The second Gauss–Bonnet
curvature of a Riemannian manifold, denoted h4, 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 h4 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 h4, 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.
Keywords
Gauss–Bonnet curvature, Einstein manifold, surgery
