Infinitesimally helicoidal motions with fixed pitch of oriented geodesics of a space form

MATEO ANARELLA; MARCOS SALVAI

ACTA APPLICANDAE MATHEMATICAE

SPRINGER

Lugar: Berlin; Año: 2022 vol. 179 p. 1 - 19

0167-8019

Let  L  be the manifold of all (unparametrized) oriented lines of  R^3.  We study the controllability of the control system in  L  given by the condition that a curve in  L 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 eq 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 joininig two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.