Sub-Riemannian geodesics on \(\mathrm{SL}(2, \mathbb{R} )\). (English) Zbl 1510.53032
Summary: We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of elliptic type defined on \(SL (2, \mathbb{R} )\). 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.
MSC:
| 53C17 | Sub-Riemannian geometry |
| 53C22 | Geodesics in global differential geometry |
| 57S20 | Noncompact Lie groups of transformations |
| 22E15 | General properties and structure of real Lie groups |
Keywords:
sub-Riemannian geometry; Lie group \(\mathrm{SL}(2, \mathbb{R} )\); symmetry reduction; optimal synthesisReferences:
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