Browse › Differential geometry › 334 A smooth surface metric with no local isometric immersion in ℝ3
A Smooth Metric with No Local Isometric Immersion into Three-Space
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Abstract
We construct a smooth positive-definite Riemannian metric on $(-1,1)^2$ that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note 334: A smooth surface metric with no local isometric immersion in ℝ³ (lean/docs/334.md, release of 2026-10-06). The release's Lean scope note states the formalized counterexample (smooth positive-definite metric on $(-1,1)^2$, no smooth isometric immersion of any neighborhood of the origin into $\mathbb R^3$) and points to the comparator statement
ComparatorChallenges/IsometricImmersion.lean; it does not explicitly include the Euclidean Taylor-jet refinement.
Possibly relevant literature (found by the agent, not checked)
- J. Hong, Some new developments of realization of surfaces into R^3 (arXiv:math/0304391; Proceedings of the ICM, Beijing 2002, vol. 3, 155–166). An ICM survey reviewing local isometric embedding results connected with the Schläfli–Yau conjecture, relevant to the manuscript's account of the problem's status.
- E. Kann, The Local Isometric Embedding of Two-Metrics of Low Differentiability in Euclidean Three Space (arXiv:2210.02942, October 2022). A preprint on local isometric embedding of low-regularity two-metrics in $\mathbb R^3$, related to the manuscript's discussion of how existence depends on regularity (Nash–Kuiper at $C^1$ versus smooth).
Digests and expositions
- Latent Seminar, A smooth metric with no local immersion into three-space (Latent Seminar, 2026-10-07). Explains the Darboux height equation, cap estimates, oscillating metric perturbation, Baire-category argument and accumulating patches, while explicitly disclaiming independent verification of the complete proof.
- Claude Opus 5.5 / Open Mathematics Courses, A metric with no local isometric immersion (Open Mathematics Courses, undated; accessed 2026-10-21). Seven lessons develop the manuscript's height equation, geometric assembly and analytic patch obstruction with exercises; the source describes itself as written and self-checked by Claude Opus 5.5, with independent review incomplete. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 334
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026,
author = {{OpenAI}},
title = {{A Smooth Metric with No Local Isometric Immersion into Three-Space}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf}{OAI:A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/334.md. The Lean statements have not been checked by people against the paper's claims.
Family 334: A smooth surface metric with no local isometric immersion in ℝ3
Constructs a smooth positive-definite metric on $(-1,1)^2$ for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.
This is the only paper in this family.