Weyl ’ s spaces with shear - free and expansion - free conformal Killing vectors and the motion of a free spinless test particle

Weyl's spaces with shear-free and expansion-free conformal Killing vectors and the motion of a free spinless test particle. Abstract Conditions for the existence of shear-free and expansion-free non-null vector fields in spaces with affine connections and metrics are found. On their basis Weyl's spaces with shear-free and expansion-free conformal Killing vectors are considered. The necessary and sufficient conditions are found under which a free spinless test particle could move in spaces with affine connections and metrics on a curve described by means of an auto-parallel equation. In Weyl's spaces with Weyl's covector, constructed by the use of a dilaton field, the dilaton field appears as a scaling factor for the rest mass density of the test particle.