Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 334-01. A Smooth Metric with No Local Isometric Immersion into Three-Space
Manuscript map · lines 8865–8874
OpenAI@2026-10-07#rem:oai-p-334-01
Authors: OpenAI. Family 334.
Title: A Smooth Metric with No Local Isometric Immersion into Three-Space.
Abstract: We construct a smooth positive-definite Riemannian metric on $(-1,1){}2$ 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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