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

About Some Quadratic Scalar Curvatures and the $h_{4}$ Yamabe Equation