Manifolds with positive second H. Weyl curvature invariant

2004/03/17 by Mohammed-Larbi Labbi, Labbi, M. -L.
Mathematics · Physics and Astronomy · #53B20 #53C21 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.math/0403286

openalex publication_date 2004/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The second H. Weyl curvature invariant of a Riemannian manifold, denoted h₄, 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 h₄ is that it is nonnegative for Einstein manifolds, hence it provides a geometric obstruction to the existence of Einstein metrics in dimensions ≥ 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.

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