Geodesics on Grushin spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912569109774336 |
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| author | Albert, Michael Borza, Samuël Gordina, Maria |
| author_facet | Albert, Michael Borza, Samuël Gordina, Maria |
| contents | We consider higher-dimensional generalizations of the $α$-Grushin plane, focusing on the problem of classification of geodesics that minimize length, also known as optimal synthesis. Solving Hamilton's equations on these spaces using the calculus of generalized trigonometric functions, we obtain explicit conjugate times for geodesics starting at a Riemannian point. From symmetries in the geodesic structure, we propose a conjectured cut time, and prove that it provides an upper bound on the first conjugate time, a key step in the extended Hadamard technique. In the three-dimensional case, we combine this method with a density argument to establish the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geodesics on Grushin spaces Albert, Michael Borza, Samuël Gordina, Maria Differential Geometry Metric Geometry 53C17, 51F99, 53B99, 28A75, 28C10 We consider higher-dimensional generalizations of the $α$-Grushin plane, focusing on the problem of classification of geodesics that minimize length, also known as optimal synthesis. Solving Hamilton's equations on these spaces using the calculus of generalized trigonometric functions, we obtain explicit conjugate times for geodesics starting at a Riemannian point. From symmetries in the geodesic structure, we propose a conjectured cut time, and prove that it provides an upper bound on the first conjugate time, a key step in the extended Hadamard technique. In the three-dimensional case, we combine this method with a density argument to establish the conjecture. |
| title | Geodesics on Grushin spaces |
| topic | Differential Geometry Metric Geometry 53C17, 51F99, 53B99, 28A75, 28C10 |
| url | https://arxiv.org/abs/2509.03411 |