2024/04/27 by Viviana del Barco, Andrei Moroianu, del Barco, Viviana +1 · 2 citations
Mathematics · #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.17956
openalex publication_date 2024/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfrakg can be characterized in terms of a closed 1-form θ∈\mathfrakg^* and a non-zero subspace \mathfraku⊂ \mathfrakg satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that \mathfrakg 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.