Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$
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| Format: | Preprint |
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2023
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| _version_ | 1866929349391810560 |
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| author | D'haene, Marie Inoguchi, Jun-ichi Van der Veken, Joeri |
| author_facet | D'haene, Marie Inoguchi, Jun-ichi Van der Veken, Joeri |
| contents | We study hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$, which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor, including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form, and of totally umbilical hypersurfaces of $\text{Sol}^4_0$. We also give a closed expression for the Riemann curvature tensor of $\text{Sol}^4_0$, using two integrable complex structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_14105 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$ D'haene, Marie Inoguchi, Jun-ichi Van der Veken, Joeri Differential Geometry Primary 53C42, Secondary 53C30 We study hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$, which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor, including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form, and of totally umbilical hypersurfaces of $\text{Sol}^4_0$. We also give a closed expression for the Riemann curvature tensor of $\text{Sol}^4_0$, using two integrable complex structures. |
| title | Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$ |
| topic | Differential Geometry Primary 53C42, Secondary 53C30 |
| url | https://arxiv.org/abs/2303.14105 |