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The subject of investigations are the almost hypercomplex manifolds withHermitian and anti-Hermitian (Norden) metrics. A linear connection D isintroduced such that the structure of these manifolds is parallel with respectto D and its torsion is totally skew-symmetric. The class of the nearly Kaehlermanifolds with respect to the first almost complex structure is of specialinterest. It is proved that D has a D-parallel torsion and is weak if it is notflat. Some curvature properties of these manifolds are studied.

A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics