Hilbert's theorem

This result is about manifolds in dimension:2

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Statement

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

Facts

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 ${\mathbb{R}}^3$.

References

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

External links

• Wikipedia page