Non-Abelian ray transform for simple Gaussian thermostats
Let $(M,g)$ be a compact oriented Riemannian surface with smooth boundary and let $E$ be a smooth vector field on $M$. The trajectories of the Gaussian thermostat $(M,g,E)$ are the unit-speed curves $\gamma$ with $$\nabla_{\dot\gamma}\dot\gamma = E - \langle E, \dot\gamma\rangle_g\, \dot\gamma.$$ Assume the thermostat is simple: $\partial M$ is strictly convex for this flow, and trajectories between points are unique and depend smoothly on their endpoints. Let $m \ge 1$ and, for $j = 1, 2$, let $A_j$ be a smooth $\mathfrak{u}(m)$-valued $1$-form and $\Phi_j$ a smooth $\mathfrak{u}(m)$-valued function on $M$. Along each maximal trajectory $\gamma : [0, \tau] \to M$ with endpoints on $\partial M$, let $U_j$ solve $$\dot U_j + \bigl(A_j(\dot\gamma) + \Phi_j(\gamma)\bigr) U_j = 0, \qquad U_j(0) = I_m .$$ If $Q : M \to U(m)$ is smooth with $Q = I_m$ on $\partial M$, the pair $(Q^{-1}A_1Q + Q^{-1}dQ,\ Q^{-1}\Phi_1Q)$ has the same endpoint matrices $U_1(\tau)$ as $(A_1, \Phi_1)$. For $E = 0$, that is on simple surfaces, these are the only pairs with the same endpoint matrices (Paternain, Salo and Uhlmann, 2012). Ainsworth (2013) proved the same for simple magnetic flows, and Jathar, Kar and Railo extended this to pairs with values in $\mathfrak{gl}(m, \mathbb{C})$. For nontrapping thermostats with strictly convex boundary and negative thermostat curvature, Assylbekov and Rea proved it for pairs with values in $\mathfrak{gl}(m, \mathbb{C})$. Jathar, Kar and Railo also showed that for Gaussian thermostats the general case would follow from injectivity, up to the natural obstruction, of the attenuated thermostat ray transform with a unitary connection and a skew-Hermitian Higgs field.
Problem. If $U_1(\tau) = U_2(\tau)$ for every such trajectory, must there be a smooth $Q : M \to U(m)$ with $Q = I_m$ on $\partial M$ such that $$A_2 = Q^{-1}A_1Q + Q^{-1}dQ, \qquad \Phi_2 = Q^{-1}\Phi_1Q \ ?$$
Source: Jathar, Kar and Railo (2025), Conjecture 1.1 (general matrix-valued pairs on nontrapping twisted geodesic flows), in the unitary case for simple Gaussian thermostats that their abstract singles out; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Gaussian thermostats, geometric inverse problems, non-Abelian ray transforms
References
@article{jathar2025,
author = {Jathar, Shubham R. and Kar, Manas and Railo, Jesse},
title = {Loop group factorization method for the magnetic and thermostatic nonabelian ray transforms},
journal = {Inverse Problems},
volume = {41},
number = {1},
year = {2025},
pages = {015006},
doi = {10.1088/1361-6420/ada08a},
eprint = {2312.06023},
archivePrefix = {arXiv},
}
@article{paternain2012,
author = {Paternain, Gabriel P. and Salo, Mikko and Uhlmann, Gunther},
title = {The attenuated ray transform for connections and {H}iggs fields},
journal = {Geom. Funct. Anal.},
volume = {22},
number = {5},
year = {2012},
pages = {1460--1489},
doi = {10.1007/s00039-012-0183-6},
eprint = {1108.1118},
archivePrefix = {arXiv},
}
@article{ainsworth2013,
author = {Ainsworth, Gareth},
title = {The attenuated magnetic ray transform on surfaces},
journal = {Inverse Probl. Imaging},
volume = {7},
number = {1},
year = {2013},
pages = {27--46},
doi = {10.3934/ipi.2013.7.27},
eprint = {1205.2392},
archivePrefix = {arXiv},
}
@misc{assylbekov2021,
author = {Assylbekov, Yernat M. and Rea, Franklin T.},
title = {The attenuated ray transforms on {G}aussian thermostats with negative curvature},
year = {2021},
eprint = {2102.04571},
archivePrefix = {arXiv},
}
@misc{aim-thermostatic-nonabelian-tomography,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Non-Abelian ray transform for simple Gaussian thermostats}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/070-thermostatic-nonabelian-tomography.md},
note = {Entry 070 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00566, "Non-Abelian ray transform for simple Gaussian thermostats", clopen.solutions.