On the irreducibility of locally metric connections

A locally metric connection on a smooth manifold $M$ is a torsion-free connection $D$ on $TM$ with compact restricted holonomy group ${\mathrm{H}\mathrm{o}\mathrm{l}}_0(D)$. If the holonomy representation of such a connection is irreducible, then $D$ preserves a conformal structure on $M$. Under some natural geometric assumption on the life-time of incomplete geodesics, we prove that conversely, a locally metric connection $D$ preserving a conformal structure on a compact manifold $M$ has irreducible holonomy representation, unless ${\mathrm{H}\mathrm{o}\mathrm{l}}_0(D)=0$ or $D$ is the Levi-Civita connection of a Riemannian metric on $M$. This result generalizes Gallot’s theorem on the irreducibility of Riemannian cones to a much wider class of connections. As an application, we give the geometric description of compact conformal manifolds carrying a tame closed Weyl connection with non-generic holonomy.