Solvable approximations of 3-dimensional almost-Riemannian structures

Philippe Jouan; Ronald Manríquez

doi:10.3934/mcrf.2021023

Article Contents

2022, Volume 12, Issue 2: 303-326. Doi: 10.3934/mcrf.2021023

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Solvable approximations of 3-dimensional almost-Riemannian structures

Philippe Jouan^1, , and
Ronald Manríquez^2,

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

Received: October 2020

Early access: March 2021

Published: June 2022

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Abstract

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.

Keywords:

Almost-Riemannian geometry,
nilpotent approximation.

Mathematics Subject Classification: Primary: 53C15, 53C17, 22E25, 53B99.

Citation:

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Figure 1. 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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Figure 2. Ball in 3-D generic case

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