A Berger-type theorem for metric connections with skew-symmetric torsion

TL;DR

This paper establishes a Berger-type theorem characterizing metric connections with skew-symmetric torsion, linking the torsion's properties to the geometric structure of the space, and refines classical results by Cartan and Schouten.

Contribution

It proves a Berger-type theorem for metric connections with skew-symmetric torsion, classifies flat canonical connections on Lie groups, and refines classical holonomy results.

Findings

Spaces with non-transitive torsion subgroup are locally isometric to Lie groups with bi-invariant metrics.

A simple Lie group admits only two flat metric connections with skew-symmetric torsion.

The holonomy group of such connections generally matches the Riemannian holonomy.

Abstract

We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. In particular, we get a refinement of a well-known theorem by Cartan and Schouten. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy.