Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

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 $n\ge 1$. 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 $k\le n$. 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 $k\le n$. 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 $\mathrm{\Phi }:{\mathcal{W}}_p({\mathbb{H}}^n)\to {\mathcal{W}}_p({\mathbb{H}}^n)$ there exists a $\psi :{\mathbb{H}}^n\to {\mathbb{H}}^n$ such that $\mathrm{\Phi }={\psi }_{\mathrm{\#}}$.