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We consider a family of Riemannian manifolds M such that for each unit speedgeodesic gamma of M there exists a distinguished bijective correspondence Lbetween infinitesimal translations along gamma and infinitesimal rotationsaround it. The simplest examples are R^3, S^3 and hyperbolic 3-space, with Ldefined in terms of the cross product. More generally, M is a connected compactsemisimple Lie group, or its non-compact dual, or Euclidean space acted ontransitively by some group which is contained properly in the full group ofrigid motions.  Let G be the identity component of the isometry group of M. A curve in G maybe thought of as a motion of a body in M. Given lambda in R, we define a leftinvariant distribution on G accounting for infinitesimal roto-translations of Mof pitch lambda. We give conditions for the controllability of the associatedcontrol system on G and find explicitly all the geodesics of the naturalsub-Riemannian structure. We also study a similar system on R^7 rtimes SO(7)involving the octonionic cross product. In an appendix we give a friendlypresentation of the non-compact dual of a compact classical group, as a set of"small rotations".

The sub-Riemannian geometry of screw motions with constant pitch