Hilbert's theorem (differential geometry)

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S {\displaystyle S} of constant negative gaussian curvature K {\displaystyle K} immersed in R 3 {\displaystyle \mathbb {R} ^{3}} . This theorem answers the question for the negative case of which surfaces in R 3 {\displaystyle \mathbb {R} ^{3}} can be obtained by isometrically immersing complete manifolds with constant curvature.

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