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 A. Swann, we give explicit descriptions of the tensors contained in the space of covariant derivatives of the fundamental form $\Omega$ and split the coderivative of $\Omega$ into its ${\sl Sp}(n){\sl Sp}(1)$-components. For $4n>8$, A. Swann also proved that all the information about the intrinsic torsion $\nabla \Omega$ is contained in the exterior derivative ${\sl d} \Omega$. Thus, we give alternative conditions, expressed in terms of ${\sl d} \Omega$, to characterize the different classes of almost quaternion-Hermitian manifolds.