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

Abstract

Let \(\mathscr{G}\) be the manifold of all (unparametrized) oriented lines of \(\mathbb{R}^{3}\). We study the controllability of the control system in \(\mathscr{G}\) given by the condition that a curve in \(\mathscr{G}\) 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 \(\mathscr{G}_{\kappa }\) of all the oriented complete geodesics of the three dimensional space form of curvature \(\kappa \): \(\mathbb{R}^{3}\) 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 \(\alpha ^{2}\neq \kappa \). In the spherical case with \(\alpha =\pm 1\), 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.