Rigidity on horocycles and hypercycles
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909211676377088 |
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| author | Lo, Cheikh Sane, Abdoul Karim |
| author_facet | Lo, Cheikh Sane, Abdoul Karim |
| contents | We show that a bijection $f:\mathbb{H}^2\rightarrow\mathbb{H}^2$ of the hyperbolic plane that sends horocycles to horocycles (respectively hypercycles to hypercycles) is an isometry. This extends a previous result of J. Jeffers on geodesics to all curves with constant curvature in $\mathbb{H}^2$. We go beyond by showing that every abstract automorphism of the geodesic graph (respectively horocycles and hypercycles graphs) is induced by an earthquake map (respectively an isometry) of $\mathbb{H}^2$. This shadowed the difference between the geometry of geodesics and that of horocycles/hypercycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17598 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity on horocycles and hypercycles Lo, Cheikh Sane, Abdoul Karim Geometric Topology We show that a bijection $f:\mathbb{H}^2\rightarrow\mathbb{H}^2$ of the hyperbolic plane that sends horocycles to horocycles (respectively hypercycles to hypercycles) is an isometry. This extends a previous result of J. Jeffers on geodesics to all curves with constant curvature in $\mathbb{H}^2$. We go beyond by showing that every abstract automorphism of the geodesic graph (respectively horocycles and hypercycles graphs) is induced by an earthquake map (respectively an isometry) of $\mathbb{H}^2$. This shadowed the difference between the geometry of geodesics and that of horocycles/hypercycles. |
| title | Rigidity on horocycles and hypercycles |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2405.17598 |