We give a complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups and × , where denotes the group of orientation preserving affine maps on the real line.
Title: Sub-Riemanian structures on 3D Lie groups
Authors: A.A. Agrachev, D. Barilari
Journal title: Dynamical and Control Systems