On special almost geodesic mappings of type $\pi_1$ of spaces with affine connection

Berezovsky, Vladimir ; Mikeš, Josef

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 43 (2004) no. 1, pp. 21-26.

Résumé

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.

Détail

Classification : 53B05, 53B99
Keywords: almost geodesic mappings; affine connection space