New Curvature Conditions for the Bochner Technique

TL;DR

This paper establishes new curvature conditions that lead to vanishing theorems for Betti numbers on closed Riemannian manifolds, generalizing previous results and highlighting limitations of Ricci flow preservation.

Contribution

It introduces generalized curvature conditions that ensure Betti number vanishing, extending classical theorems and addressing the non-preservation of certain positivity conditions under Ricci flow.

Findings

Vanishing of Betti numbers under new curvature bounds

Extension of classical results to broader curvature conditions

Identification of limitations of Ricci flow in preserving curvature positivity

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 B\"ohm-Wilking observed, $3$-positivity of the curvature operator is not preserved by the Ricci flow.