Almost Hermitian structures and quaternionic geometries

Gray and Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we obtain a number of relations that give hyper-Kahler or locally conformal hyper-Kahler structures, thus generalising a result of Hitchin. We also study the types of almost quaternion-Hermitian geometries that arise and tabulate the results.