Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group
Zoltán M. Balogh
Universität Bern, SwitzerlandTamás Titkos
Corvinus University of Budapest, Hungary; HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, HungaryDániel Virosztek
HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

Abstract
Our purpose in this paper is to study isometries and isometric embeddings of the -Wasserstein space over the Heisenberg group for all and for all . First, we create a link between optimal transport maps in the Euclidean space and the Heisenberg group . Then we use this link to understand isometric embeddings of and into for . That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results, we determine the metric rank of . Namely, we show that can be embedded isometrically into for if and only if . As a consequence, we conclude that and can be embedded isometrically into if and only if . In the second part of the paper, we study the isometry group of for . We find that these spaces are all isometrically rigid, meaning that for every isometry , there exists an isometry such that .
Cite this article
Zoltán M. Balogh, Tamás Titkos, Dániel Virosztek, Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group. Rev. Mat. Iberoam. 41 (2025), no. 6, pp. 2055–2084
DOI 10.4171/RMI/1576