Betti numbers and the curvature operator of the second kind

TL;DR

This paper investigates the geometric implications of curvature operators of the second kind on compact Riemannian manifolds, establishing conditions under which certain Betti numbers vanish or the manifold is flat, thus linking curvature positivity to topological properties.

Contribution

It introduces new curvature positivity conditions related to the curvature operator of the second kind that imply topological restrictions like Betti number vanishing or flatness of the manifold.

Findings

Manifolds with (n+2)/2-nonnegative curvature operators of the second kind are either rational homology spheres or flat.

Vanishing of the p-th Betti number occurs under C(p,n)-positive curvature operator conditions.

Existence of algebraic curvature operators with negative Ricci curvature under certain positivity conditions.

Abstract

We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number provided that the curvature operator of the second kind is $C(p,n)$-positive. Our curvature conditions become weaker as $p$ increases. For $p=\frac{n}{2}$ we have $C(p,n)= \frac{3n}{2} \frac{n+2}{n+4} $, and for $5 \leq p \leq \frac{n}{2}$ we exhibit a $C(p,n)$-positive algebraic curvature operator of the second kind with negative Ricci curvatures.