117.194.87.199 at 16:51, 12 November 2013

2013-11-12T16:51:50Z

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In [[differential geometry]], '''Hilbert's theorem''' (1901) states that there exists no complete [[regular surface]] <math>S</math> of constant negative [[gaussian curvature]] <math>K</math> [[immersion (mathematics)|immersed]] in <math>\mathbb{R}^{3}</math>. This theorem answers the question for the negative case of which surfaces in <math>\mathbb{R}^{3}</math> can be obtained by isometrically immersing [[complete manifold]]s 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 negative," (1902).

==Proof==
The [[proof (mathematics)|proof]] of Hilbert's theorem is elaborate and requires several [[lemma (mathematics)|lemma]]s. The idea is to show the nonexistence of an isometric [[immersion (mathematics)|immersion]]
:<math>\varphi = \psi \circ \exp_p: S' \longrightarrow \mathbb{R}^{3}</math>

of a plane <math>S'</math> to the real space <math>\mathbb{R}^{3}</math>. This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and [[Michael Spivak|Spivak]].

''Observations'': In order to have a more manageable treatment, but without loss of generality, the [[curvature]] may be considered equal to minus one, <math>K=-1</math>. There is no loss of generality, since it is being dealt with constant curvatures, and similarities of <math>\mathbb{R}^{3}</math> multiply <math>K</math> by a constant. The [[exponential map]] <math>\exp_p: T_p(S) \longrightarrow S</math> is a [[local diffeomorphism]] (in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an [[inner product]] in the [[tangent space]] of <math>S</math> at <math>p</math>: <math>T_p(S)</math>. Furthermore, <math>S'</math> denotes the geometric surface <math>T_p(S)</math> with this inner product. If <math>\psi:S \longrightarrow \mathbb{R}^{3}</math> is an isometric immersion, the same holds for
:<math>\varphi = \psi \circ \exp_o:S' \longrightarrow \mathbb{R}^{3}</math>.

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 <math>S'</math> is infinite. 
''Proof's Sketch:'' 
The idea of the proof is to create a [[global isometry]] between <math>H</math> and <math>S'</math>. Then, since <math>H</math> has an infinite area, <math>S'</math> will have it too. 
The fact that the [[Hyperbolic manifold|hyperbolic plane]] <math>H</math> has an infinite area comes by computing the [[surface integral]] with the corresponding [[coefficient]]s 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 <math>q\in \mathbb{R}^{2}</math> with coordinates <math>(u,v)</math> 
:<math>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} </math> 

Since the hyperbolic plane is unbounded, the limits of the integral are [[Infinity|infinite]], and the area can be calculated through
:<math>\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{u} du dv = \infty</math>

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 <math>S'</math>, i.e. a global isometry. <math>\varphi: H \rightarrow S'</math> will be the map, whose domain is the hyperbolic plane and image the [[2-dimensional manifold]] <math>S'</math>, which carries the inner product from the surface <math>S</math> with negative curvature. <math>\varphi</math> will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces,
:<math>\psi:T_p(H) \rightarrow T_{p'}(S')</math>.

That is
:<math>\varphi = \exp_{p'} \circ \psi \circ \exp_p^{-1}</math>,

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

The following step involves the use of [[polar coordinates]], <math>(\rho, \theta)</math> and <math>(\rho', \theta')</math>, around <math>p</math> and <math>p'</math> respectively. The requirement will be that the axis are mapped to each other, that is <math>\theta=0</math> goes to <math>\theta'=0</math>. Then <math>\varphi</math> preserves the first fundamental form. 
In a geodesic polar system, the [[Gaussian curvature]] <math>K</math> can be expressed as
:<math>K = - \frac{(\sqrt{G})_{\rho \rho}}{\sqrt{G}}</math>.

In addition K is constant and fulfills the following differential equation
:<math>(\sqrt{G})_{\rho \rho} + K\cdot \sqrt{G} = 0</math>

Since <math>H</math> and <math>S'</math> have the same constant Gaussian curvature, then they are locally isometric ([[Minding's Theorem]]). That means that <math>\varphi</math> is a local isometry between <math>H</math> and <math>S'</math>. Furthermore, from the Hadamard's theorem it follows that <math>\varphi</math> is also a covering map. 
Since <math>S'</math> is simply connected, <math>\varphi</math> is a homeomorphism, and hence, a (global) isometry. Therefore, <math>H</math> and <math>S'</math> are globally isometric, and because <math>H</math> has an infinite area, then <math>S'=T_p(S)</math> has an infinite area, as well. <math>\square</math>

'''Lemma 2''': For each <math>p\in S'</math> exists a parametrization <math>x:U \subset \mathbb{R}^{2} \longrightarrow S', \qquad p \in x(U)</math>, such that the coordinate curves of <math>x</math> are asymptotic curves of <math> x(U) = V'</math> and form a Tchebyshef net.

'''Lemma 3''': Let <math>V' \subset S'</math> be a coordinate [[neighborhood]] of <math>S'</math> such that the coordinate curves are asymptotic curves in <math>V'</math>. Then the area A of any quadrilateral formed by the coordinate curves is smaller than <math>2\pi</math>.

The next goal is to show that <math>x</math> is a parametrization of <math>S'</math>.

'''Lemma 4''': For a fixed <math>t</math>, the curve <math>x(s,t), -\infty < s < +\infty </math>, is an asymptotic curve with <math>s</math> as arc length.

The following 2 lemmas together with lemma 8 will demonstrate the existence of a [[parametrization]] <math>x:\mathbb{R}^{2} \longrightarrow S'</math>

'''Lemma 5''': <math>x</math> is a local diffeomorphism.

'''Lemma 6''': <math>x</math> is [[surjective]].

'''Lemma 7''': On <math>S'</math> there are two differentiable linearly independent vector fields which are tangent to the [[asymptotic curve]]s of <math>S'</math>.

'''Lemma 8''': <math>x</math> is [[injective]]. 

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

As stated in the observations, the tangent plane <math>T_p(S)</math> is endowed with the metric induced by the exponential map <math>\exp_p: T_p(S) \longrightarrow S</math> . Moreover, <math>\varphi = \psi \circ \exp_p:S' \longrightarrow \mathbb{R}^{3}</math> is an isometric immersion and Lemmas 5,6, and 8 show the existence of a parametrization <math>x:\mathbb{R}^{2} \longrightarrow S'</math> of the whole <math>S'</math>, such that the coordinate curves of <math>x</math> are the asymptotic curves of <math>S'</math>. This result was provided by Lemma 4. Therefore, <math>S'</math> can be covered by a union of "coordinate" quadrilaterals <math>Q_{n}</math> with <math> Q_{n} \subset Q_{n+1}</math>. By Lemma 3, the area of each quadrilateral is smaller than <math>2 \pi </math>. On the other hand, by Lemma 1, the area of <math>S'</math> is infinite, therefore has no bounds. This is a contradiction and the proof is concluded. <math>\square</math>

==References==
* {{aut|Do Carmo, Manfredo}}, ''Differential Geometry of Curves and Surfaces'', Prentice Hall, 1976.
* {{aut|[[Michael Spivak|Spivak, Michael]]}}, ''A Comprenhensive Introduction to Differential Geometry'', Publish or Perish, 1999.

{{DEFAULTSORT:Hilberts theorem}}
[[Category:Hyperbolic geometry]]
[[Category:Theorems in differential geometry]]
[[Category:Articles containing proofs]]

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117.194.87.199 at 16:51, 12 November 2013