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Smooth isometric immersion of every closed Riemannian surface into R^4

Nash's theorem places a compact Riemannian nn-manifold smoothly and isometrically in Euclidean space of high dimension (17 for surfaces), and Gromov's methods give smooth isometric immersions of closed surfaces into R5\mathbb R^5. In R3\mathbb R^3 the topology and curvature obstruct global immersion and even local smooth immersion is delicate; C1C^1 immersions exist by Nash-Kuiper, and Poznyak treated compact planar domains in R4\mathbb R^4. Gromov recorded the question of smooth isometric immersion of arbitrary Riemannian surfaces into R4\mathbb R^4 (2000) and later attributed it to Chern around 1950. Does every closed smooth Riemannian surface, orientable or not and of any curvature, admit a C∞C^\infty isometric immersion into R4\mathbb R^4?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Isometric immersions, geometric PDE
Posed by
Mikhael Gromov (attributing the question to Shiing-Shen Chern)
Year posed
2000
Years open
26y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every closed smooth Riemannian surface (compact, without boundary, not necessarily orientable or connected, any Gaussian curvature) admits a C∞C^\infty isometric immersion into R4\mathbb R^4; self-intersections are allowed. This lowers Gromov's R5\mathbb R^5 immersion dimension for closed surfaces to four. Not shown: embeddings, noncompact or bounded surfaces (the paper says it treats the compact boundaryless case of Gromov's question), or analytic regularity.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (September 23, 2026) is the whole family.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the question as the manuscript cites it from Gromov. The construction (positive rank-one metric decomposition, oscillatory corrections with controlled normal geometry, smoothing) was not refereed. The challenge lean/ComparatorChallenges/SurfaceImmersion.json (theorem OAI.ClosedSurfaceR4.FiniteOrderSmoothing.smooth_isometric_immersion, solution module OAI.Geometry.SurfaceImmersion.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; it is found through lean/docs/333.md. Its statement was read here: for every compact Hausdorff second-countable smooth manifold modelled on R2\mathbb R^2 without boundary and every smooth Riemannian metric, there is a smooth map to R4\mathbb R^4 whose differential preserves the inner product on every tangent space. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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