On the G 2 bundle of a Riemannian 4 - manifold

We expose the theory of the construction of a natural G 2 structure on the unit sphere tangent bundle SM of any given orientable Riemannian 4-manifold M , as laid in [4]. This time we work in the context of metric connections, or geometry with torsion. 1 Recalling the theory By a G 2 manifold it is understood a 7 dimensional Riemannian manifold with holonomy group contained in G 2 = Aut(O). The structure is characterized by a smooth 3-form which at each point is given by φ(u, v, w) = uv, w, where uv is the octonionic product, and it is known that there exists a non-degenerate φ if the first two Stiefel-Whitney classes vanish. Besides this topological determination, by general principles of Riemannian geometry, the holonomy is in G 2 if and only if φ is parallel. It was proved the latter differential condition corresponds with φ being harmonic and it is a particularly difficult problem to