Sub-Riemannian structures on 3D Lie groups

Andrei Agrachev (1) , Davide Barilari (1)

1 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies

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 corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and $A^+(\R)\times S^1$, where $A^+(\R)$ denotes the group of orientation preserving affine maps on the real line.

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]

Andrei Agrachev, Davide Barilari. Sub-Riemannian structures on 3D Lie groups. 2011. ⟨hal-00672261⟩