Lie groups with all left-invariant semi-Riemannian metrics complete
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916471364386816 |
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| author | Elshafei, Ahmed Ferreira, Ana Cristina Sánchez, Miguel Zeghib, Abdelghani |
| author_facet | Elshafei, Ahmed Ferreira, Ana Cristina Sánchez, Miguel Zeghib, Abdelghani |
| contents | For each left-invariant semi-Riemannian metric $g$ on a Lie group $G$, we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of $g$. When the adjoint representation of $G$ satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any $g$. We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products $K \ltimes_ρ\mathbb{R}^n$ , where $K$ is the direct product of a compact and an abelian Lie group and $ρ(K)$ is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the the absence of linear growth and suggest new questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16513 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lie groups with all left-invariant semi-Riemannian metrics complete Elshafei, Ahmed Ferreira, Ana Cristina Sánchez, Miguel Zeghib, Abdelghani Differential Geometry For each left-invariant semi-Riemannian metric $g$ on a Lie group $G$, we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of $g$. When the adjoint representation of $G$ satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any $g$. We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products $K \ltimes_ρ\mathbb{R}^n$ , where $K$ is the direct product of a compact and an abelian Lie group and $ρ(K)$ is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the the absence of linear growth and suggest new questions. |
| title | Lie groups with all left-invariant semi-Riemannian metrics complete |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2308.16513 |