On the G₂ bundle of a Riemannian 4-manifold

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}.