Geodesic mappings and their generalizations
Diffeomorphisms and automorphisms of geometrically generalized spaces constitute one of the contemporary actual directions in differential geometry. A large number of works is devoted to geodesic, quasigeodesic, holomorphically projective, almost geodesic, F-planar
and other mappings, transformations and deformations.
This lecture is dedicated to some results concerning the fundamental equations of these mappings and deformations. Obviously the existence of a solution of these fundamental equations imply the existence of the mentioned mappings, transformations and deformations.
These fundamental equations were found in several forms.
Among these forms there is the particularly important form of a system of differential equations of Cauchy type. For their linear forms the question of solvability can be answered by algebraic methods.