Weyl connections with special holonomy on compact conformal manifolds

We consider compact conformal manifolds (M,[g]) endowed with a closed Weyl connection ∇ , 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 LCP metrics.