Curvature of quaternionic skew-Hermitian manifolds and bundle constructions

Abstract

This paper is devoted to a description of the second-order differential geometry of torsion-free almost quaternionic skew-Hermitian manifolds, that is, of quaternionic skew-Hermitian manifolds $(M,Q,\omega )$. We provide a curvature characterization of such integrable geometric structures, based on the holonomy theory of symplectic connections and we study qualitative properties of the induced Ricci tensor. Then, we proceed with bundle constructions over such a manifold $(M,Q,\omega )$. In particular, we prove the existence of almost hypercomplex skew-Hermitian structures on the Swann bundle over M and investigate their integrability.