The Hilbert theorem states that there exists no complete regular surface S of constant negative gaussian curvature $K$ immersed in $R^3$.
Ok.. so I'm guessing that the surface of revolution of the Tractrix, which has constant negative curvature, is not complete or regular. Furthermore, in Minkowski 3d space we can define a surface with the following equation:
i.e. the locus of points equidistant of the origin. This surface has constant negative curvature. In fact, a simple projection of this surface on the $y_1y_2$ plane gives us the Poincaré disc, a commonly used hyperbolic model, and other models. Is this surface also not complete or regular? If so, what about the Poincaré disc then?
I'm having a hard time understanding Hilbert's theorem. Also, I understand that the surface of revolution of the Tractrix is "smaller" than the surface defined in the Minkowski 3d space (and consequently the Poincaré disc and the other hyperbolic models), in the sense that you can fit one inside the other. Is this relevant? (sorry for the lack of technical terms ;)