  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 $G_2$.
