Classification of 6-dimensional splittable flat solvmanifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866914675889799168 |
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| author | Tolcachier, Alejandro |
| author_facet | Tolcachier, Alejandro |
| contents | A flat solvmanifold is a compact quotient $Γ\backslash G$ where $G$ is a simply-connected solvable Lie group endowed with a flat left invariant metric and $Γ$ is a lattice of $G$. Any such Lie group can be written as $G=\mathbb{R}^k\ltimes_ϕ \mathbb{R}^m$ with $\mathbb{R}^m$ the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting $G$ by a lattice $Γ$ that can be decomposed as $Γ=Γ_1\ltimes_ϕΓ_2$, where $Γ_1$ and $Γ_2$ are lattices of $\mathbb{R}^k$ and $\mathbb{R}^m$, respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_08002 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Classification of 6-dimensional splittable flat solvmanifolds Tolcachier, Alejandro Differential Geometry Group Theory 20H15, 22E25, 22E40, 53C29 A flat solvmanifold is a compact quotient $Γ\backslash G$ where $G$ is a simply-connected solvable Lie group endowed with a flat left invariant metric and $Γ$ is a lattice of $G$. Any such Lie group can be written as $G=\mathbb{R}^k\ltimes_ϕ \mathbb{R}^m$ with $\mathbb{R}^m$ the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting $G$ by a lattice $Γ$ that can be decomposed as $Γ=Γ_1\ltimes_ϕΓ_2$, where $Γ_1$ and $Γ_2$ are lattices of $\mathbb{R}^k$ and $\mathbb{R}^m$, respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5. |
| title | Classification of 6-dimensional splittable flat solvmanifolds |
| topic | Differential Geometry Group Theory 20H15, 22E25, 22E40, 53C29 |
| url | https://arxiv.org/abs/2105.08002 |