In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface
of constant negative gaussian curvature
This theorem answers the question for the negative case of which surfaces in
can be obtained by isometrically immersing complete manifolds with constant curvature.
The proof of Hilbert's theorem is elaborate and requires several lemmas.
The idea is to show the nonexistence of an isometric immersion of a plane
This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and Spivak.
Observations: In order to have a more manageable treatment, but without loss of generality, the curvature may be considered equal to minus one,
There is no loss of generality, since it is being dealt with constant curvatures, and similarities of
is a local diffeomorphism (in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an inner product in the tangent space of
is an isometric immersion, the same holds for 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.
has an infinite area comes by computing the surface integral with the corresponding coefficients of the First fundamental form.
Since the hyperbolic plane is unbounded, the limits of the integral are infinite, and the area can be calculated through 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
will be the map, whose domain is the hyperbolic plane and image the 2-dimensional manifold
will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces, That is where
Then travels from one tangent plane to the other through the isometry
The following step involves the use of polar coordinates,
In a geodesic polar system, the Gaussian curvature
can be expressed as In addition K is constant and fulfills the following differential equation Since
have the same constant Gaussian curvature, then they are locally isometric (Minding's Theorem).
and form a Tchebyshef net.
Then the area A of any quadrilateral formed by the coordinate curves is smaller than
there are two differentiable linearly independent vector fields which are tangent to the asymptotic curves of
Proof of Hilbert's Theorem: First, it will be assumed that an isometric immersion from a complete surface
As stated in the observations, the tangent plane
is endowed with the metric induced by the exponential map
is an isometric immersion and Lemmas 5,6, and 8 show the existence of a parametrization
can be covered by a union of "coordinate" quadrilaterals
By Lemma 3, the area of each quadrilateral is smaller than
On the other hand, by Lemma 1, the area of