# Hilbert's theorem (differential geometry)

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface $S$ of constant negative gaussian curvature $K$ immersed in $\mathbb {R} ^{3}$ . This theorem answers the question for the negative case of which surfaces in $\mathbb {R} ^{3}$ 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 négative," (1902).

## Proof

The proof of Hilbert's theorem is elaborate and requires several lemmas. The idea is to show the nonexistence of an isometric immersion

$\varphi =\psi \circ \exp _{p}:S'\longrightarrow \mathbb {R} ^{3}$ of a plane $S'$ to the real space $\mathbb {R} ^{3}$ . This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and Spivak.

$\varphi =\psi \circ \exp _{o}:S'\longrightarrow \mathbb {R} ^{3}$ .

The first lemma is independent from the other ones, and will be used at the end as the counter statement to reject the results from the other lemmas.

Lemma 1: The area of $S'$ is infinite.
Proof's Sketch:
The idea of the proof is to create a global isometry between $H$ and $S'$ . Then, since $H$ has an infinite area, $S'$ will have it too.
The fact that the hyperbolic plane $H$ has an infinite area comes by computing the surface integral with the corresponding coefficients of the First fundamental form. To obtain these ones, the hyperbolic plane can be defined as the plane with the following inner product around a point $q\in \mathbb {R} ^{2}$ with coordinates $(u,v)$ $E=\left\langle {\frac {\partial }{\partial u}},{\frac {\partial }{\partial u}}\right\rangle =1\qquad F=\left\langle {\frac {\partial }{\partial u}},{\frac {\partial }{\partial v}}\right\rangle =\left\langle {\frac {\partial }{\partial v}},{\frac {\partial }{\partial u}}\right\rangle =0\qquad G=\left\langle {\frac {\partial }{\partial v}},{\frac {\partial }{\partial v}}\right\rangle =e^{u}$ Since the hyperbolic plane is unbounded, the limits of the integral are infinite, and the area can be calculated through

$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{u}dudv=\infty$ Next it is needed to create a map, which will show that the global information from the hyperbolic plane can be transfer to the surface $S'$ , i.e. a global isometry. $\varphi :H\rightarrow S'$ will be the map, whose domain is the hyperbolic plane and image the 2-dimensional manifold $S'$ , which carries the inner product from the surface $S$ with negative curvature. $\varphi$ will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces,

$\psi :T_{p}(H)\rightarrow T_{p'}(S')$ .

That is

$\varphi =\exp _{p'}\circ \psi \circ \exp _{p}^{-1}$ ,

where $p\in H,p'\in S'$ . That is to say, the starting point $p\in H$ goes to the tangent plane from $H$ through the inverse of the exponential map. Then travels from one tangent plane to the other through the isometry $\psi$ , and then down to the surface $S'$ with another exponential map.

$K=-{\frac {({\sqrt {G}})_{\rho \rho }}{\sqrt {G}}}$ .

In addition K is constant and fulfills the following differential equation

$({\sqrt {G}})_{\rho \rho }+K\cdot {\sqrt {G}}=0$ The following 2 lemmas together with lemma 8 will demonstrate the existence of a parametrization $x:\mathbb {R} ^{2}\longrightarrow S'$ Lemma 7: On $S'$ there are two differentiable linearly independent vector fields which are tangent to the asymptotic curves of $S'$ .

Proof of Hilbert's Theorem:
First, it will be assumed that an isometric immersion from a complete surface with negative curvature$S$ exists: $\psi :S\longrightarrow \mathbb {R} ^{3}$ 