Nonbranching of analytic equiregular sub-Riemannian geodesics
Let $(M, \Delta, g)$ be a real-analytic sub-Riemannian manifold: $\Delta$ is a real-analytic bracket-generating distribution and $g$ a real-analytic inner product on $\Delta$. Let $\Delta^1 = \Delta$ and let $\Delta^{j+1}$ be spanned by $\Delta^j$ and the brackets of local sections of $\Delta$ with local sections of $\Delta^j$; the structure is equiregular if the rank of each $\Delta^j$ is constant on $M$. In a real-analytic sub-Riemannian manifold normal geodesics do not branch. Mietton and Rizzi gave the first branching sub-Riemannian manifolds, with strictly normal geodesics, by a gluing that is not real-analytic. Rossi, Schiavoni Piazza and Socionovo constructed a polynomial structure on $\mathbb{R}^3$ with two strictly abnormal minimizing geodesics that coincide on an initial interval and then separate; there the branching occurs in the Martinet region, where the structure is not equiregular.
Problem. For every equiregular real-analytic sub-Riemannian manifold, are length-minimizing geodesics nonbranching: if $\gamma_1, \gamma_2 : [0, L] \to M$ are unit-speed horizontal length minimizers that agree on $[0, \varepsilon]$ for some $\varepsilon > 0$, must they agree on all of $[0, L]$?
Source: Rossi, Schiavoni Piazza and Socionovo (2026), §1.4, the second open problem of the first version (arXiv:2603.23068v1); with the definition of branching made explicit by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MG, OC
- Keywords
- AIM Open Applied Problems, branching geodesics, sub-Riemannian geodesics
References
@misc{rossi2026,
author = {Rossi, Tommaso and Schiavoni Piazza, Alec Jacopo Almo and Socionovo, Alessandro},
title = {Interior singularity and branching of geodesics in real-analytic sub-{R}iemannian manifolds},
year = {2026},
eprint = {2603.23068},
archivePrefix = {arXiv},
note = {Version 1, 24 March 2026, §1.4},
}
@article{mietton2020,
author = {Mietton, Thomas and Rizzi, Luca},
title = {Branching geodesics in sub-{R}iemannian geometry},
journal = {Geom. Funct. Anal.},
volume = {30},
number = {4},
year = {2020},
pages = {1139--1151},
doi = {10.1007/s00039-020-00539-z},
}
@misc{aim-equiregular-analytic-nonbranching,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Nonbranching of analytic equiregular sub-Riemannian geodesics}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/125-equiregular-analytic-nonbranching.md},
note = {Entry 125 at commit a153f46, 2026-10-13},
}
Resolution claims
We study the regularity and branching of strictly abnormal minimizing geodesics in sub-Riemannian geometry. We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions. Moreover, using a lifting procedure, we provide the existence of non-smooth and branching minimizing geodesics also in Carnot …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: arXiv:2603.23068v3. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00491, "Nonbranching of analytic equiregular sub-Riemannian geodesics", clopen.solutions.