In this paper we define generalized Kählerian spaces of the first kind (\( G_1^K N \)) given by (2.1)–(2.3). For them we consider hollomorphically projective mappings with invariant complex structure. Also, we consider equitorsion geodesic mapping between these two spaces (\( G_1^K N \) and \( G_1^{\bar K} N \)) and for them we find invariant geometric objects.

It was shown in [15, 16] that there does not exist any warped product submanifold of a Kaehler manifold such that the spherical manifold of the warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function. We show that there exists a class of nontrivial warped product submanifolds of a Kaehler manifold such that the spherical manifold is pointwise slant by giving an example and a characterization theorem. We also prove that if the warped product is mixed totally geodesic then the warping function is constant.

In this paper, we study warped product bi-slant submanifolds of cosymplectic manifolds. It is shown that there is no proper warped product bi-slant submanifold other than pseudo-slant warped product. Finally, we give an example of warped product pseudo-slant submanifolds.

We study G-almost geodesic mappings of the second type \(\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2\), θ = 1,2 between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider e-structures that generate mappings of type \(\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2\), θ = 1,2. For a mapping \(\mathop {{\pi _2}}\limits_\theta (e,F),\theta = 1,2\), θ = 1,2, we determine the basic equations which generate them.