OpenAI · 334-01 · 2026-09-24 · Differential geometry
A Smooth Metric with No Local Isometric Immersion into Three-Space
preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026
Family 334. A smooth surface metric with no local immersion in .
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Constructs a smooth positive-definite metric on for which no neighborhood of the origin admits a smooth isometric immersion into . This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.
We construct a smooth positive-definite Riemannian metric on 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 334: A smooth surface metric with no local isometric immersion in ℝ<sup>3</sup>. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.