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Math-1679 · mathematics · Geometry

Geometry of Curves and Surfaces — Problem 1.9

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $g$ be a $C^\infty$ Riemannian metric on a neighborhood of a point $p$ in a smooth two-manifold. In local coordinates $(u^1,u^2)$ write $g=\sum_{i,j=1}^2g_{ij}\,du^i\,du^j$, with $(g_{ij})$ positive definite. A local smooth isometric embedding is a $C^\infty$ embedding $f:U\to\mathbb R^3$ on some open neighborhood $U$ of $p$ satisfying \[ \left\langle\frac{\partial f}{\partial u^i}, \frac{\partial f}{\partial u^j}\right\rangle =g_{ij}\qquad(1\le i,j\le2). \] Does such a neighborhood and embedding exist for every smooth positive-definite metric and every point $p$?

The neighborhood may be shrunk as much as necessary; no global embedding is requested. There is no assumption that the Gauss curvature is nonzero, has a fixed sign, or vanishes to finite order. In particular, difficult behavior at flat points is part of the statement.

A theorem for analytic metrics or metrics with additional curvature hypotheses establishes only a special case. An immersion of rank two is locally an embedding after further restriction, so the local issue is the smooth solvability of the metric equations, not global self-intersection.

Short English statement

Can every smooth metric on a surface, including near degenerate zero-curvature points, be realized locally by a smooth surface in ordinary three-dimensional space?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1679-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1679

A smooth three-space immersion obstruction: A smooth positive-definite metric with no immersion on any neighborhood of its origin also has no embedding there, exactly negating the universal local smooth embedding assertion. Local immersion/embedding equivalence is explicitly part of our definition.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Geometry of Curves and Surfaces — Problem 1.9

OpenAI's claim

A smooth three-space immersion obstruction: A smooth positive surface metric is claimed to have no isometric immersion into $\mathbb R^3$ on any neighborhood of the origin. All its derivatives at the origin nevertheless agree with the Euclidean metric.

Mathematical statement of the claimed result

Mathematical claim 1.
There exists a positive-definite smooth Riemannian metric $g$ on $(-1,1)^2$ such that for every open neighborhood $U$ of $(0,0)$ there is no $C^\infty$ immersion $F:U\to\mathbb R^3$ satisfying \[F^*(dx_1^2+dx_2^2+dx_3^2)=g|_U.\] Consequently smooth local isometric embeddings also fail for this metric.

Outline of the argument

A smooth three-space immersion obstruction: The necessary Darboux equation for an immersion height is obstructed on a sequence of small square patches. A candidate height with an appropriate Hessian component would violate the patch estimate. The metric is arranged to produce such a height from any hypothetical local immersion, while the patch amplitudes vanish fast enough to preserve smooth flatness at the origin.

Formal verification

Family 334 describes Lean coverage for a smooth metric for which every neighborhood of the origin fails three-space isometric immersion.

A smooth three-space immersion obstruction: The manuscript links IsometricImmersion.lean (declaration OAI.SmoothLocal.Geometry.exists_local_metric_without_local_immersion).

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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