Local scalar geodesic tomography on smooth surfaces
Let $(M,g)$ be a smooth Riemannian surface with boundary whose boundary is strictly convex at a point $p \in \partial M$; the metric is smooth, with no analyticity assumption. For a neighborhood $U$ of $p$, only geodesic segments lying in $U \cap M$ with both endpoints on $U \cap \partial M$ are used. For the Euclidean disk the answer to the question below is yes, as Paternain, Salo and Uhlmann note. Uhlmann and Vasy proved the analogous local injectivity in dimension at least three, near any boundary point where the boundary is strictly convex (2016). On surfaces, Ilmavirta, Lehtonen and Salo proved it for piecewise constant functions (2020). By a layer-stripping argument, a positive answer would give injectivity of the geodesic X-ray transform on every compact nontrapping surface with strictly convex boundary (Paternain, Salo and Uhlmann).
Problem. Is there a sufficiently small neighborhood $U$ of $p$ such that for every $f \in C^\infty(\overline{U} \cap M)$: if $$\int_\gamma f \, ds_g = 0$$ for every geodesic segment $\gamma$ lying in $U \cap M$ whose two endpoints lie on $U \cap \partial M$, then $f$ vanishes in some neighborhood of $p$ in $M$?
Source: Paternain, Salo and Uhlmann (2023), Chapter 15, Open problem 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, geodesic X-ray transforms, geometric inverse problems, integral geometry
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},
}
@article{uhlmann2016,
author = {Uhlmann, Gunther and Vasy, Andr{\'a}s},
title = {The inverse problem for the local geodesic ray transform},
journal = {Invent. Math.},
volume = {205},
number = {1},
year = {2016},
pages = {83--120},
doi = {10.1007/s00222-015-0631-7},
eprint = {1210.2084},
archivePrefix = {arXiv},
}
@article{ilmavirta2020,
author = {Ilmavirta, Joonas and Lehtonen, Jere and Salo, Mikko},
title = {Geodesic {X}-ray tomography for piecewise constant functions on nontrapping manifolds},
journal = {Math. Proc. Cambridge Philos. Soc.},
volume = {168},
number = {1},
year = {2020},
pages = {29--41},
doi = {10.1017/S0305004118000543},
eprint = {1702.07622},
archivePrefix = {arXiv},
}
@misc{aim-local-ray-injectivity-surfaces,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Local scalar geodesic tomography on smooth surfaces}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/119-local-ray-injectivity-surfaces.md},
note = {Entry 119 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-00533, "Local scalar geodesic tomography on smooth surfaces", clopen.solutions.