Hilbert's theorem (differential geometry)

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface ${\displaystyle S}$ of constant negative gaussian curvature ${\displaystyle K}$ 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.