Optimal Synthesis in a Radially Symmetric Grushin Space
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910105265504256 |
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| author | Albert, Michael |
| author_facet | Albert, Michael |
| contents | We study the geometry of $\mathbb{R}^3$ equipped with a rotationally invariant Carnot-Carthéodory metric obtained by weighting motion in the $z$-direction by a function $f(r)$ of the cylindrical radius. When $f$ vanishes only at $r=0$, the space exhibits a Grushin--type singularity along the vertical axis. We provide sufficient conditions on $f$ ensuring a Grushin--like structure and describe the full optimal synthesis at singular points. For Riemannian points, we propose a candidate cut time determined by a discrete symmetry of the Hamiltonian flow. In the integrable case $f(r)=r$, we prove that this candidate coincides with the true cut time and give an explicit description of the cut locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04201 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Synthesis in a Radially Symmetric Grushin Space Albert, Michael Differential Geometry Primary: 53C17 Secondary: 53C22, 70H20 We study the geometry of $\mathbb{R}^3$ equipped with a rotationally invariant Carnot-Carthéodory metric obtained by weighting motion in the $z$-direction by a function $f(r)$ of the cylindrical radius. When $f$ vanishes only at $r=0$, the space exhibits a Grushin--type singularity along the vertical axis. We provide sufficient conditions on $f$ ensuring a Grushin--like structure and describe the full optimal synthesis at singular points. For Riemannian points, we propose a candidate cut time determined by a discrete symmetry of the Hamiltonian flow. In the integrable case $f(r)=r$, we prove that this candidate coincides with the true cut time and give an explicit description of the cut locus. |
| title | Optimal Synthesis in a Radially Symmetric Grushin Space |
| topic | Differential Geometry Primary: 53C17 Secondary: 53C22, 70H20 |
| url | https://arxiv.org/abs/2604.04201 |