Solenoidal injectivity of the geodesic ray transform on 2-tensors
Let $(M, g)$ be a smooth compact simple Riemannian manifold of dimension $n \ge 3$: its boundary is strictly convex, and any two points are joined by a unique geodesic, depending smoothly on its endpoints. For a smooth symmetric covariant 2-tensor field $f$ on $M$, its geodesic ray transform is $$I_2 f(\gamma) = \int_0^{\ell_\gamma} f_{\gamma(t)}\bigl(\dot\gamma(t), \dot\gamma(t)\bigr)\,dt$$ on every maximal unit-speed geodesic $\gamma \colon [0, \ell_\gamma] \to M$ joining boundary points. For a smooth 1-form $v$ put $(d^s v)_{ij} = (\nabla_i v_j + \nabla_j v_i)/2$. If $v$ vanishes on $\partial M$, integrating along each geodesic gives $I_2(d^s v) = 0$. Paternain, Salo and Uhlmann proved that on simple surfaces these are the only elements of the kernel, and likewise for symmetric tensor fields of every order. For simple manifolds of any dimension, the corresponding statement for $I_2$ was proved by Pestov and Sharafutdinov under negative sectional curvature, by Stefanov and Uhlmann for generic simple metrics, real-analytic ones included, and, in dimension $n \ge 3$, by Stefanov, Uhlmann and Vasy for manifolds foliated by strictly convex hypersurfaces.
Problem. For every such $(M, g)$, is $I_2 f = 0$ equivalent to $f = d^s v$ for a smooth 1-form $v$ vanishing on $\partial M$?
Source: Sharafutdinov (1994), Problem 1.1.2, for symmetric tensor fields of every order (see Paternain, Salo and Uhlmann (2013)); for 2-tensors in dimension at least three, Paternain, Salo and Uhlmann (2014), §11, open problem 2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, geodesic ray transforms, simple manifolds, tensor tomography
References
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}
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}
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archivePrefix = {arXiv},
}
@article{pestov1988,
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@article{stefanov2005,
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}
@article{stefanov2018,
author = {Stefanov, Plamen and Uhlmann, Gunther and Vasy, Andr{\'a}s},
title = {Inverting the local geodesic {X}-ray transform on tensors},
journal = {J. Anal. Math.},
volume = {136},
number = {1},
year = {2018},
pages = {151--208},
doi = {10.1007/s11854-018-0058-3},
eprint = {1410.5145},
archivePrefix = {arXiv},
}
@misc{aim-tensor-ray-injectivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Solenoidal injectivity of the geodesic ray transform on 2-tensors}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/056-tensor-ray-injectivity.md},
note = {Entry 056 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-00565, "Solenoidal injectivity of the geodesic ray transform on 2-tensors", clopen.solutions.