A Geometric Study of Wasserstein Spaces: Isometric Rigidity in Negative Curvature

Abstract

Abstract. -Given a metric space X, one defines its Wasserstein space W 2 (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 W 2 (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 W 2 (X) comes from an isometry of X.

Geodesically complete spaces with an upper curvature bound

“…Applications to topological questions, geometric group theory and sphere theorems will be discussed in forthcoming papers. Results and ideas of preliminary versions of this work which we have circulated in the last 10 years have already been used, for instance in [Kap07], [Kra11], [BK16], [KK17].…”

On isometric embeddings of Wasserstein spaces – the discrete case

“…First we recall that for a given Polish space M, a map f : W p (M) → W p (M) is called shape preserving if for all µ ∈ W p (M) there exists an isometry ψ µ : M → M (depending on µ), such that f (µ) is the push-forward measure of µ by the map ψ µ . Note that the isometries discussed in [2,10,16] are all shape preserving. A distance preserving map is called exotic if it is not shape preserving.…”

“…The most important developments in our considerations have been done by Bertrand and Kloeckner in connection with the Wasserstein metric [2,3,12]. Besides that the Wasserstein distance metrises the weak convergence of probability measures, its importance also lies in its role in geometric investigations of metric spaces, for more information see [14,17,18,19,20] and the references therein.…”

“…In the study of isometries it is a quite usual phenomenon that isometries also preserve some other underlying structure of the metric space. For instance, in [2,4,9,10] and [16] it turned out -as a consequence of the main theorems -that every isometry is automatically affine, i.e. preserves convex combinations of measures; and in [12] that every isometry of W 2 (R) preserves geodesically convex combinations of measures.…”

On geodesic extendibility and the space of compact balls of length spaces

“…For instance, in such spaces any pair of points can be joined by a unique distance realizing geodesic segment and are homeomorphic to an Euclidean space [6,9,1], and thus, they may be considered the closest analogs to the classical Euclidean and hyperbolic geometries. There has been a significant amount of research on geodesically complete metric length spaces with upper curvature bounds, specially from the perspective of geometric measure theory [15,4] and geometric group theory [8,7].…”