The characteristic group of locally conformally product structures
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916093416701952 |
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| author | Flamencourt, Brice |
| author_facet | Flamencourt, Brice |
| contents | A compact manifold $M$ together with a Riemannian metric $h$ on its universal cover $\tilde M$ for which $π_1(M)$ acts by similarities is called a similarity structure. In the case where $π_1(M) \not\subset \mathrm{Isom}(\tilde M, h)$ and $(\tilde M, h)$ is reducible but not flat, this is a Locally Conformally Product (LCP) structure. The so-called characteristic group of these manifolds, which is a connected abelian Lie group, is the key to understand how they are built. We focus in this paper on the case where this group is simply connected, and give a description of the corresponding LCP structures. It appears that they are quotients of trivial $\mathbb{R}^p$-principal bundle over simply-connected manifolds by certain discrete subgroups of automorphisms. We prove that, conversely, it is always possible to endow such quotients with an LCP structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08372 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The characteristic group of locally conformally product structures Flamencourt, Brice Differential Geometry 53C05 (Primary) 53C18, 53C29 (Secondary) A compact manifold $M$ together with a Riemannian metric $h$ on its universal cover $\tilde M$ for which $π_1(M)$ acts by similarities is called a similarity structure. In the case where $π_1(M) \not\subset \mathrm{Isom}(\tilde M, h)$ and $(\tilde M, h)$ is reducible but not flat, this is a Locally Conformally Product (LCP) structure. The so-called characteristic group of these manifolds, which is a connected abelian Lie group, is the key to understand how they are built. We focus in this paper on the case where this group is simply connected, and give a description of the corresponding LCP structures. It appears that they are quotients of trivial $\mathbb{R}^p$-principal bundle over simply-connected manifolds by certain discrete subgroups of automorphisms. We prove that, conversely, it is always possible to endow such quotients with an LCP structure. |
| title | The characteristic group of locally conformally product structures |
| topic | Differential Geometry 53C05 (Primary) 53C18, 53C29 (Secondary) |
| url | https://arxiv.org/abs/2401.08372 |