Transversely Kähler almost contact metric Lie algebras
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911592086503424 |
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| author | Dileo, Giulia Poyraz, Deniz Şahin, Bayram |
| author_facet | Dileo, Giulia Poyraz, Deniz Şahin, Bayram |
| contents | We study transversely Kähler almost contact metric Lie algebras $(\mathfrak{g},φ,ξ,η,g)$ such that the structure $1$-form $η$ is a contact form. They include both quasi Sasakian and anti-quasi-Sasakian Lie algebras of maximal rank. In the case where the center of the Lie algebra is nontrivial, they are $1$-dimensional central extensions of Kähler Lie algebras via a symplectic form.
We investigate the $5$-dimensional case, obtaining a classification of $η$-Einstein transversely Kähler almost contact metric Lie algebras of maximal rank. If the center is trivial, the structure is always $α$-Sasakian. If the center is nontrivial and the Kähler quotient $\mathfrak{g}/\mathfrak{z(g)}$ is not abelian, the structure is quasi Sasakian; it is $α$-Sasakian on central extensions of Kähler-Einstein $4$-dimensional Lie algebras, and not conversely. Up to isomorphisms, the Heisenberg Lie algebra $\mathfrak{h}_5$ is the only $5$-dimensional Lie algebra admitting $η$-Einstein transversely Kähler structures which are not quasi Sasakian, including anti-quasi-Sasakian structures. In fact, we show that any $5$-dimensional anti-quasi-Sasakian Lie algebra is isomorphic to $\mathfrak{h}_5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12538 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transversely Kähler almost contact metric Lie algebras Dileo, Giulia Poyraz, Deniz Şahin, Bayram Differential Geometry 53C25, 53C15, 53D15 We study transversely Kähler almost contact metric Lie algebras $(\mathfrak{g},φ,ξ,η,g)$ such that the structure $1$-form $η$ is a contact form. They include both quasi Sasakian and anti-quasi-Sasakian Lie algebras of maximal rank. In the case where the center of the Lie algebra is nontrivial, they are $1$-dimensional central extensions of Kähler Lie algebras via a symplectic form. We investigate the $5$-dimensional case, obtaining a classification of $η$-Einstein transversely Kähler almost contact metric Lie algebras of maximal rank. If the center is trivial, the structure is always $α$-Sasakian. If the center is nontrivial and the Kähler quotient $\mathfrak{g}/\mathfrak{z(g)}$ is not abelian, the structure is quasi Sasakian; it is $α$-Sasakian on central extensions of Kähler-Einstein $4$-dimensional Lie algebras, and not conversely. Up to isomorphisms, the Heisenberg Lie algebra $\mathfrak{h}_5$ is the only $5$-dimensional Lie algebra admitting $η$-Einstein transversely Kähler structures which are not quasi Sasakian, including anti-quasi-Sasakian structures. In fact, we show that any $5$-dimensional anti-quasi-Sasakian Lie algebra is isomorphic to $\mathfrak{h}_5$. |
| title | Transversely Kähler almost contact metric Lie algebras |
| topic | Differential Geometry 53C25, 53C15, 53D15 |
| url | https://arxiv.org/abs/2604.12538 |