On a special almost geodesic mapping of third type of affine spaces.

N. S. Sinyukov [5] introduced the concept of an of a space $A_{n}$ with an affine connection without torsion onto ${\bar{A}}_{n}$ and found three types: $π_{1}$ , $π_{2}$ and $π_{3}$ . The authors of [1] proved completness of that classification for $n > 5$ .By definition, special types of mappings $π_{1}$ are characterized by equations $P_{i j, k}^{h} + P_{i j}^{α} P_{α k}^{h} = a_{i j} δ_{k}^{h},$ where $P_{i j}^{h} \equiv {\bar{Γ}}_{i j}^{h} - Γ_{i j}^{h}$ is the deformation tensor of affine connections of the spaces $A_{n}$ and ${\bar{A}}_{n}$ .In this paper geometric objects which preserve these mappings are found and also closed classes of such spaces are described. ...

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