Special Riemannian geometries and the Magic Square of Lie algebras

TL;DR

This paper explores special Riemannian geometries inspired by symmetric spaces connected to the Magic Square of Lie algebras, extending known structures and analyzing those with compatible skew torsion connections.

Contribution

It extends the classification of Riemannian geometries related to the Magic Square by identifying additional structures with skew torsion.

Findings

Extended the collection of geometries from four to twelve structures.

Identified geometries admitting compatible connections with skew torsion.

Provided insights into the geometric properties linked to Lie algebra symmetries.

Abstract

We investigate nonintegrable Riemannian geometries modelled after certain symmetric spaces related to the Freudenthal-Tits Magic Square. The collection of four such structures found by Nurowski is extended by further eight. A focus is given to those admitting a compatible connection with completely skew torsion.