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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.17292 |
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| _version_ | 1866915997186785280 |
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| author | Boucetta, Mohamed |
| author_facet | Boucetta, Mohamed |
| contents | In this paper, we establish a complete structural description of flat Lorentzian Lie groups, i.e., Lie groups endowed with a flat left invariant Lorentzian metric, thereby resolving a long-standing open problem in the theory of pseudo-Riemannian Lie groups. Our main result shows that any flat Lorentzian Lie group either admits a timelike parallel left-invariant vector field or is of Kundt type, and that in both cases the underlying Lie algebra falls into one of six explicit classes. A key ingredient of the proof is a refined analysis of the double extension process, which reveals that all flat Lorentzian Lie algebras arise - directly or in a generalized sense - from flat Euclidean ones. As a consequence, we obtain easily a complete classification in dimensions three and four, recovering and unifying several previously known partial results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17292 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Flat Lorentzian Lie groups: A complete description Boucetta, Mohamed Differential Geometry In this paper, we establish a complete structural description of flat Lorentzian Lie groups, i.e., Lie groups endowed with a flat left invariant Lorentzian metric, thereby resolving a long-standing open problem in the theory of pseudo-Riemannian Lie groups. Our main result shows that any flat Lorentzian Lie group either admits a timelike parallel left-invariant vector field or is of Kundt type, and that in both cases the underlying Lie algebra falls into one of six explicit classes. A key ingredient of the proof is a refined analysis of the double extension process, which reveals that all flat Lorentzian Lie algebras arise - directly or in a generalized sense - from flat Euclidean ones. As a consequence, we obtain easily a complete classification in dimensions three and four, recovering and unifying several previously known partial results. |
| title | Flat Lorentzian Lie groups: A complete description |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2604.17292 |