On the G_2 bundle of a Riemannian 4-manifold

TL;DR

This paper investigates the G_2 structure on the unit tangent sphere bundle of a Riemannian 4-manifold, analyzing torsion components to classify the structure within the framework of G_2 geometry.

Contribution

It introduces a new perspective on the G_2 structure on sphere bundles and characterizes torsion components relevant for classification in G_2 geometry.

Findings

Identification of torsion components that influence G_2 structure equations

Classification of G_2 structures based on torsion tensors

Extension of G_2 geometry to tangent sphere bundles of 4-manifolds

Abstract

We study the natural G_2 structure on the unit tangent sphere bundle SM of any given orientable Riemannian 4-manifold M, as it was discovered in \cite{AlbSal}. A name is proposed for the space. We work in the context of metric connections, or so called geometry with torsion, and describe the components of the torsion of the connection which imply certain equations of the G_2 structure. This article is devoted to finding the G_2-torsion tensors which classify our structure according to the theory in \cite{FerGray}.