On the irreducibility of locally metric connections

TL;DR

This paper investigates the conditions under which locally metric connections on compact manifolds have irreducible holonomy, extending known results from Riemannian geometry to a broader class of connections and providing geometric classifications.

Contribution

It generalizes Gallot's theorem on Riemannian cones to locally metric connections with conformal structures, under natural geometric assumptions.

Findings

Irreducible holonomy implies preservation of a conformal structure.

Connections with reducible holonomy are either trivial or Levi-Civita of a Riemannian metric.

Provides a geometric classification of certain conformal manifolds with Weyl connections.

Abstract

A locally metric connection on a smooth manifold $M$ is a torsion-free connection $D$ on $TM$ with compact restricted holonomy group $\mathrm{Hol}_0(D)$. 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 $\mathrm{Hol}_0(D)=0$ 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.