The sub-Riemannian geometry of screw motions with constant pitch

Note the non-canonical time!

Speaker: Marcos Salvai, Professor, National University of Cordoba, Argentina.     

Abstract: 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. When  λ ≠ k^2, there is a canonical sub-Riemannian structure on  D^λ. We describe its geodesics. 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, when  M  has positive injectivity radius.

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 compact Lie groups. We define sub-Riemannian structures analogous to the above and find some of their geodesis.