Tachibana-type Theorems and special Holonomy

TL;DR

This paper establishes rigidity results for compact Riemannian manifolds, especially those with special holonomy, showing conditions under which they are locally symmetric or conformally equivalent to spheres.

Contribution

It extends Tachibana-type theorems to manifolds with special holonomy, including results on Kähler and quaternion Kähler manifolds with divergence-free tensors.

Findings

Manifolds with divergence-free Weyl tensors and nonnegative curvature operators are locally symmetric or sphere quotients.

Results for Kähler manifolds with divergence-free Bochner tensor.

Partial progress on the LeBrun-Salamon conjecture for quaternion Kähler manifolds.

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 $\lfloor \frac{n-1}{2} \rfloor$-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 K\"ahler manifolds with divergence free Bochner tensor. For quaternion K\"ahler manifolds we obtain a partial result towards the LeBrun-Salamon conjecture.