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)×S 1, where A +(ℝ) denotes the group of orientation preserving affine maps on the real line. © 2012 Springer Science+Business Media, LLC.
Sub-Riemannian structures on 3D lie groups
Agrachev A.;Barilari D.
2012
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 consequence, 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. © 2012 Springer Science+Business Media, LLC.
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