Special almost Hermitian geometry

Abstract

We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results to almost hyperhermitian geometry. Thus, we show that the exterior derivatives of the three K{\"a}hler forms of an almost hyperhermitian manifold are sufficient to determine the three covariant derivatives of such forms, i.e., the three mentioned exterior derivatives determine the intrinsic torsion of the ${\sl Sp}(n)$-structure.