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Abstract
The article is devoted to the study of a special type of diffeomorphisms of pseudo-Riemannian spaces with an affinor structure. In [4] we studied mappings of pseudo-Riemannian spaces, which are quasi-geodesic [2] and at the same time almost geodesic of the second type [3]. By definition, for a quasi-geodesic mapping corresponding to the affinor $F^h_i$, each geodesic curve in the space $(V_n, g_{ij})$ is mapped onto the so-called quasi-geodesic curve in another space $(\overline{V}_n, \overline{g}_{ij}, F^h_i)$. In [4], [8] it was assumed that the quasi-geodesic mapping $V_n$ onto $\overline{V}_n$ satisfies the condition of reciprocity, i.e. the inverse mapping is also quasi-geodesic, corresponding to the same affinor $F^h_i$. In this case, the affinor satisfies the purely algebraic conditions (consistency with the metric tensors $V_n$ and $\overline{V}_n$). With an almost geodesic mapping of the second type, by definition, each geodesic curve in $(V_n, g_{ij}, F^h_i)$ is mapped onto an almost geodesic curve in $(\overline{V}_n, \overline{g}_{ij})$, if the affinor $F^h_i$ in $V_n$ satisfies a certain differential equation. In \cite {Kurbatova1980} it is proved that the set of the specified algebraic and differential conditions leads to the fact that the affinor $F^h_i$ necessarily determines the $e$-structure in $V_n$, and elliptic and hyperbolic cases are considered. We call an affinor with such conditions a generalized-recurrent structure (and $V_n$ with such a structure, respectively, a generalized-recurrent space).
In what follows, we study quasi-geodesic mappings of parabolic type generalized-recurrent spaces. In this paper, we obtain the properties of the Riemannian tensor of generalized-recurrence space associated with the generalized-recurrence vector. It is proved that the class of pseudo-Riemannian spaces with generalized-recurrent structure of parabolic type is closed with respect to the considered mappings, but the vectors of generalized recurrence of spaces $V_n$ and $\overline{V}_n$ may differ. If the generalized recurrence vector is gradient, there is a $K$-structure in the generalized- recurrence space. It is proved that if $K$-space admits a quasi-geodesic mapping, which preserves an integrable parabolic type $K$-structure , then this $K$-structure is Kähler (note that an integrable $K$-structure of parabolic type may not be Kähler). The structure of the Riemannian tensor of parabolic type generalized-recurrent space , which admits quasi-geodesic mapping onto a flat space, is found. The components of the metric tensor of such a space in a special coordinate system are given.
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