Remarks on exact G2-structures on compact manifolds

An important open question related to the study of $G_2$-holonomy manifolds concerns whether or not a compact seven-manifold can support an exact $G_2$-structure. To provide insight into this question, we identify various relationships between the two-form underlying an exact $G_2$-structure, the torsion of the $G_2$-structure, and the curvatures of the associated metric. In addition to establishing identities valid for any hypothetical exact $G_2$-structure, we also consider exact $G_2$-structures subject to additional constraints, for instance proving incompatibility between the exact $G_2$ and Extremally Ricci-Pinched conditions and establish new identities for soliton solutions of the Laplacian flow.