Conformal geometry of quasi-umbilical timelike surfaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909807378694144 |
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| author | Musso, Emilio Nicolodi, Lorenzo Pember, Mason |
| author_facet | Musso, Emilio Nicolodi, Lorenzo Pember, Mason |
| contents | The paper focuses on the conformal Lorentz geometry of quasi-umbilical timelike surfaces in the $(1+2)$-Einstein universe, the conformal compactification of Minkowski 3-space realized as the space of oriented null lines through the origin of $\mathbb{R}^{2,3}$. A timelike immersion of a surface $X$ in the Einstein universe is quasi-umbilical if its shape operator at any point of $X$ is non-diagonalizable over the complex numbers. We prove that quasi-umbilical surfaces are isothermic, that their conformal deformations depend on one arbitrary function in one variable, and show that their conformal Gauss map is harmonic. We then investigate their geometric structure and show how to construct all quasi-umbilical surfaces from null curves in the 4-dimensional neutral space form $S^{2,2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformal geometry of quasi-umbilical timelike surfaces Musso, Emilio Nicolodi, Lorenzo Pember, Mason Differential Geometry 53C50, 53A30, 53B30 The paper focuses on the conformal Lorentz geometry of quasi-umbilical timelike surfaces in the $(1+2)$-Einstein universe, the conformal compactification of Minkowski 3-space realized as the space of oriented null lines through the origin of $\mathbb{R}^{2,3}$. A timelike immersion of a surface $X$ in the Einstein universe is quasi-umbilical if its shape operator at any point of $X$ is non-diagonalizable over the complex numbers. We prove that quasi-umbilical surfaces are isothermic, that their conformal deformations depend on one arbitrary function in one variable, and show that their conformal Gauss map is harmonic. We then investigate their geometric structure and show how to construct all quasi-umbilical surfaces from null curves in the 4-dimensional neutral space form $S^{2,2}$. |
| title | Conformal geometry of quasi-umbilical timelike surfaces |
| topic | Differential Geometry 53C50, 53A30, 53B30 |
| url | https://arxiv.org/abs/2410.10330 |