The Sub-Riemannian Geometry of Screw Motions with Constant Pitch

Abstract

We consider a family of Riemannian manifolds M such that for each unit speed geodesic $\varvec{\gamma }$ of M there exists a distinguished bijective correspondence $\varvec{L}$ between infinitesimal translations along $\varvec{\gamma }$ and infinitesimal rotations around it. The simplest examples are ${\mathbb {R}}^{3}$, $\varvec{S}^{3}$ and hyperbolic 3-space, with $\varvec{L}$ defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, 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. Given ${\varvec{\lambda }} \in {\mathbb {R}}$, we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch $\varvec{\lambda }$. We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on ${\mathbb {R}}^{7}\rtimes SO\left( 7\right)$ involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of “small rotations”.