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Hilbert's theorem

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative gaussian curvature K immersed in ${\\displaystyle \\mathbb {R} ^{3}}$ . This theorem answers the question for the negative case of which surfaces in ${\\displaystyle \\mathbb {R} ^{3}}$ can be obtained by isometrically immersing complete manifolds with constant curvature.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Hilbert%27s_theorem_(differential_geometry))

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