Hilbert's theorem on regular surfaces states that no regular surface in 
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 
(Hilbert 1901).
Hilbert's Theorem on Regular Surfaces
See also
Breather Surface, Complete Riemannian Metric, Gaussian Curvature, Hyperbolic Plane, Regular SurfaceExplore with Wolfram|Alpha
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