Hilbert's theorem

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Template:Curvature result for surfaces

Template:Relating curvature with embedding

This result is about manifolds in dimension:2

This article or section of article is sourced from:Wikipedia


There exists no complete regular surface of constant negative Gaussian curvature, embedded in \R^3.


Note that there could exist regular surfaces of constant negative curvature, but that such regular surfaces must have boundary points or cusp points at which the curvature no longer remains negative.

ALso Hilbert's theorem is of relevance only for orientable surfaces because non-orientable surfaces anyway cannot be embedded in \R^3.



  • Über Flächen von konstanter Krümmung by David Hilbert, Trans. Amer. Math. Soc. 2 (1901), 87-99

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