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

Abstract

Let G be the manifold of all (unparametrized) oriented lines of R3. 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 α. Actually, we pose the analogous more general problem by means of a control system on the manifold Gκ of all the oriented complete geodesics of the three dimensional space form of curvature κ: R3 for κ= 0 , S3 for κ= 1 and hyperbolic 3-space for κ= − 1. We obtain that the system is controllable if and only if α2≠ κ. In the spherical case with α= ± 1 , an admissible curve remains in the set of fibers of a fixed Hopf fibration of S3. 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.