Florin Belgun
Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

On a compact conformal manifold, we consider as well a compatible Riemannian metric and a closed, non-exact Weyl connection (which is detemined by a compatible metric on the universal covering, for which the fundmental group acts by homotheties). Our aim is to classify locally the structure of a compact conformal manifold for which both a Riemannian and a closed Weyl connection have special (i.e. non-generic) holonomies. If the Weyl connection has irreducible holonomy, the Riemannian metric turns out to be Vaisman or a mapping torus of an isometry of a Nearly Kähler or a nearly parallel $G2$ metric, while if the Weyl structure is Locally Conformally a Product (LCP), then it turns out that the Riemannian metric is also (locally) a product, and can be described locally as a special kind of double warped product.