Issue |
ESAIM: COCV
Volume 28, 2022
|
|
---|---|---|
Article Number | 76 | |
Number of page(s) | 30 | |
DOI | https://doi.org/10.1051/cocv/2022068 | |
Published online | 22 December 2022 |
Sub-Riemannian geodesics on SL(2,ℝ)
1
Department of Mathematics, Iowa State University,
Ames,
IA
50011,
USA
2
Department of Mathematics, University of California Santa Barbara,
Santa Barbara,
CA
93106-3080,
USA
* Corresponding author: daless@iastate.edu
Received:
1
May
2022
Accepted:
13
October
2022
We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on SL(2, ℝ). Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups to identify exactly automorphic structures on which our results apply.
Mathematics Subject Classification: 54C17 / 53C22 / 57S20 / 22E15
Key words: Sub-Riemannian Geometry / Lie group SL(2,ℝ) / symmetry reduction / optimal synthesis
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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