  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. 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 lambda in R, we define a left
invariant distribution on G accounting for infinitesimal roto-translations of M
of pitch 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 R^7 rtimes SO(7)
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".
