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Math-2656 · mathematics · Geometry

Smooth isometric immersion of closed surfaces into four-space

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each compact smooth surface M without boundary and each smooth Riemannian metric g on M, is there a smooth map $F:M\to R^{4}$ with injective differential at every point and $F*\delta =g$? The map may have self-intersections, and M may be nonorientable.

Short English statement

For each compact smooth surface M without boundary and each smooth Riemannian metric g on M, is there a smooth map $F:M\to R^{4}$ with injective differential at every point and $F*\delta =g$? The map may have self-intersections, and M may be nonorientable.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2656-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2656

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. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Smooth isometric immersion of closed surfaces into four-space

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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