New curvature conditions for the Bochner Technique

Abstract

We show that manifolds with $$ \lceil \frac{n}{2} \rceil $$ -positive curvature operators are rational homology spheres. This follows from a more general vanishing and estimation theorem for the pth 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.