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

TL;DR

This paper explores quadratic scalar curvatures and the $h_4$ Yamabe equation, establishing formulas, positivity properties, and characterizations of special metrics on closed Riemannian manifolds of dimension at least 4.

Contribution

It introduces a simple formula relating $h_4$ and $\sigma_2$ curvatures, studies their positivity, and characterizes special metrics using these curvatures, extending Newton transformations and identities.

Findings

Established a formula linking $h_4$ and $\sigma_2$ curvatures.

Analyzed positivity properties of quadratic curvatures.

Characterized space forms, Einstein, and conformally flat metrics.

Abstract

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.