}_p$, for $p>1$.