Left-invariant Codazzi tensors and harmonic curvature on Lie groups endowed with a left invariant Lorentzian metric
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| Format: | Preprint |
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2024
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| _version_ | 1866929256686157824 |
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| author | Aberaouze, Ilyes Boucetta, Mohamed |
| author_facet | Aberaouze, Ilyes Boucetta, Mohamed |
| contents | A Lorentzian Lie group is a Lie group endowed with a left invariant Lorentzian metric. We study left-invariant Codazzi tensors on Lorentzian Lie groups. We obtain new results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator. In contrast to the Riemannian case, the Ricci operator of a let-invariant Lorentzian metric can be of four types: diagonal, of type $\{n-2,z\bar{z}\}$, of type $\{n,a2\}$ and of type $\{n,a3\}$. We first describe Lorentzian Lie algebras with a non-diagonal Codazzi operator and with these descriptions in mind, we study three classes of Lorentzian Lie groups with harmonic curvature. Namely, we give a complete description of the Lie algebra of Lorentzian Lie groups having harmonic curvature and where the Ricci operator is non-diagonal and its diagonal part consists of one real eigenvalue $α$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Left-invariant Codazzi tensors and harmonic curvature on Lie groups endowed with a left invariant Lorentzian metric Aberaouze, Ilyes Boucetta, Mohamed Differential Geometry A Lorentzian Lie group is a Lie group endowed with a left invariant Lorentzian metric. We study left-invariant Codazzi tensors on Lorentzian Lie groups. We obtain new results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator. In contrast to the Riemannian case, the Ricci operator of a let-invariant Lorentzian metric can be of four types: diagonal, of type $\{n-2,z\bar{z}\}$, of type $\{n,a2\}$ and of type $\{n,a3\}$. We first describe Lorentzian Lie algebras with a non-diagonal Codazzi operator and with these descriptions in mind, we study three classes of Lorentzian Lie groups with harmonic curvature. Namely, we give a complete description of the Lie algebra of Lorentzian Lie groups having harmonic curvature and where the Ricci operator is non-diagonal and its diagonal part consists of one real eigenvalue $α$. |
| title | Left-invariant Codazzi tensors and harmonic curvature on Lie groups endowed with a left invariant Lorentzian metric |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.16381 |