Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

  • Zoltán M. Balogh

    Universität Bern, Switzerland
  • Tamás Titkos

    Corvinus University of Budapest, Hungary; HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Dániel Virosztek

    HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group cover
Download PDF

This article is published open access under our Subscribe to Open model.

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