Torsion-free connections on G-structures

Abstract

We prove that for a Lie group ${\mathrm{SO}}_n\left(R\right)\subset G\subset {\mathrm{GL}}_n\left(R\right)$, any G-structure on a smooth manifold can be endowed with a torsion free connection which is locally the Levi-Civita connection of a Riemannian metric in a given conformal class. In this process, we classify the admissible groups.