Explicit formulas for the $G_2$-components of the Riemannian curvature tensor on a manifold with a $G_2$ structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact $G_2$ manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a $G_2$ structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed $G_2$ manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one $G_2$ manifolds. As a consequence every Fern\'andez-Gray type of $G_2$ structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in $G_2$.

The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex and symplectic form are isomorphic. This provides a class of examples of "weak mirror symmetry" as suggested by Merkulov. For nilpotent algebras in dimension 4 and 6 the isomorphism classes of the semi-direct products g+g and g+g^* are listed. A one-parameter family of inequivalent pseudo-K\"ahler structures is given.