Existence of a flexible closed C² surface in three-space
Let $S$ be a compact connected surface without boundary, and let $g_{\mathrm{Eucl}}$ be the Euclidean metric on $\mathbb{R}^3$. A closed convex surface cannot be bent: by the rigidity theorem going back to Cauchy for polyhedra and extended to all convex surfaces by Alexandrov and Pogorelov, isometric closed convex surfaces are congruent. Connelly constructed a closed polyhedral surface that flexes, and closed surfaces do flex in the class $C^1$: Bleecker showed that the unit sphere can be squeezed into an arbitrarily small ball through a continuous family of isometric $C^1$ immersions.
Problem. Is there such a surface $S$ and a continuous path $t \mapsto f_t$, $0 \le t \le 1$, in $C^2(S, \mathbb{R}^3)$ such that every $f_t$ is an embedding, $$f_t^* g_{\mathrm{Eucl}} = f_0^* g_{\mathrm{Eucl}} \quad \text{for all } t,$$ and $f_1(S)$ is not congruent to $f_0(S)$? No convexity is imposed.
Source: Traced by Ghomi (2019), Problem 1.1, to Euler (1776) for polyhedral surfaces and to Maxwell (1856) for smooth surfaces.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, flexible surfaces, isometric deformations, rigidity
References
@misc{ghomi2019,
author = {Ghomi, Mohammad},
title = {Open problems in geometry of curves and surfaces},
year = {2019},
url = {https://ghomi.math.gatech.edu/Papers/op.pdf},
note = {Last revised September 2, 2019},
}
@article{maxwell1856,
author = {Maxwell, James Clerk},
title = {On the transformation of surfaces by bending},
journal = {Trans. Cambridge Philos. Soc.},
volume = {9},
year = {1856},
pages = {445--470},
note = {Read 13 March 1854},
}
@article{connelly1977,
author = {Connelly, Robert},
title = {A counterexample to the rigidity conjecture for polyhedra},
journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.},
volume = {47},
year = {1977},
pages = {333--338},
doi = {10.1007/BF02684342},
}
@article{bleecker1997,
author = {Bleecker, David},
title = {Isometric deformations of compact hypersurfaces},
journal = {Geom. Dedicata},
volume = {64},
number = {2},
year = {1997},
pages = {193--227},
doi = {10.1023/A:1017999111399},
}
@incollection{connelly1993,
author = {Connelly, Robert},
title = {Rigidity},
booktitle = {Handbook of Convex Geometry, Vol. A},
editor = {Gruber, Peter M. and Wills, J{\"o}rg M.},
publisher = {North-Holland},
address = {Amsterdam},
year = {1993},
pages = {223--271},
doi = {10.1016/B978-0-444-89596-7.50012-2},
}
@misc{aim-smooth-flexible-closed-shell,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Existence of a flexible closed C² surface in three-space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/249-smooth-flexible-closed-shell.md},
note = {Entry 249 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-00561, "Existence of a flexible closed C² surface in three-space", clopen.solutions.