Integral Curvature Bounds and Betti Numbers

TL;DR

This paper establishes new upper bounds on Betti numbers of compact Riemannian manifolds using integral curvature bounds and introduces a curvature condition for Betti number vanishing, extending previous results.

Contribution

It provides a novel integral curvature bound approach and a new curvature condition for Betti number vanishing, generalizing earlier theorems.

Findings

Betti numbers are bounded by integral curvature conditions.

A new curvature condition ensures Betti number vanishing.

Generalizes results from Gallot and Petersen-Wink.

Abstract

We introduce an upper bound of the Betti numbers of a compact Riemannian manifold given integral bounds on the average of the lowest eigenvalues of the curvature operator. We then establish a new curvature condition for the Betti numbers to vanish using the Bochner technique. This generalizes results from Gallot and Petersen-Wink.