Torsion-free connections on $G$-structures

TL;DR

The paper proves that for certain groups G, any G-structure on a smooth manifold admits a torsion-free connection compatible with a conformal class of Riemannian metrics, and classifies the admissible groups.

Contribution

It establishes the existence of torsion-free connections on G-structures within a specified group class and classifies the groups for which this is possible.

Findings

Existence of torsion-free connections on G-structures.

Classification of admissible groups G.

Connections are locally Levi-Civita of conformal metrics.

Abstract

We prove that for a group $\mathrm{SO}_n(\mathrm{R}) \subset G \subset \mathrm{GL}_n (\mathrm{R})$, 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.