Geodesic-preserving bijections of the Thurston geometries
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918177856815104 |
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| author | Dickmann, Ryan Lideros, Palani Narayanan, Akash |
| author_facet | Dickmann, Ryan Lideros, Palani Narayanan, Akash |
| contents | We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder $\mathbb{S}^1 \times \mathbb{R}$ and the bijections of the hyperbolic plane $\mathbb{H}^2$ that preserve constant curvature curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geodesic-preserving bijections of the Thurston geometries Dickmann, Ryan Lideros, Palani Narayanan, Akash Geometric Topology Metric Geometry 57M50, 51M10 We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder $\mathbb{S}^1 \times \mathbb{R}$ and the bijections of the hyperbolic plane $\mathbb{H}^2$ that preserve constant curvature curves. |
| title | Geodesic-preserving bijections of the Thurston geometries |
| topic | Geometric Topology Metric Geometry 57M50, 51M10 |
| url | https://arxiv.org/abs/2510.25585 |