Remarks on Exact G$_{2}$-Structures on Compact Manifolds

Abstract: An important open question in G${2}$ geometry concerns whether or not a compact seven-manifold can support an exact G${2}$-Structure. Given the significance of this question we initiate a study of exact G${2}$-Structures on compact manifolds. We focus on exact G$_2$-Structures subject to no additional constraints but we also consider the relationship between the exact condition and other conditions for closed G${2}$-Structures such as the Extremally Ricci-Pinched and Laplacian Soliton conditions.