Hilbert's Theorem on Regular Surfaces

Hilbert's theorem on regular surfaces states that no regular surface in R^3 having constant negative Gaussian curvature can be complete with respect to its induced Riemannian metric. Equivalently, the hyperbolic plane has no regular isometric immersion into R^3 (Hilbert 1901).

See also

Breather Surface, Complete Riemannian Metric, Gaussian Curvature, Hyperbolic Plane, Regular Surface

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References

Hilbert, D. "Ueber Flächen von constanter Gaussscher Krümmung." Trans. Amer. Math. Soc. 2, 87-99, 1901. https://doi.org/10.1090/S0002-9947-1901-1500557-5.

Cite this as:

Weisstein, Eric W. "Hilbert's Theorem on Regular Surfaces." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HilbertsTheoremonRegularSurfaces.html

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