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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.04201 |
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Table of 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.