Sub-Riemannian structures on 3D lie groups

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+($ \mathbb{R} $) × S1, where A+($ \mathbb{R} $) denotes the group of orientation preserving affine maps on the real line.

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