Mapping of trajectories of test bodies in the pseudo-Riemann spaces $V_4$, and $\overline V_4$

In general coordination and for identical initial conditions all time-like geodesies of the pseudo-Riemann space $V_4$ are mapped with time-like world lines of the space $\overline V_4$ due to a pseudo-Lorentzian force. The mapping of all time-like world lines of $\overline V_4$ with the geodesies of the space $V_4$, is also considered. Necessary and sufficient conditions for the existence of such mappings are found.