On pseudo-Riemannian Ricci-parallel Lie groups which are not Einstein
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910417471668224 |
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| author | An, Huihui Yan, Zaili |
| author_facet | An, Huihui Yan, Zaili |
| contents | In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and Bérard Bergery, there are two typical types of such metrics, which are characterized by the minimal polynomial of the Ricci operator. Namely, its form is either $(X-α)(X-\barα)$ (type I), where $α\in \mathbb{C}\setminus \mathbb{R}$, or $X^{2}$ (type II). Firstly, we obtain a complete description of Ricci-parallel metrics of type I. In particular, such a Ricci-parallel metric is uniquely determined by an Einstein metric and an invariant symmetric parallel complex structure up to isometry and scaling. Then we study Ricci-parallel metric Lie algebras of type II by using double extension process. Surprisingly, we find that every double extension of a metric Abelian Lie algebra is Ricci-parallel and the converse holds for Lorentz Ricci-parallel metric nilpotent Lie algebras of type II. Moreover, we construct infinitely many new explicit examples of Ricci-parallel metric Lie algebras which are not Einstein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13797 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On pseudo-Riemannian Ricci-parallel Lie groups which are not Einstein An, Huihui Yan, Zaili Differential Geometry 53C50, 53C25, 22E25 In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and Bérard Bergery, there are two typical types of such metrics, which are characterized by the minimal polynomial of the Ricci operator. Namely, its form is either $(X-α)(X-\barα)$ (type I), where $α\in \mathbb{C}\setminus \mathbb{R}$, or $X^{2}$ (type II). Firstly, we obtain a complete description of Ricci-parallel metrics of type I. In particular, such a Ricci-parallel metric is uniquely determined by an Einstein metric and an invariant symmetric parallel complex structure up to isometry and scaling. Then we study Ricci-parallel metric Lie algebras of type II by using double extension process. Surprisingly, we find that every double extension of a metric Abelian Lie algebra is Ricci-parallel and the converse holds for Lorentz Ricci-parallel metric nilpotent Lie algebras of type II. Moreover, we construct infinitely many new explicit examples of Ricci-parallel metric Lie algebras which are not Einstein. |
| title | On pseudo-Riemannian Ricci-parallel Lie groups which are not Einstein |
| topic | Differential Geometry 53C50, 53C25, 22E25 |
| url | https://arxiv.org/abs/2404.13797 |