  In this work we are interested in studying deformations of the
$\sigma_2$-curvature and the volume. For closed manifolds, we relate critical
points of the total $\sigma_2$-curvature functional to the $\sigma_2$-Einstein
metrics and, as a consequence of results of H. J. Gursky and J. A. Viaclovsky
(2001) and Z. Hu and H. Li (2004), we obtain a sufficient and necessary
condition for a critical metric to be Einstein. Moreover, we show a volume
comparison result for Einstein manifolds with respect to $\sigma_2$-curvature
which shows that the volume can be controlled by the $\sigma_2$-curvature under
certain conditions. Next, for compact manifold with nonempty boundary, we study
variational properties of the volume functional restricted to the space of
metrics with constant $\sigma_2$-curvature and with fixed induced metric on the
boundary. We characterize the critical points to this functional as the
solutions of an equation and show that in space forms they are geodesic balls.
Studying second order properties of the volume functional we show that there is
a variation for which geodesic balls are indeed local minimum in a natural
direction.
