On the G2G2 bundle of a Riemannian 4-manifold

We study the natural G2G2 structure on the unit tangent sphere bundle SMSM of any given orientable Riemannian 4-manifold MM, as was discovered in Albuquerque and Salavessa (2009,2010) [9] and [10]. 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 G2G2 structure. This article is devoted to finding the G2G2 torsion tensors which classify our structure according to the theory in Fernandez and Gray (1982) [4].