 
Summary: SubRiemannian structures on 3D Lie groups
Andrei Agrachev
SISSA, Trieste, Italy and MIAN, Moscow, Russia  agrachev@sissa.it
Davide Barilari
SISSA, Trieste, Italy  barilari@sissa.it
July 28, 2010
Abstract
We give a complete classification of leftinvariant subRiemannian structures on three
dimensional Lie groups in terms of the basic differential invariants. As a corollary we
explicitly find a subRiemannian isometry between the nonisomorphic Lie groups SL(2)
and A+
(R) × S1
, where A+
(R) denotes the group of orientation preserving affine maps on
the real line.
1 Introduction
In this paper, by a subRiemannian manifold we mean a triple (M, , g), where M is a con
nected smooth manifold of dimension n, is a smooth vector distribution of constant rank
k < n, and g is a Riemannian metric on , smoothly depending on the point.
In the following we always assume that the distribution satisfies the bracket generating
