Canonical almost geodesic mappings of the first type of manifolds with affine connection

In this paper we study special cases of canonical almost geodesic mappings of the first type of manifolds with affine connection. We reduce basic equations of these mappings to a closed Cauchy-type system with respect to covariant derivatives. We obtain estimates for essential parameters that affect the general solution. We give examples of such mappings of some flat space onto another one. Such mappings transform straight lines in one space into parabolas in another one. The mentioned almost geodesic mappings of the first type differ from almost geodesic mappings of the second and third types.