Sub-Riemannian structures on 3D Lie groups

We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the literature, in particular the one obtained in [Falbel-Gorodski, 1996] in terms of curvature invariants of a canonical connection. Moreover, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^{+}(\mathbb{R})\times S^1$, where $A^{+}(\mathbb{R})$ denotes the group of orientation preserving affine maps on the real line.