On types of non-integrable geometries

TL;DR

This paper classifies non-integrable G-structures on Riemannian manifolds, focusing on those with unique connections having totally skew-symmetric torsion, especially in 8-dimensional Spin(7) geometries.

Contribution

It characterizes geometric types with such connections and proves uniqueness under certain conditions, highlighting special properties of Spin(7) structures.

Findings

8-dimensional Spin(7) structures admit a unique connection with skew-symmetric torsion

Such geometries are characterized by specific algebraic conditions

The automorphism group of these structures is analyzed

Abstract

We study the types of non-integrable $\mathrm{G}$-structures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skew-symmetric torsion are characterized. 8-dimensional manifolds equipped with a $\Spin(7)$-structure play a special role. Any geometry of that type admits a unique connection with totally skew-symmetric torsion. Under weak conditions on the structure group we prove that this geometry is the only one with this property. Finally, we discuss the automorphism group of a Riemannian manifold with a fixed non-integrable $\mathrm{G}$-structure.