The G2 sphere of a 4-manifold

We bring to light a G ₂ structure existing on the unit sphere tangent bundle S _M of any given orientable Riemannian 4-manifold M. The associated 3-form φ is co-calibrated if, and only if, M is an Einstein manifold—a result which leads to new examples of co-calibrated G ₂ spaces. We hope to be contributing both to the knowledge of special geometries and to the study of 4-manifolds.