Boundary rigidity of simple manifolds in dimensions at least three
A compact Riemannian manifold with boundary is called simple here if its boundary is strictly convex and any two points are joined by a unique geodesic depending smoothly on the endpoints. Let $M$ be a compact smooth manifold with boundary of dimension $n \ge 3$, let $g_1, g_2$ be smooth simple metrics on $M$, and let $d_g$ be the intrinsic Riemannian distance of $g$ on $M$. If $F : M \to M$ is a diffeomorphism with $F|_{\partial M} = \mathrm{Id}$, then $g$ and $F^* g$ have the same boundary distances. Pestov and Uhlmann proved that compact simple surfaces are boundary rigid. In dimensions $n \ge 3$, Burago and Ivanov proved rigidity when one metric is close to the Euclidean metric on a region of $\mathbb{R}^n$, Stefanov and Uhlmann proved it locally near generic simple metrics, and Stefanov, Uhlmann and Vasy proved it when $g_1$ has non-positive sectional curvature, non-negative sectional curvature or no focal points.
Problem. Does $$d_{g_1}(x, y) = d_{g_2}(x, y) \quad (x, y \in \partial M) \quad \Longrightarrow \quad g_1 = F^* g_2$$ hold for some diffeomorphism $F : M \to M$ with $F|_{\partial M} = \mathrm{Id}$, for every such $M$, $g_1$ and $g_2$? No curvature sign or convex-foliation hypothesis is assumed.
Source: Michel (1981), who conjectured that simple manifolds are boundary rigid; see Stefanov, Uhlmann and Vasy (2021), §1. Stated for dimensions n ≥ 3 by AIM — Open Applied Problems; the case n = 2 is Pestov and Uhlmann's theorem (2005).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, boundary rigidity, simple manifolds, travel-time tomography
References
@article{michel1981,
author = {Michel, Ren{\'e}},
title = {Sur la rigidit{\'e} impos{\'e}e par la longueur des g{\'e}od{\'e}siques},
journal = {Invent. Math.},
volume = {65},
number = {1},
year = {1981},
pages = {71--83},
doi = {10.1007/BF01389295},
}
@article{stefanov2021,
author = {Stefanov, Plamen and Uhlmann, Gunther and Vasy, Andr{\'a}s},
title = {Local and global boundary rigidity and the geodesic {X}-ray transform in the normal gauge},
journal = {Ann. of Math. (2)},
volume = {194},
number = {1},
year = {2021},
pages = {1--95},
doi = {10.4007/annals.2021.194.1.1},
eprint = {1702.03638},
archivePrefix = {arXiv},
}
@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},
eprint = {math/0305280},
archivePrefix = {arXiv},
}
@article{burago2010,
author = {Burago, Dmitri and Ivanov, Sergei},
title = {Boundary rigidity and filling volume minimality of metrics close to a flat one},
journal = {Ann. of Math. (2)},
volume = {171},
number = {2},
year = {2010},
pages = {1183--1211},
doi = {10.4007/annals.2010.171.1183},
}
@article{stefanov2005,
author = {Stefanov, Plamen and Uhlmann, Gunther},
title = {Boundary rigidity and stability for generic simple metrics},
journal = {J. Amer. Math. Soc.},
volume = {18},
number = {4},
year = {2005},
pages = {975--1003},
doi = {10.1090/S0894-0347-05-00494-7},
}
@misc{aim-simple-boundary-rigidity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Boundary rigidity of simple manifolds in dimensions at least three}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/055-simple-boundary-rigidity.md},
note = {Entry 055 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-00556, "Boundary rigidity of simple manifolds in dimensions at least three", clopen.solutions.