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

arXiv comment22 pages, references updated

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Cited by in corpus (2)

• Remarks on the Quadratic Orthogonal Bisectional Curvature
• Curvature operators and rational cobordism