Betti numbers and the curvature operator of the second kind
Jan Nienhaus, Peter Petersen, Matthias Wink
Abstract
Abstract We show that compact, ‐dimensional Riemannian manifolds with ‐nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the th Betti number provided that the curvature operator of the second kind is ‐positive. Our curvature conditions become weaker as increases. For , we have , and for , we exhibit a ‐positive algebraic curvature operator of the second kind with negative Ricci curvatures.
Topics & Concepts
Betti numberMathematicsScalar curvatureCurvatureRicci curvaturePure mathematicsSectional curvatureOperator (biology)Mathematical analysisCurvature of Riemannian manifoldsRiemann curvature tensorAlgebraic numberHomology (biology)GeometryGeneRepressorTranscription factorBiochemistryChemistryGeometric Analysis and Curvature FlowsGeometry and complex manifoldsPoint processes and geometric inequalities