A smooth surface metric with no local immersion in R^3 at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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A smooth surface metric with no local immersion in $\mathbb R^3$

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	curvy surfaces	Levels:	1
Category:	Differential geometry	Lean version:	YES! ✔
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A smooth surface metric with no local isometric immersion in ℝ³. 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.

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

released 2026-09-24 | 7 lemmas · 10 proofs · 8,579 words | PLAY LEVEL 1 » (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.

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