  Connections with (skew-symmetric) torsion on non-symmetric Riemannian
manifold satisfying the Einstein metricity condition (NGT with torsion) are
considered. It is shown that an almost Hermitian manifold is an NGT with
torsion if and only if it is a Nearly K\"ahler manifold. In the case of an
almost contact metric manifold the NGT with torsion spaces are characterized
and a possibly new class of almost contact metric manifolds is extracted.
Similar considerations lead to a definition of a particular classes of almost
para-Hermitian and almost paracontact metric manifolds. The conditions are
given in terms of the corresponding Nijenhuis tensors and the exterior
derivative of the skew-symmetric part of the non-symmetric Riemannian metric.
