Levi-Civita regularization and geodesic flows for the `curved' Kepler problem

We introduce the regularization Levi-Civita parameter for the `curved Kepler', i.e., motion under the `Kepler-Coulomb' potential in a configuration space with any constant curvature and metric of any signature type. Consistent use of this parameter allows to solve the problem of motion (orbit shape and time evolution along the orbit), thereby extending the use of the Levi-Civita parameter beyond the usual Kepler problem in a flat Euclidean configuration space. A `universal' description, where all relations are applicable to the motions in any space and with any energy follow from our approach, with no need to discuss separately the cases where the configuration space is flat or where energy vanishes. We also discuss the connection of this `curved Kepler' problem with a geodesic flow. The well known results by Moser, Osipov and Belbruno are shown to hold essentially unchanged beyond the flat Euclidean configuration space. `Curved' Kepler motions with a fixed value of the constant of motion $\sigma:=-(2E - \kappa_1\kappa_2 J^2)$ on any curved configuration space with constant curvature $\kappa_1$ and metric of signature type $\kappa_2$ can be identified with the geodesic flow on a space with curvature $\sigma$ and metric of the same signature type.