A smooth surface metric with no local isometric immersion in ℝ^(3)
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Constructs a smooth positive-definite metric on (-1,1)^2 for which no neighborhood of the origin admits a smooth isometric immersion into ℝ^(3). This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.
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MANUSCRIPT 01
A Smooth Metric with No Local Isometric Immersion into Three-Space
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…
Full source 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.
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