Manifolds with positive second H. Weyl curvature invariant

The second H. Weyl curvature invariant of a Riemannian manifold, denoted $h_4$, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of $h_4$ is that it is nonnegative for Einstein manifolds, hence it provides a geometric obstruction to the existence of Einstein metrics in dimensions $\ge 4$, independently from the sign of the Einstein constant. This motivates our study of the positivity of this invariant. Here in this paper, we prove many constructions of metrics with positive second H. Weyl curvature invariant, generalizing similar well known results for the scalar curvature.