Weyl structures with special holonomy on compact conformal manifolds

Início: 08/03/2024 17:00

Término: 08/03/2024 18:00

Palestrante: Andrei Moroianu (Paris-Saclay/CNRS)

E-mail do Palestrante: andrei.moroianu@math.cnrs.fr

Resumo: 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. This is joint work with Florin Belgun and Brice Flamencourt.