Journal of Dynamical and Control Systems

, Volume 18, Issue 1, pp 21–44 | Cite as

Sub-Riemannian structures on 3D lie groups

Article

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.

Key words and phrases

Sub-Riemannian geometry Lie groups left-invariant structures 

2000 Mathematics Subject Classification

53C17 22E30 49J15 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Steklov Mathematical InstituteMoscowRussia

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