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almost geodesic mappings; affine connection space
N.~S.~Sinyukov [5] introduced the concept of an {\em almost geodesic mapping} of a space $A_n$ with an affine connection without torsion onto $\overline{A}_n$ and found three types: $\pi _1$, $\pi _2$ and~$\pi _3$. The authors of [1] proved completness of that classification for $n>5$.\par By definition, special types of mappings $\pi _1$ are characterized by equations $$ P_{ij,k}^h+P_{ij}^\alpha P_{\alpha k}^h =a_{ij} \delta_{k}^h , $$ where $P_{ij}^h\equiv \overline{\Gamma }_{ij}^h-\Gamma _{ij}^h$ is the deformation tensor of affine connections of the spaces $A_n$ and $\overline{A}_n$.\par In this paper geometric objects which preserve these mappings are found and also closed classes of such spaces are described.
[1] Berezovsky V. E., Mikeš J.: On the classification of almost geodesic mappings of affine-connected spaces. In: Proc. Conf., Dubrovnik (Yugoslavia) 1988, 41–48 (1989). MR 1040054
[2] Berezovsky V. E., Mikeš J.: On almost geodesic mappings of the type $\pi _1$ of Riemannian spaces preserving a system $n$-orthogonal hypersurfaces. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 59, 103–108 (1999). MR 1692261
[3] Chernyshenko V. M.: Räume mit einem speziellen Komplex von geodätischen Linien. Tr. Semin. Vektor. Tenzor. Anal. 11 (1961), 253–268 (in Russian). Zbl 0156.41804
[4] Mikeš J.: Holomorphically projective mappings and their generalizations. J. Math. Sci., New York 89, 3 (1998), 1334–1353. MR 1619720 | Zbl 0983.53013
[5] Sinyukov N. S.: On geodesic mappings of Riemannian spaces. : Nauka, Moscow. 1979 (in Russian). MR 0552022
[6] Sinyukov N. S.: Almost geodesic mappings of affine connected and Riemannian spaces. Itogi Nauki Tekh., Ser. Probl. Geom. 13 (1982), 3–26 (in Russian); J. Sov. Math. 25 (1984), 1235–1249. MR 0674123 | Zbl 0498.53010
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