The sub-Riemannian geometry of screw motions with constant pitch

Let M be an oriented three-dimensional Riemannian manifold of constant sectional curvature k = 0 , 1, -1 and let SO(M) be its direct orthonormal frame bundle (direct refers to positive orientation), which has dimension six and may be thought of as the set of all positions of a small body in M. Given λ ∈ R, there is a three-dimensional distribution D^λ on SO(M) accounting for infinitesimal rototranslations of constant pitch λ. We show that the associated system is controllable when λ ≠ k^2. In this case, we describe the geodesics of the canonical sub-Riemannian structure. For k = 0, -1, we compute the lengths of all periodic geodesics of ( SO(M) , D^λ ) in terms of the lengths and the holonomies of the periodic geodesics of M.

It turns out that the notion of rototranslating with constant pitch makes sense for some higher dimensional Riemannian manifolds, for instance, for R^7 via the octonionic cross product, or for orthogonal and unitary matrix groups. We define sub-Riemannian structures analogous to the above and find some of their geodesics.