Title: Sub-Riemanian structures on 3D Lie groups
Authors: A.A. Agrachev, D. Barilari
Journal title: Dynamical and Control Systems
Year: 2012
Volume: 18
Pages: 21-44
Abstract:

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 $$SL(2)$$ and $$A^+ (R)$$ × $$S^1$$, where $$A^+ (R)$$ denotes the group of orientation preserving affine maps on the real line.