Result 334, Differential geometry

A smooth surface metric with no local isometric immersion in ℝ3

Constructs a smooth positive-definite metric on (−1,1)2(-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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

An unreviewed manuscript reports a smooth geometry on a two-dimensional patch that cannot be realized as a smooth surface in ordinary three-dimensional space, even after shrinking to an arbitrarily small neighborhood of one point.

What changes?

The construction gives a smooth, positive-definite metric on the open square with both coordinates between -1 and 1. A metric specifies lengths and angles; positive-definite means every nonzero direction has positive squared length. At the origin, it agrees with the Euclidean metric in value and in derivatives of every order. Nevertheless, the manuscript claims that no neighborhood of the origin admits a smooth isometric immersion into three-dimensional Euclidean space: a smooth realization preserving those lengths and angles, even allowing self-intersections.

What does that help mathematicians do?

The striking consequence is that all derivative data at one point can match flat geometry without guaranteeing a local surface realization. The ordinary Euclidean metric has such a realization, while this metric reportedly does not. Thus, that pointwise data alone cannot decide local realizability. Researchers cannot infer existence merely from smoothness, positive-definiteness, or agreement with flat geometry to every order at the chosen point.

Are there practical applications?

The immediate value is foundational: the example separates an internally specified rule for measuring a surface from its possible realization in three-dimensional space. It rules out a universal smooth realization guarantee and shows why existence results need additional restrictions. The reported obstruction concerns smooth, length-preserving immersions into three dimensions, not every possible representation of the metric.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Smooth Metric with No Local Isometric Immersion into Three-Space

September 24, 2026 19 pages Main result formalized in Lean

We construct a smooth positive-definite Riemannian metric on (−1,1)2(-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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/334.md.

A smooth surface metric with no local isometric immersion in ℝ3

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The local isometric-immersion problem asks whether every smooth surface metric can be realized locally in Euclidean three-space. The formalized counterexample is a smooth positive-definite metric on (−1,1)2(-1,1)^2 for which no neighborhood of the origin admits a smooth isometric immersion into R3\mathbb R^3. The obstruction applies to every smooth candidate map on every such neighborhood.

Comparator links

Result Comparator statement
Smooth local isometric-immersion obstruction IsometricImmersion.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.