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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2401.05977 |
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| _version_ | 1866916088527192064 |
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| author | D'haene, Marie |
| author_facet | D'haene, Marie |
| contents | In this survey we focus on a special class of homogeneous manifolds called Thurston geometries. We give special attention to the four-dimensional Thurston geometries with 4 or 5-dimensional isometry group which are not a product (except for $\mathrm F^4$). These are the manifolds $\mathrm{Sol}^4_0$, $\mathrm{Sol}^4_1$, $\mathrm{Sol}^4_{m,n}$ and $\mathrm{Nil}^4$. We give a description of each of the spaces and we exhibit all Riemannian metrics which are invariant under the action of their associated group (this last part includes some small new results). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05977 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Thurston geometries in dimension four from a Riemannian perspective D'haene, Marie Differential Geometry Primary 53C42 In this survey we focus on a special class of homogeneous manifolds called Thurston geometries. We give special attention to the four-dimensional Thurston geometries with 4 or 5-dimensional isometry group which are not a product (except for $\mathrm F^4$). These are the manifolds $\mathrm{Sol}^4_0$, $\mathrm{Sol}^4_1$, $\mathrm{Sol}^4_{m,n}$ and $\mathrm{Nil}^4$. We give a description of each of the spaces and we exhibit all Riemannian metrics which are invariant under the action of their associated group (this last part includes some small new results). |
| title | Thurston geometries in dimension four from a Riemannian perspective |
| topic | Differential Geometry Primary 53C42 |
| url | https://arxiv.org/abs/2401.05977 |