Tachibana-type theorems and special Holonomy

Abstract

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence-free Weyl tensors and -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.