A note on flat metric connections with antisymmetric torsion

TL;DR

This paper revisits flat metric connections with antisymmetric torsion, providing a new proof that spaces of this type split into simple Lie groups or special $S^7$ connections, with implications for $G_2$-structures.

Contribution

It offers a new proof for the classification of flat metric connections with antisymmetric torsion, independent of symmetric space classification.

Findings

Spaces split into simple Lie groups or special $S^7$ connections.

The special $S^7$ connection relates to $G_2$-structures.

Provides analysis of flat metric connections of vectorial type.

Abstract

In this short note we study flat metric connections with antisymmetric torsion $T \neq 0$. The result has been originally discovered by Cartan/Schouten in 1926 and we provide a new proof not depending on the classification of symmetric spaces. Any space of that type splits and the irreducible factors are compact simple Lie group or a special connection on $S^7$. The latter case is interesting from the viewpoint of $G_2$-structures and we discuss its type in the sense of the Fernandez-Gray classification. Moreover, we investigate flat metric connections of vectorial type.