Sub-Riemannian structures on 3D lie groups

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 + ( )  × S 1 , where A + ( ) d...

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Veröffentlicht in:Journal of dynamical and control systems 2012, Vol.18 (1), p.21-44
Hauptverfasser: Agrachev, A., Barilari, D.
Format: Artikel
Sprache:eng
Schlagworte:
Calculus of Variations and Optimal Control
Optimization
Control
Dynamical Systems
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
Systems Theory
Vibration
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Zusammenfassung: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 + ( )  × S 1 , where A + ( ) denotes the group of orientation preserving affine maps on the real line.
ISSN:1079-2724
1573-8698
DOI:10.1007/s10883-012-9133-8