Result 333, Differential geometry

Smooth isometric immersions of surfaces into ℝ4

Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.

Lean formalization Proof

The bigger picture

Why it matters

A surface's internal geometry records lengths measured along it. The manuscript claims that every closed smooth surface can realize that geometry in four-dimensional Euclidean space, provided self-intersections are allowed.

What changes?

The unreviewed manuscript reports a smooth isometric immersion for every closed smooth Riemannian surface: a compact two-dimensional surface without boundary, equipped with smoothly varying rules for measuring lengths and angles. An immersion maps the surface without collapsing any tangent direction; isometric means it preserves those measurements, not straight-line distances between image points. The target is Euclidean four-space. The claim includes nonorientable surfaces and arbitrary Gaussian curvature, but does not require the image to avoid self-intersections.

What does that help mathematicians do?

If correct, the result rules out orientability and restrictions on Gaussian curvature as barriers to such an immersion in four dimensions. Researchers could therefore represent any metric on a closed smooth surface as geometry inherited from a smooth map into that space. This connects abstract surface geometry to its realization in Euclidean space, without settling whether self-intersections can be eliminated or whether three dimensions suffice.

Are there practical applications?

The immediate value is foundational: it would let geometers study any closed surface metric through a smooth Euclidean realization while retaining its intrinsic lengths and angles. That offers a common setting for relating internal geometry to how a surface sits in a surrounding space. The supplied abstract describes no numerical construction or practical performance guarantee.

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

Smooth isometric immersions of closed surfaces into Euclidean four-space

September 23, 2026 41 pages

Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, without an orientability assumption. This resolves the closed-surface form of the classical four-dimensional isometric-immersion problem.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026,
  author = {{OpenAI}},
  title = {{Smooth isometric immersions of closed surfaces into Euclidean four-space}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf}{OAI:Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/333.md.

Smooth isometric immersions of surfaces into ℝ4

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

Scope

The formalization proves that every closed smooth Riemannian surface admits a smooth isometric immersion into R4\mathbb R^4. The differential preserves the Riemannian inner product on every tangent space. No orientability assumption is imposed, and the statement asks for an immersion rather than an embedding.

Comparator links

Result Comparator statement
Smooth isometric immersion of every closed surface into four-space SurfaceImmersion.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.