# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021023

## Solvable approximations of 3-dimensional almost-Riemannian structures

 1 Lab. R. Salem, CNRS UMR 6085, Université de Rouen, avenue de l'université BP 12, 76801 Saint Étienne-du-Rouvray, France 2 Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France

* Corresponding author: philippe.jouan@univ-rouen.fr

Received  October 2020 Published  March 2021

In some cases, the nilpotent approximation of an almost-Riemannian structure can degenerate into a constant rank sub-Riemannian one. In those cases, the nilpotent approximation can be replaced by a solvable one that turns out to be a linear ARS on a nilpotent Lie group or a homogeneous space.

The distance defined by the solvable approximation is analyzed in the 3D-generic cases. It is shown that it is a better approximation of the original distance than the nilpotent one.

Citation: Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021023
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##### References:
Geodesics for $\theta\in\left\{0,\frac{\pi}{3},\frac{2\pi}{3},\pi,\frac{4\pi}{6},\frac{5\pi}{6}\right\}$ when $r = 0$
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