Tachibana-type theorems and special Holonomy

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that with divergence free Weyl tensors and $\lfloor \frac{n-1}{2} \rfloor$-nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient sphere. The main focus paper is similar special holonomy. In particular, consider K\"ahler Bochner tensor. quaternion obtain partial result towards LeBrun-Salamon conjecture.