Local smooth isometric immersion of surfaces in three-space

Theorem A: there is a $C^\infty$ positive definite metric $g$ on $(-1,1)^2$ such that no open neighbourhood of the origin admits a $C^\infty$ isometric immersion into $\mathbb R^3$ ; the metric can have the full Taylor jet of the Euclidean metric at the origin, so local realisability is not decided by the jet. The curvature takes both signs. Not settled: the smooth local problem under $K\ge0$ or $K\le0$ , finite-regularity versions, and realisation in $\mathbb R^4$ .

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Local smooth isometric immersion of surfaces in three-space · Local isometric immersion of smooth surfaces in R3

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed
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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: Local smooth isometric immersion of surfaces in three-space

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstract and Theorem A of 'A Smooth Metric with No Local Isometric Immersion into Three-Space', read against the problem as quoted from Yau and Ghomi. The proof (a Darboux-equation obstruction on caps, an oscillating perturbation, Baire category and the Nadirashvili-Yuan boundary-saddle argument) was not refereed. Lean: formalization.yaml lists OAI.SmoothLocal.Geometry.exists_local_metric_without_local_immersion (lean/OAI/Geometry/IsometricImmersion/Main.lean, comparator ComparatorChallenges/IsometricImmersion.lean). Its statement was read: a smooth positive-definite matrix field g on the open square (-1,1)^2 such that for every open U inside the square containing 0, no smooth F : U -> R^3 has inner products of its derivatives equal to g. That is the headline. Not rebuilt here. A reader should know that a 2002 arXiv preprint of Nadirashvili and Yuan stated a stronger counterexample, which their 2008 paper and later accounts treat as leaving the question open; the manuscript notes this.

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Ghomi, Open Problems in Geometry of Curves and Surfaces (Problem 8.1)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Local smooth isometric immersion of surfaces in three-space

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean: smooth metric with no local isometric immersion

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean: smooth metric with no local isometric immersion

lean artifact · pending

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Compute record

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Lineage and corrections

VibeMathed record: Local smooth isometric immersion of surfaces in three-space evidence for this event

Ghomi, Open Problems in Geometry of Curves and Surfaces (Problem 8.1) evidence for this event

This event attributed to Unreleased internal OpenAI model

Local smooth isometric immersion of surfaces in three-space parent of this event

Lean: smooth metric with no local isometric immersion evidence for this event

Local smooth isometric immersion of surfaces in three-space evidence for this event

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