Sub-Riemannian geodesics on the Heisenberg 3D nil-manifold

Abstract We study the projection of the left-invariant sub-Riemannian structure on the 3D Heisenberg group G to the Heisenberg 3D nil-manifold M — the compact homogeneous space of G by the discrete Heisenberg group. First we describe dynamical properties of the geodesic flow for M : periodic and dense orbits, a dynamical characterization of the normal Hamiltonian flow of Pontryagin maximum principle and its integrability properties. We show that it is Liouville integrable on a nonzero level hypersurface Σ of the Hamiltonian outside an appropriate smaller proper hypersurface in Σ and has no nontrivial analytic integrals on all of Σ. Then we obtain sharp twoside bounds of sub-Riemannian balls and distance in G , and on this basis we estimate the cut time for sub-Riemannian geodesics in M .