Manifolds with positive second Gauss-Bonnet curvature

• Autores: Mohhamed Larbi Labbi
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 227, Nº 2, 2006, págs. 295-310
• Idioma: inglés
• DOI: 10.2140/pjm.2006.227.295
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• Resumen

  • 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.