Left-invariant Pseudo-Riemannian metrics on Lie groups: The null cone
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915006658904064 |
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| author | Hervik, Sigbjorn |
| author_facet | Hervik, Sigbjorn |
| contents | We study left-invariant pseudo-Riemannian metrics on Lie groups using the bracket flow of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the $G=O(p,q)$-action; i.e., Lie algebras $μ$ where zero is in the closure of the orbits: $0\in\overline{G\cdot μ}$. We provide examples of such Lie groups in various signatures and give some general results. For signatures $(1,q)$ and $(2,q)$ we classify all cases belonging to the null cone. More generally, we show that all nilpotent and completely solvable Lie algebras are in the null cone of some $O(p,q)$ action. In addition, several examples of non-trivial Levi-decomposable Lie algebras in the null cone are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03536 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Left-invariant Pseudo-Riemannian metrics on Lie groups: The null cone Hervik, Sigbjorn Differential Geometry Mathematical Physics Group Theory We study left-invariant pseudo-Riemannian metrics on Lie groups using the bracket flow of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the $G=O(p,q)$-action; i.e., Lie algebras $μ$ where zero is in the closure of the orbits: $0\in\overline{G\cdot μ}$. We provide examples of such Lie groups in various signatures and give some general results. For signatures $(1,q)$ and $(2,q)$ we classify all cases belonging to the null cone. More generally, we show that all nilpotent and completely solvable Lie algebras are in the null cone of some $O(p,q)$ action. In addition, several examples of non-trivial Levi-decomposable Lie algebras in the null cone are given. |
| title | Left-invariant Pseudo-Riemannian metrics on Lie groups: The null cone |
| topic | Differential Geometry Mathematical Physics Group Theory |
| url | https://arxiv.org/abs/2402.03536 |