Weyl structures with special holonomy on compact conformal manifolds
Florin Belgun, Brice Flamencourt, Andrei Moroianu
TL;DR
The paper addresses the problem of classifying compact conformal manifolds $(M,[g])$ carrying a closed non-exact Weyl connection $ abla$ when both $ abla$ and the Levi-Civita connection $ abla^g$ have special holonomy. By lifting to the universal cover and applying holonomy theory, the authors separate into irreducible and locally conformally product (LCP) cases. In the irreducible case, they show that $(M,[g], abla)$ is either Vaisman or a mapping torus of an isometry of a compact nearly Kähler or nearly parallel $ ext{G}_2$ manifold; in the LCP/reducible case, they prove that $g$ is not Kähler nor Einstein and provide a local classification in terms of adapted metrics, with the universal cover $ ilde{M}$ splitting as products involving $R^q$ and an incomplete irreducible factor. They further show that LCP structures cannot occur on compact Kähler or Einstein manifolds. The results thus yield a near-complete picture of how constrained conformal Weyl geometries with two special holonomies can be, linking Vaisman, nearly Kähler, and G2 geometries to explicit product and cone structures on universal covers.
Abstract
We consider compact conformal manifolds $(M,[g])$ endowed with a closed Weyl structure $\nabla$, 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 metrics.