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Metric lines in
the special Euclidean group on the plane
Yuyang Wang, Sean Ku and Alejandro Bravo-Doddoli |
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Vol. 19 (2026), No. 3, 471–490
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Abstract |
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The special Euclidean group on the plane has a left-invariant sub-Riemannian structure. Every sub-Riemannian manifold possesses a Hamiltonian function governing the sub-Riemannian geodesic flow. Two natural questions are: What are the necessary conditions for sub-Riemannian geodesics to be periodic? What type of geodesics are the metric lines in ? In this article, we answer both questions, and our method to answering the second is using the Hamilton–Jacobi theory. |
Keywords
sub-Riemannian geometry, globally minimizing geodesics,
Hamilton–Jacoby equations, calibration functions
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Mathematical Subject Classification
Primary: 22E60, 35F21, 53C17, 53C22
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Milestones
Received: 25 August 2024
Revised: 13 December 2024
Accepted: 15 December 2024
Published: 12 June 2026
Communicated by Michael Dorff
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Authors |
| Yuyang Wang | |
| Department of Mathematics University of Michigan Ann Arbor, MI United States |
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| Sean Ku | |
| Courant Institute of Mathematical
Sciences New York University New York, NY United States |
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| Alejandro Bravo-Doddoli | |
| Department of Mathematics University of Michigan Ann Arbor, MI United States |
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| © 2026 MSP (Mathematical Sciences Publishers). |
