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Math-1678 · mathematics · Geometry

Geometry of Curves and Surfaces — Problem 1.8

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $(M,g)$ be any smooth two-dimensional Riemannian manifold, with no boundary unless separately specified. Both the manifold and metric are $C^\infty$; compactness and completeness are not assumed. A smooth isometric embedding into $\mathbb R^4$ is an injective smooth immersion $f$ that is a homeomorphism onto its image and satisfies \[ g_u(v,w)=\langle df_u(v),df_u(w)\rangle_{\mathbb R^4} \quad(u\in M,\ v,w\in T_uM). \] Does every such $(M,g)$ admit a $C^\infty$ isometric embedding $f:M\to\mathbb R^4$?

The assertion is global: local embeddings on separate coordinate patches must arise from one injective map on all of $M$. The metric equality preserves lengths of curves and the induced intrinsic distance, rather than claiming that intrinsic distance equals the ambient straight-line distance between image points. For noncompact $M$, properness is not included in the definition unless imposed as an additional condition.

Increasing the ambient dimension, dropping injectivity to obtain an immersion, or lowering the regularity to $C^1$ changes the question. The fixed target dimension four and smooth regularity must therefore be retained in either a construction or an obstruction.

Short English statement

Can every smooth two-dimensional Riemannian manifold be embedded globally and smoothly in four-dimensional Euclidean space with its metric exactly preserved?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1678-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1678

Closed-surface immersion in four-space: The paper requires a closed surface and supplies an immersion only. Our definition expressly allows noncompact incomplete surfaces and requires one globally injective embedding; neither injectivity nor that broader domain is asserted.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Geometry of Curves and Surfaces — Problem 1.8

OpenAI's claim

Closed-surface immersion in four-space: The paper claims a smooth isometric immersion of every closed smooth Riemannian surface into $\mathbb R^4$. Nonorientable surfaces are included, and no embedding conclusion is asserted.

Mathematical statement of the claimed result

Mathematical claim 1.
For every closed smooth Riemannian surface $(M,g)$ there is a smooth immersion $F:M\to\mathbb R^4$ with \[F^*\left(\sum_{i=1}^4dx_i^2\right)=g.\] Closedness means compactness without boundary. The assertion is an isometric immersion, which allows self-intersections; it does not assert a globally injective isometric embedding.

Outline of the argument

Closed-surface immersion in four-space: A short initial immersion into a small three-sphere gives a global radial normal and a finite decomposition of the metric deficit into rank-one terms. Local primitive additions use velocity loops and exact corrections to realize those terms. The corrections preserve the geometric and boundary conditions required for the next addition, completing the finite smooth construction.

Formal verification

Family 333 describes Lean coverage for closed-surface smooth isometric immersion into four-space, including nonorientable surfaces.

Closed-surface immersion in four-space: The manuscript links SurfaceImmersion.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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