Sub-Riemannian Geodesics on SL(2, ℝ)

2022/03/09 by Domenico D'Alessandro, Domenico D’Alessandro, D'Alessandro, Domenico +2
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.2203.05073

arxiv created 2022/03/09 · openalex publication_date 2022/03/09 · arxiv updated 2022/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 obtained in \citeBoscaRossi. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups \citeAGBD, \citeBiggs, \citeHB2 to identify exactly automorphic structures on which our results apply.

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