Lens rigidity for nontrapping surfaces with conjugate points
Let $M$ be a compact connected smooth surface with boundary, and let $g_1$ and $g_2$ be smooth Riemannian metrics on $M$ that induce the same metric on $\partial M$, for each of which $\partial M$ is strictly convex and every maximal unit-speed geodesic leaves $M$ in finite forward and backward time. Conjugate points are allowed. For an inward-pointing unit vector $v$ at a boundary point, let $\tau_g(v)$ be its exit time and $S_g(v)$ its exit point and exit unit vector; incoming and outgoing boundary unit vectors of the two metrics are identified through their tangential components and the inward or outward normal. Pestov and Uhlmann proved that compact simple surfaces are boundary distance rigid (2005). Guillarmou, Mazzucchelli and Tzou proved that compact nontrapping surfaces without conjugate points are lens rigid, without assuming the boundary convex (2021).
Problem. Does $$(\tau_{g_1}, S_{g_1}) = (\tau_{g_2}, S_{g_2})$$ imply $g_1 = F^* g_2$ for a smooth diffeomorphism $F : M \to M$ fixing $\partial M$ pointwise?
Source: Uhlmann (2004), Conjecture 4.1, that the scattering relation determines the metric up to isometry; stated for nontrapping surfaces with strictly convex boundary by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, conjugate points, lens rigidity, scattering relations
References
@incollection{uhlmann2004,
author = {Uhlmann, Gunther},
title = {The {C}auchy data and the scattering relation},
booktitle = {Geometric Methods in Inverse Problems and {PDE} Control},
editor = {Croke, Christopher B. and Lasiecka, Irena and Uhlmann, Gunther and Vogelius, Michael S.},
series = {IMA Vol. Math. Appl.},
volume = {137},
publisher = {Springer},
address = {New York},
year = {2004},
pages = {263--287},
doi = {10.1007/978-1-4684-9375-7_10},
}
@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{guillarmou2021,
author = {Guillarmou, Colin and Mazzucchelli, Marco and Tzou, Leo},
title = {Boundary and lens rigidity for non-convex manifolds},
journal = {Amer. J. Math.},
volume = {143},
number = {2},
year = {2021},
pages = {533--575},
doi = {10.1353/ajm.2021.0012},
eprint = {1711.10059},
archivePrefix = {arXiv},
}
@misc{aim-nontrapping-surface-lens-rigidity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Lens rigidity for nontrapping surfaces with conjugate points}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/120-nontrapping-surface-lens-rigidity.md},
note = {Entry 120 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-00541, "Lens rigidity for nontrapping surfaces with conjugate points", clopen.solutions.