关于非零常曲率的洛伦兹李群
On Lorentzian Lie groups of Nonzero Constant Curvature
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中文总结 AI 辅助
本文研究非零常曲率左不变洛伦兹度量的李群，刻画其李代数并分类低维情形，证明SL(2,R)上度量双不变且完备，半单李群存在此类度量当且仅当局部同构于SL(2,R)。
中文摘要 AI 辅助
我们研究具有非零常截面曲率的左不变洛伦兹度量的李群。首先，我们重新审视Heintze理论，并得到负常曲率黎曼李群李代数的一个刻画。我们还将Nomizu和Barnet的经典结果推广到伪黎曼情形，通过构造一大类具有常截面曲率的不完备左不变度量的李群。然后，我们根据其中心和导出理想的因果性质来描述非零常曲率的洛伦兹李代数。这一描述是完备的，除非导出理想是洛伦兹的，对此我们在四维情形得到了完全分类。我们证明，在SL(2,R)上，每一个非零常曲率的左不变洛伦兹度量都是双不变的且完备的。此外，一个半单李群承认一个完备的非零常曲率左不变洛伦兹度量，当且仅当它局部同构于SL(2,R)。我们还通过非中心类空左不变Killing向量场的存在性来刻画SL(2,R)。最后，我们在至多四维情形下分类了非零常曲率的洛伦兹李代数。
英文摘要
We investigate Lie groups endowed with left-invariant Lorentzian metrics of nonzero constant sectional curvature. We first revisit Heintze's theory and obtain a characterization of the Lie algebras of Riemannian Lie groups of negative constant curvature. We also extend classical results of Nomizu and Barnet to the pseudo-Riemannian setting by constructing a large family of Lie groups carrying incomplete left-invariant metrics of constant sectional curvature. We then describe Lorentzian Lie algebras of nonzero constant curvature according to the causal nature of their center and derived ideal. This description is complete except when the derived ideal is Lorentzian, for which we obtain a complete classification in dimension four. We show that every left-invariant Lorentzian metric of nonzero constant curvature on \(\mathrm{SL}(2,\mathbb R)\) is bi-invariant and complete. Moreover, a semisimple Lie group admits a complete left-invariant Lorentzian metric of nonzero constant curvature if and only if it is locally isomorphic to \(\mathrm{SL}(2,\mathbb R)\). We also characterize \(\mathrm{SL}(2,\mathbb R)\) through the existence of a noncentral spacelike left-invariant Killing vector field. Finally, we classify Lorentzian Lie algebras of nonzero constant curvature in dimensions at most four.
发表机构
- Cadi-Ayyad University（卡迪·阿亚德大学）
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