  We investigate rigidity and stability properties of critical points of
quadratic curvature functionals on the space of Riemannian metrics. We show it
is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion,
to become elliptic. Fredholm theory may then be used to describe local
properties of the moduli space of critical metrics. We show a number of compact
examples are infinitesimally rigid, and consequently, are isolated critical
points in the space of unit-volume Riemannian metrics. We then give examples of
critical metrics which are strict local minimizers (up to diffeomorphism and
scaling). A corollary is a local "reverse Bishop's inequality" for such
metrics. In particular, any metric $g$ in a $C^{2,\alpha}$-neighborhood of the
round metric $(S^n,g_S)$ satisfying $Ric(g) \leq Ric(g_S)$ has volume $Vol(g)
\geq Vol(g_S)$, with equality holding if and only if $g$ is isometric to $g_S$.
