Connections with (skew-symmetric) torsion on a non-symmetric Riemannian manifold satisfying the Einstein metricity condition (non-symmetric gravitation theory (NGT) with torsion) are considered. It is shown that an almost Hermitian manifold is NGT with torsion if and only if it is a nearly Kähler 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 class of almost para-Hermitian and almost paracontact metric manifolds. 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.

Starting from the definition of generalized Riemannian space (GR N ) [5], in which a non-symmetric basic tensor g ij is introduced, in the present paper a generalized Kählerian space GK 2 N of the second kind is defined, as a GR N with almost complex structure F h i , that is covariantly constant with respect to the second kind of covariant derivative (equation (2.3)).We observe hollomorphically projective mapping of the spaces GK 2 N and GK 2 N with invariant complex structure. Also, we consider equitorsion geodesic mapping between these two spaces, and for them we find invariant geometric objects.

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In this paper we consider concircular vector fields of manifolds with non-symmetric metric tensor. The subject of our paper is an equitorsion concircular mapping. A mapping f : GR N → GR N is an equitorsion if the torsion tensors of the spaces GR N and GR N are equal. For an equitorsion concircular mapping of two generalized Riemannian spaces GR N and GR N , we obtain some invariant curvature tensors of this mapping Z θ , θ = 1, 2,. .. , 5, given by equations (3.14, 3.21, 3.28, 3.31, 3.38). These quantities are generalizations of the concircular tensor Z given by equation (2.5). i jk = ip Γ p. jk respectively, where (i j) = (i j) −1 and i j denotes symmetrization with division of the indices i and j. Generally the generalized Christoffel symbols 2010 Mathematics Subject Classification. Primary 53B05;

-In the papers [19], [20] several Ricci type identities are obtained by using non-symmetric affine connection. In these identities appear 12 curvature tensors, 5 of which being independent [21], while the rest can be expressed as linear combinations of the others.In the general case of a geodesic mapping f of two non-symmetric affine connection spaces GA N and GA N it is impossible to obtain a generalization of the Weyl projective curvature tensor. In the present paper we study the case when GA N and GA N have the same torsion in corresponding points. Such a mapping we name``equitorsion mapping''.With respect to each of mentioned above curvature tensors we have obtained quantities i