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Concept# Hilbert's theorem (differential geometry)

Summary

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface of constant negative gaussian curvature immersed in . This theorem answers the question for the negative case of which surfaces in 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 in "Sur les surfaces à courbure constante négative" (1902).
A far-leading generalization was obtained by Nikolai Efimov in 1975.
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 to the real space . 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 multiply by a constant. The exponential map is a local diffeomorphism (in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an inner product in the tangent space of at : . Furthermore, denotes the geometric surface with this inner product. If 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.
Lemma 1: The area of is infinite.
Proof's Sketch:
The idea of the proof is to create a global isometry between and . Then, since has an infinite area, will have it too.
The fact that the hyperbolic plane has an infinite area comes by computing the surface integral with the corresponding coefficients of the First fundamental form.

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