Flows of G2-Structures, I

This is a foundational paper on flows of G2-structures. We use local coordinates to describe the four torsion forms of a G2-structure and derive the evolution equations for a general flow of a G2-structure φ on a 7-manifold M. Specifically, we compute the evolution of the metric g, the dual 4-form ψ and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernández–Gray. As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G2-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G2-geometry.