Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866916991097372672 |
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| author | Balogh, Zoltán M. Titkos, Tamás Virosztek, Dániel |
| author_facet | Balogh, Zoltán M. Titkos, Tamás Virosztek, Dániel |
| contents | 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 $n\geq 1$. 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 $k\leq n$. 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 $Φ:\mathcal{W}_p(\mathbb{H}^n)\to\mathcal{W}_p(\mathbb{H}^n)$ there exists a $ψ:\mathbb{H}^n\to\mathbb{H}^n$ such that $Φ=ψ_{\#}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15095 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group Balogh, Zoltán M. Titkos, Tamás Virosztek, Dániel Metric Geometry Mathematical Physics Functional Analysis 46E27, 49Q22, 54E40 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 $n\geq 1$. 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 $k\leq n$. 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 $Φ:\mathcal{W}_p(\mathbb{H}^n)\to\mathcal{W}_p(\mathbb{H}^n)$ there exists a $ψ:\mathbb{H}^n\to\mathbb{H}^n$ such that $Φ=ψ_{\#}$. |
| title | Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group |
| topic | Metric Geometry Mathematical Physics Functional Analysis 46E27, 49Q22, 54E40 |
| url | https://arxiv.org/abs/2303.15095 |