Weyl connections with special holonomy on compact conformal manifolds

Abstract

We consider compact conformal manifolds $(M,[g])$ endowed with a closed Weyl connection $\nabla$, 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 $\nabla$ and ${\nabla }^g$, the Levi-Civita connection of $g$, have special holonomy. In such a setting, $(M,[g],\nabla )$ 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 $\nabla$ 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 ${\mathrm{G}}_2$ 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 LCP metrics.