Characterizing realizable geodesic scattering data
Let $M$ be a smooth compact disk and $h$ a smooth Riemannian metric on $\partial M$. A Riemannian metric $g$ on $M$ is simple if $\partial M$ is strictly convex and any two points of $M$ are joined by a unique geodesic, depending smoothly on its endpoints. Write $$B_h^*\partial M = \{(x,\eta) \in T^*\partial M : |\eta|_h < 1\}.$$ Every smooth simple metric $g$ on $M$ inducing $h$ on $\partial M$ has a scattering map $\alpha_g : B_h^*\partial M \to B_h^*\partial M$: extend $\eta$ to the inward-pointing unit covector at $x$, follow its geodesic, and record the exit point and the tangential part of the exit covector. Pestov and Uhlmann proved that compact simple surfaces are boundary distance rigid: the distances between boundary points determine the metric up to the natural ambiguity. For the non-Abelian X-ray transform of a fixed simple surface, Bohr and Paternain characterized the range.
Problem. Give necessary and sufficient conditions, expressed entirely in terms of a proposed boundary map $A$ and $h$, for $A = \alpha_g$ for some smooth simple metric $g$ on $M$ inducing $h$. The conditions must give an intrinsic characterization rather than restate the existence of an interior metric.
Source: Paternain, Salo and Uhlmann (2023), §15.1, Open problem 2, as the range of g ↦ α_g on simple metrics; stated for a fixed boundary metric, with intrinsic conditions, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, geometric inverse problems, scattering relation, 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 = {Section 15.1, Open problem 2},
}
@article{pestov2005,
author = {Pestov, Leonid and Uhlmann, Gunther},
title = {Two dimensional compact simple {R}iemannian manifolds are boundary distance rigid},
journal = {Ann. of Math. (2)},
volume = {161},
number = {2},
year = {2005},
pages = {1093--1110},
doi = {10.4007/annals.2005.161.1093},
}
@article{bohr2023,
author = {Bohr, Jan and Paternain, Gabriel P.},
title = {The transport {O}ka--{G}rauert principle for simple surfaces},
journal = {J. \'Ec. polytech. Math.},
volume = {10},
year = {2023},
pages = {727--769},
doi = {10.5802/jep.231},
}
@misc{aim-range-of-geodesic-scattering,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Characterizing realizable geodesic scattering data}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/063-range-of-geodesic-scattering.md},
note = {Entry 063 at commit a153f46, 2026-10-13},
}
Resolution claims
We give necessary and sufficient conditions for a boundary map to be the geodesic scattering map of a smooth simple Riemannian metric on a fixed disk with a prescribed boundary metric. The conditions depend only on the proposed map and that boundary metric. Boundary invariant functions first determine a normalized conformal boundary coordinate. In the resulting Euclidean disk, length inequalities …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/geodesic_scattering.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00767, "Characterizing realizable geodesic scattering data", clopen.solutions.