Weyl connections with special holonomy on compact conformal manifolds

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.