Manifolds with positive second gauss-bonnet curvature

The second Gauss-Bonnet curvature of a Riemannian manifold, denoted h(4), is a generalization of the four-dimensional Gauss-Bonnet integrand to higher dimensions. It coincides with the second curvature invariant, which appears in the well known Weyl's tube formula. A crucial property of h(4) is that it is nonnegative for Einstein manifolds; hence it provides, independently of the sign of the Einstein constant, a geometric obstruction to the existence of Einstein metrics in dimensions >= 4. This motivates our study of the positivity of this invariant. We show that positive sectional curvature implies the positivity of h(4), and so does positive isotropic curvature in dimensions >= 8. Also, we prove many constructions of metrics with positive second Gauss-Bonnet curvature that generalize similar well known results for the scalar curvature.