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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In differential geometry, Hilbert''s theorem (1901) states that there exists no complete regular surface S of constant negative Gaussian curvature K immersed in R3. This theorem answers the question for the negative case of which surfaces in R3 can be obtained by isometrically immersing complete manifolds with constant curvature. Hilbert''s theorem was first treated by David Hilbert in, "Über Flächen von konstanter Krümmung" (Trans. Amer. Math. Soc. 2 (1901), 87-99). A different proof was given shortly after by E. Holmgren, "Sur les surfaces à courbure constante negative," (1902).