AbstractWe prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence-free Weyl tensors and "Equation missing"-nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider Kähler manifolds with divergence-free Bochner tensors. For quaternion Kähler manifolds, we obtain a partial result towards the LeBrun–Salamon conjecture.

Abstract We show that compact Kähler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the Kähler curvature operator is positive. This follows from a more general vanishing and estimation theorem for the individual Hodge numbers. We also prove an analogue of Tachibana’s theorem for Kähler manifolds.

Abstract: We show that compact, ‐dimensional Riemannian manifolds with ‐nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the th Betti number provided that the curvature operator of the second kind is ‐positive. Our curvature conditions become weaker as increases. For , we have , and for , we exhibit a ‐positive algebraic curvature operator of the second kind with negative Ricci curvatures.