Injectivity with attenuation of finite angular degree
Let $(M, g)$ be a smooth compact oriented simple Riemannian surface, with strictly convex boundary and unique geodesics between any two points, depending smoothly on the endpoints, and let $SM$ be its unit tangent bundle. A function $a \in C^\infty(SM; \mathbb{C})$ has finite angular degree if there is an integer $N \ge 0$ such that its Fourier coefficients $a_k(x)$ in the angle of $v \in S_xM$ vanish for every $x \in M$ and every $|k| > N$. For every maximal unit-speed geodesic $\gamma : [0, \ell] \to M$ between boundary points put $$I_a f(\gamma) = \int_0^\ell \exp\left(\int_0^t a(\gamma(s), \dot\gamma(s))\,ds\right) f(\gamma(t))\,dt.$$ Salo and Uhlmann proved that $I_a$ is injective on $C^\infty(M)$ when $a$ depends only on the position, and Paternain, Salo and Uhlmann extended this to every attenuation of angular degree at most one, $a(x, v) = h(x) + \theta_x(v)$ with $h$ a smooth complex function and $\theta$ a smooth complex 1-form. For an arbitrary attenuation in $C^\infty(SM; \mathbb{C})$ injectivity fails, already in the Euclidean case, by an example of Boman.
Problem. For every such $a$ of finite angular degree, does $I_a f = 0$ imply $f = 0$ for smooth scalar functions $f$ on $M$?
Source: Paternain, Salo and Uhlmann (2023), §12.4 and §15.1, Open problem 4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, attenuated ray transforms, geometric inverse problems, simple surfaces
References
@book{paternain2023,
author = {Paternain, Gabriel P. and Salo, Mikko and Uhlmann, Gunther},
title = {Geometric Inverse Problems: With Emphasis on Two Dimensions},
series = {Cambridge Studies in Advanced Mathematics},
volume = {204},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2023},
doi = {10.1017/9781009039901},
note = {\S 12.4 and \S 15.1, Open problem 4; Theorem 12.2.7},
}
@article{salo2011,
author = {Salo, Mikko and Uhlmann, Gunther},
title = {The attenuated ray transform on simple surfaces},
journal = {J. Differential Geom.},
volume = {88},
number = {1},
year = {2011},
pages = {161--187},
doi = {10.4310/jdg/1317758872},
eprint = {1004.2323},
archivePrefix = {arXiv},
}
@article{boman1993,
author = {Boman, Jan},
title = {An example of non-uniqueness for a generalized {R}adon transform},
journal = {J. Anal. Math.},
volume = {61},
year = {1993},
pages = {395--401},
doi = {10.1007/BF02788850},
}
@misc{aim-finite-angular-attenuation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Injectivity with attenuation of finite angular degree}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/062-finite-angular-attenuation.md},
note = {Entry 062 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-00517, "Injectivity with attenuation of finite angular degree", clopen.solutions.