Proceedings of the International Geometry Center

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Квазі-геодезичні відображення спеціальних псевдоріманових просторів

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Irina Kurbatova
М. І. Піструіл

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.

Keywords:
affinor structure, quasi-geodesic maps

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How to Cite
Kurbatova, I., & Піструіл, М. (2020). Квазі-геодезичні відображення спеціальних псевдоріманових просторів. Proceedings of the International Geometry Center, 13(3), 18-32. https://doi.org/10.15673/tmgc.v13i3.1770
Section
Papers
Author Biographies

Irina Kurbatova, Odessa I.I.Mechnikov National University

Odesa I. I. Mechnikov National University

М. І. Піструіл, Odesa I. I. Mechnikov National University

Odesa I. I. Mechnikov National University