A geometric study of Wasserstein spaces: isometric rigidity in negative curvature

Jérôme Bertrand (1) , Benoît Kloeckner (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 IF - Institut Fourier

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.

Mots clés: optimal transport; Hadamard spaces; metric geometry

Domaines: Géométrie métrique [math.MG]; Géométrie différentielle [math.DG]

Jérôme Bertrand, Benoît Kloeckner. A geometric study of Wasserstein spaces: isometric rigidity in negative curvature. International Mathematics Research Notices, 2016, 2016 (5), pp.1368-1386. ⟨10.1093/imrn/rnv177⟩. ⟨hal-00974554v3⟩