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$D$ is introduced such that the structure of these manifolds is parallel with respect to D$D$. Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly constant almost complex structures and the case when the torsion of D$D$ is D$D$-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.