  This article consists of some loosely related remarks about the geometry of
G_2-structures on 7-manifolds and is partly based on old unpublished joint work
with two other people: F. Reese Harvey and Steven Altschuler. Much of this work
has since been subsumed in the work of Hitchin \cite{MR02m:53070} and Joyce
\cite{MR01k:53093}. I am making it available now mainly because of interest
expressed by others in seeing these results written up since they do not seem
to have all made it into the literature.
  A formula is derived for the scalar curvature and Ricci curvature of a
G_2-structure in terms of its torsion. When the fundamental 3-form of the
G_2-structure is closed, this formula implies, in particular, that the scalar
curvature of the underlying metric is nonpositive and vanishes if and only if
the structure is torsion-free. This version contains some new results on the
pinching of Ricci curvature for metrics associated to closed G_2-structures.
  Some formulae are derived for closed solutions of the Laplacian flow that
specify how various related quantities, such as the torsion and the metric,
evolve with the flow. These may be useful in studying convergence or long-time
existence for given initial data.
