On isometries of compact Lp–Wasserstein spaces

Abstract

Let (X,d,m) be a compact non-branching metric measure space equipped with a qualitatively non-degenerate measure m. The study of properties of the Lp–Wasserstein space (Pp(X),Wp) associated to X has proved useful in describing several geometrical properties of X. In this paper we focus on the study of isometries of Pp(X) for p∈(1,∞) under the assumption that there is some characterization of optimal maps between measures, the so called Good Transport Behaviour GTBp. Our first result states that the set of Dirac deltas is invariant under isometries of the Lp–Wasserstein space. Additionally, for Riemannian manifolds we obtain that the isometry groups of the Lp–Wasserstein space and of the base space coincide under geometric assumptions on the manifold; namely, for p=2 that the sectional curvature is strictly positive and for general p∈(1,∞) that the Riemannian manifold is a Compact Rank One Symmetric Space.