Hilbert's theorem on the hyperbolic plane

Remarks

After scaling, one may assume $K = -1$; one may also pass to the universal cover, which is the hyperbolic plane. For an immersed surface with $K < 0$, the two asymptotic directions at each point give coordinates in which the angle $\omega$ between the coordinate curves satisfies the sine-Gordon equation $\omega_{uv} = \sin\omega$. Completeness lets these coordinates cover the whole plane, and the equation then bounds the area of every coordinate quadrilateral by $2\pi$. This contradicts the fact that the hyperbolic plane has infinite area.

The pseudosphere has constant curvature $-1$ and lies in $\R^3$, but it is not complete: it ends in a sharp edge. The smoothness hypothesis matters: by the Nash–Kuiper theorem there are $C^1$ isometric embeddings of the hyperbolic plane into $\R^3$. For constant positive curvature the situation is different, since round spheres are complete and lie in $\R^3$.

Going deeper

For a student

The theorem explains why the hyperbolic plane is studied as an abstract Riemannian manifold rather than as a surface in space. Efimov later showed that a complete surface with curvature bounded above by a negative constant cannot be smoothly immersed in $\R^3$ either. In contrast, the Nash embedding theorems show that every Riemannian manifold embeds isometrically in some higher-dimensional Euclidean space. The proof is a good example of how the Gauss equation turns a geometric problem into a partial differential equation.