Family 334Differential geometryLeanNot yet digested
A smooth surface metric with no local isometric immersion in ℝ
Constructs a smooth positive-definite metric on for which no neighborhood of the origin admits a smooth isometric immersion into ℝ. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.
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A Smooth Metric with No Local Isometric Immersion into Three-Space
Sep 24, 2026·paper.pdfLean
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.