The sub-Riemannian geometry of screw motions with constant pitch

Eduardo Hulett, Paola Ruth Moas, Marcos Salvai

Published 2023-02-07Version 1

We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it. The simplest examples are R^3, S^3 and hyperbolic 3-space, with L defined in terms of the cross product. Such an L exists even if the codimension of gamma is not equal to 2, for instance when M is a connected compact semisimple Lie group, like SO(n) and SU(n), or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M with reference state at a point in M. For lambda in R (the pitch), we define a left invariant distribution D^lambda on G in such a way that a curve in G is always tangent to D^lambda if, at each instant, at the infinitesimal level, translating the body along a direction entails rotating it according to lambda L around that direction. We give conditions for the controllability of the control system on G determined by D^lambda and find explicitly all the geodesics of the natural sub-Riemannian structure on (G,D^lambda). We also study a similar system on R^7 \rtimes SO(7) with L involving the octonionic cross product. In the appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of "small rotations".