  This is a paper based on a talk given at the conference on Conformal Geometry
which held at Roscoff in France in the 2008 summer. We study some aspects of
the equation arising from the problem of the existence on a given closed
Riemannian manifold of dimension at leat 4, of a conformal metric with constant
$h_4$ curvature. We establish a simple formula relating the second Gauss-Bonnet
curvature $h_4$ to the $\sigma_2$ curvature and we study some positivity
properties of these two quadratic curvatures. We use different quadratic
curvatures to characterize space forms, Einstein metrics and conformally flat
metrics. In the appendix we introduce natural generalizations of Newton
transformations, the corresponding Newton identities are used to obtain Avez
type formulas for all the Gauss-Bonnet curvatures.
