Left-Invariant Riemannian Distances on Higher-Rank Sol-Type Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909493747515392 |
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| author | Levitin, Daniel N. |
| author_facet | Levitin, Daniel N. |
| contents | In this paper, we generalize the results of ($\textit{Groups, Geom. Dyn.}$, forthcoming) to describe the split left-invariant Riemannian distances on higher-rank Sol-type groups $G=\mathbf{N}\rtimes \mathbb{R}^k$. We show that the rough isometry type of such a distance is determined by a specific restriction of the metric to $\mathbb{R}^k$, and therefore the space of rough similarity types of distances is parameterized by the symmetric space $SL_k(\mathbb{R})/SO_k(\mathbb{R})$. In order to prove this result, we describe a family of uniformly roughly geodesic paths, which arise by way of the new technique of $\textit{Euclidean curve surgery}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Left-Invariant Riemannian Distances on Higher-Rank Sol-Type Groups Levitin, Daniel N. Group Theory Differential Geometry Metric Geometry 20F69, 22E25, 53C23, In this paper, we generalize the results of ($\textit{Groups, Geom. Dyn.}$, forthcoming) to describe the split left-invariant Riemannian distances on higher-rank Sol-type groups $G=\mathbf{N}\rtimes \mathbb{R}^k$. We show that the rough isometry type of such a distance is determined by a specific restriction of the metric to $\mathbb{R}^k$, and therefore the space of rough similarity types of distances is parameterized by the symmetric space $SL_k(\mathbb{R})/SO_k(\mathbb{R})$. In order to prove this result, we describe a family of uniformly roughly geodesic paths, which arise by way of the new technique of $\textit{Euclidean curve surgery}$. |
| title | Left-Invariant Riemannian Distances on Higher-Rank Sol-Type Groups |
| topic | Group Theory Differential Geometry Metric Geometry 20F69, 22E25, 53C23, |
| url | https://arxiv.org/abs/2412.11290 |