A smooth surface metric with no local isometric immersion in ℝ3

What OpenAI claims

Constructs a smooth positive-definite metric on (-1,1)² for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood. (Lean)

Paper

1. A Smooth Metric with No Local Isometric Immersion into Three-Space

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.