Inserted: 10 mar 2012
Last Updated: 10 mar 2012
Journal: J. Dynam. Control Systems
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 consequence, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^+(R) \times S^1$, where $A^+(R)$ denotes the group of orientation preserving affine maps on the real line.