Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

Abstract

Our purpose in this paper is to study isometries and isometric embeddings of the p -Wasserstein space \mathcal{W}_{p}(\mathbb{H}^{n}) over the Heisenberg group \mathbb{H}^{n} for all p>1 and for all n\geq1 . First, we create a link between optimal transport maps in the Euclidean space \mathbb{R}^{2n} and the Heisenberg group \mathbb{H}^{n} . Then we use this link to understand isometric embeddings of \mathbb{R} and \mathbb{R}_{+} into \mathcal{W}_{p}(\mathbb{H}^{n}) for p>1 . That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results, we determine the metric rank of \mathcal{W}_{p}(\mathbb{H}^{n}) . Namely, we show that \mathbb{R}^{k} can be embedded isometrically into \mathcal{W}_{p}(\mathbb{H}^{n}) for p>1 if and only if k\leq n . As a consequence, we conclude that \mathcal{W}_{p}(\mathbb{R}^{k}) and \mathcal{W}_{p}(\mathbb{H}^{k}) can be embedded isometrically into \mathcal{W}_{p}(\mathbb{H}^{n}) if and only if k\leq n . In the second part of the paper, we study the isometry group of \mathcal{W}_{p}(\mathbb{H}^{n}) for p>1 . We find that these spaces are all isometrically rigid, meaning that for every isometry \Phi\colon \mathcal{W}_{p}(\mathbb{H}^{n})\to\mathcal{W}_{p}(\mathbb{H}^{n}) , there exists an isometry \psi\colon\mathbb{H}^{n}\to\mathbb{H}^{n} such that \Phi=\psi_{\#} .