Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form

Let G be the manifold of all (unparametrized) oriented lines of R-3. We study the controllability of the control system in G given by the condition that a curve in G describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed alpha. Actually, we pose the analogous more general problem by means of a control system on the manifold G(kappa) of all the oriented complete geodesics of the three dimensional space form of curvature kappa: R-3 for kappa = 0, S-3 for kappa = 1 and hyperbolic 3-space for kappa = -1. We obtain that the system is controllable if and only if alpha(2) not equal kappa. In the spherical case with alpha = +/- 1, an admissible curve remains in the set of fibers of a fixed Hopf fibration of S-3. We also address and solve a sort of Kendall's (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.