Hilbert's theorem (differential geometry)
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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative
gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
K immersed in \mathbb^. This theorem answers the question for the negative case of which surfaces in \mathbb^ can be obtained by isometrically immersing
complete manifold Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
s with
constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...
.


History

* Hilbert's theorem was first treated by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
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 Nikolai Vladimirovich Yefimov (russian: Никола́й Влади́мирович Ефи́мов; 31 May 1910 in Orenburg – 14 August 1982 in Moscow) was a Soviet mathematician. He is most famous for his work on generalized Hilbert's problem o ...
in 1975.Ефимов, Н. В. Непогружаемость полуплоскости Лобачевского. Вестн. МГУ. Сер. мат., мех. — 1975. — No 2. — С. 83—86.


Proof

The proof of Hilbert's theorem is elaborate and requires several lemmas. The idea is to show the nonexistence of an isometric
immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux * Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
:\varphi = \psi \circ \exp_p: S' \longrightarrow \mathbb^ of a plane S' to the real space \mathbb^. 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 ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
, the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
may be considered equal to minus one, K=-1. There is no loss of generality, since it is being dealt with constant curvatures, and similarities of \mathbb^ multiply K by a constant. The exponential map \exp_p: T_p(S) \longrightarrow S is a
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Form ...
(in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
in the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
of S at p: T_p(S). Furthermore, S' denotes the geometric surface T_p(S) with this inner product. If \psi:S \longrightarrow \mathbb^ is an isometric immersion, the same holds for :\varphi = \psi \circ \exp_o:S' \longrightarrow \mathbb^. 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
H has an infinite area comes by computing the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
with the corresponding
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s of the
First fundamental form In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and me ...
. To obtain these ones, the hyperbolic plane can be defined as the plane with the following inner product around a point q\in \mathbb^ with coordinates (u,v)
:E = \left\langle \frac, \frac \right\rangle = 1 \qquad F = \left\langle \frac, \frac \right\rangle = \left\langle \frac, \frac \right\rangle = 0 \qquad G = \left\langle \frac, \frac \right\rangle = e^ Since the hyperbolic plane is unbounded, the limits of the integral are
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
, and the area can be calculated through :\int_^ \int_^ e^ du dv = \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_(S'). That is :\varphi = \exp_ \circ \psi \circ \exp_p^, 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. The following step involves the use of
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, (\rho, \theta) and (\rho', \theta'), around p and p' respectively. The requirement will be that the axis are mapped to each other, that is \theta=0 goes to \theta'=0. Then \varphi preserves the first fundamental form.
In a geodesic polar system, the
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
K can be expressed as :K = - \frac. In addition K is constant and fulfills the following differential equation :(\sqrt)_ + K\cdot \sqrt = 0 Since H and S' have the same constant Gaussian curvature, then they are locally isometric ( Minding's Theorem). That means that \varphi is a local isometry between H and S'. Furthermore, from the Hadamard's theorem it follows that \varphi is also a covering map.
Since S' is simply connected, \varphi is a homeomorphism, and hence, a (global) isometry. Therefore, H and S' are globally isometric, and because H has an infinite area, then S'=T_p(S) has an infinite area, as well. \square Lemma 2: For each p\in S' exists a parametrization x:U \subset \mathbb^ \longrightarrow S', \qquad p \in x(U), such that the
coordinate curves In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
of x are asymptotic curves of x(U) = V' and form a Tchebyshef net. Lemma 3: Let V' \subset S' be a coordinate
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of S' such that the coordinate curves are asymptotic curves in V'. Then the area A of any quadrilateral formed by the coordinate curves is smaller than 2\pi. The next goal is to show that x is a parametrization of S'. Lemma 4: For a fixed t, the curve x(s,t), -\infty < s < +\infty , is an asymptotic curve with s as arc length. The following 2 lemmas together with lemma 8 will demonstrate the existence of a parametrization x:\mathbb^ \longrightarrow S' Lemma 5: x is a local diffeomorphism. Lemma 6: x is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
. Lemma 7: On S' there are two differentiable linearly independent vector fields which are tangent to the
asymptotic curve In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a line. Definitions An asymp ...
s of S'. Lemma 8: x is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. ''Proof of Hilbert's Theorem:''
First, it will be assumed that an isometric immersion from a
complete surface Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
S with negative curvature exists: \psi:S \longrightarrow \mathbb^ As stated in the observations, the tangent plane T_p(S) is endowed with the metric induced by the exponential map \exp_p: T_p(S) \longrightarrow S . Moreover, \varphi = \psi \circ \exp_p:S' \longrightarrow \mathbb^ is an isometric immersion and Lemmas 5,6, and 8 show the existence of a parametrization x:\mathbb^ \longrightarrow S' of the whole S', such that the coordinate curves of x are the asymptotic curves of S'. This result was provided by Lemma 4. Therefore, S' can be covered by a union of "coordinate" quadrilaterals Q_ with Q_ \subset Q_. By Lemma 3, the area of each quadrilateral is smaller than 2 \pi . On the other hand, by Lemma 1, the area of S' is infinite, therefore has no bounds. This is a contradiction and the proof is concluded. \square


See also

*
Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
, states that every Riemannian manifold can be isometrically embedded into some Euclidean space.


References

* , ''Differential Geometry of Curves and Surfaces'', Prentice Hall, 1976. * , ''A Comprehensive Introduction to Differential Geometry'', Publish or Perish, 1999. {{DEFAULTSORT:Hilberts theorem Hyperbolic geometry Theorems in differential geometry Articles containing proofs