The structure of locally conformally product Lie algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909183748603904 |
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| author | del Barco, Viviana Moroianu, Andrei |
| author_facet | del Barco, Viviana Moroianu, Andrei |
| contents | A locally conformally product (LCP) structure on a compact conformal manifold is a closed non-exact Weyl connection (i.e.~a linear connection which is locally but not globally the Levi-Civita connection of Riemannian metrics in the conformal class), with reducible holonomy. A left-invariant LCP structure on a compact quotient $Γ\backslash G$ of a simply connected Riemannian Lie group $(G,g)$ with Lie algebra $\mathfrak{g}$ can be characterized in terms of a closed 1-form $θ\in\mathfrak{g}^*$ and a non-zero subspace $\mathfrak{u}\subset \mathfrak{g}$ satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that $\mathfrak{g}$ is isomorphic to a semidirect product of a non-unimodular Lie algebra acting on an abelian one by a conformal representation. This extends to the general case previous results holding for solvmanifolds. In addition, we construct explicit examples of compact LCP manifolds which are not solvmanifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The structure of locally conformally product Lie algebras del Barco, Viviana Moroianu, Andrei Differential Geometry A locally conformally product (LCP) structure on a compact conformal manifold is a closed non-exact Weyl connection (i.e.~a linear connection which is locally but not globally the Levi-Civita connection of Riemannian metrics in the conformal class), with reducible holonomy. A left-invariant LCP structure on a compact quotient $Γ\backslash G$ of a simply connected Riemannian Lie group $(G,g)$ with Lie algebra $\mathfrak{g}$ can be characterized in terms of a closed 1-form $θ\in\mathfrak{g}^*$ and a non-zero subspace $\mathfrak{u}\subset \mathfrak{g}$ satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that $\mathfrak{g}$ is isomorphic to a semidirect product of a non-unimodular Lie algebra acting on an abelian one by a conformal representation. This extends to the general case previous results holding for solvmanifolds. In addition, we construct explicit examples of compact LCP manifolds which are not solvmanifolds. |
| title | The structure of locally conformally product Lie algebras |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2404.17956 |