A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds to the hyper-Kähler manifolds of the investigated type and the case when the torsion of D is D-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.