New Curvature Conditions for the Bochner Technique

Abstract

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5,6$ we obtain vanishing of the Betti numbers provided that the curvature operator is $3$-positive. As Böhm-Wilking observed, $3$-positivity of the curvature operator is not preserved by the Ricci flow.

An edited version to appear in Invent. Math. Corollary 3.3 and Proposition 3.5 to be published elsewhere

References in corpus (1)

• Compact manifolds of dimension $n\ge 12$ with positive isotropic curvature

Cited by in corpus (7)

• Betti numbers and the curvature operator of the second kind
• Estimation and Vanishing Results for Hodge numbers
• Tachibana-type Theorems and special Holonomy
• The Bochner Technique and Weighted Curvatures
• Remarks on the Quadratic Orthogonal Bisectional Curvature
• Curvature operators and rational cobordism
• Curvature operator and Euler number