A geometric study of Wasserstein spaces: isometric rigidity in negative curvature
Metric Geometry
2019-10-15 v3 Differential Geometry
Abstract
Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X) when X is a simply connected, nonpositively curved metric spaces by considering its isometry group. When X is Euclidean, the second named author proved that this isometry group is larger than the isometry group of X. In contrast, we prove here a rigidity result: when X is negatively curved, any isometry of W2(X) comes from an isometry of X.
Cite
@article{arxiv.1404.1734,
title = {A geometric study of Wasserstein spaces: isometric rigidity in negative curvature},
author = {Jérôme Bertrand and Benoît Kloeckner},
journal= {arXiv preprint arXiv:1404.1734},
year = {2019}
}
Comments
v2: several typos corrected.v3: addition of a missing hypothesis in the secondary result Theorem 1.2