Weyl structures with special holonomy on compact conformal manifolds

2023/05/11 by Florin Belgun, Belgun, Florin, Brice Flamencourt +3 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2305.06637

openalex publication_date 2023/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider compact conformal manifolds (M,[g]) endowed with a closed Weyl structure ∇, i.e. a torsion-free connection preserving the conformal structure, which is locally but not globally the Levi-Civita connection of a metric in [g]. Our aim is to classify all such structures when both ∇ and ∇^g, the Levi-Civita connection of g, have special holonomy. In such a setting, (M,[g],∇) is either flat, or irreducible, or carries a locally conformally product (LCP) structure. Since the flat case is already completely classified, we focus on the last two cases. When ∇ has irreducible holonomy we prove that (M,g) is either Vaisman, or a mapping torus of an isometry of a compact nearly Kähler or nearly parallel G₂ manifold, while in the LCP case we prove that g is neither Kähler nor Einstein, thus reducible by the Berger-Simons Theorem, and we obtain the local classification of such structures in terms of adapted metrics.

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• Reducible Riemannian manifolds with conformal product structures

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