Almost quaternion-Hermitian manifolds

Following the point of view of Gray and Hervella, we derive detailed conditions which characterize each one of the classes of almost quaternion-Hermitian 4n-manifolds, n > 1. Previously, by completing a basic result of Swann, we give explicit descriptions of the tensors contained in the space of covariant derivatives of the fundamental form Ohm and split the coderivative of Ohm into its Sp(n) Sp(1)-components. For 4n > 8, Swann also proved that all the information about the intrinsic torsion delOhm is contained in the exterior derivative dOhm. Thus, we give alternative conditions, expressed in terms of dOhm, to characterize the different classes of almost quaternion-Hermitian manifolds.