Abstract

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

Mathematical Subject Classification

Primary: 53C21, 53B20

Authors