On Lorentzian Lie groups of Nonzero Constant Curvature
Abstract
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 SL(2,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 SL(2,ℝ). We also characterize SL(2,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.
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BibTeX
@misc{boucetta2026lorentzian,
title = {{On Lorentzian Lie groups of Nonzero Constant Curvature}},
author = {Mohamed Boucetta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.40017},
eprint = {2609.40017},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40017},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Lorentzian Lie groups of Nonzero Constant Curvature
AU - Mohamed Boucetta
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.40017
UR - https://arxiv.org/abs/2609.40017
AB - 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 SL(2,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 SL(2,ℝ). We also characterize SL(2,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.
ER -