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 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 ...
.
History
* 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
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 on ...
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
:
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
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 canon ...
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
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 ' ...
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 m ...
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 ...
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
with coordinates
:
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
, i.e. a global isometry.
will be the map, whose domain is the hyperbolic plane and image the
2-dimensional manifold , which carries the inner product from the surface
with negative curvature.
will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces,
:
.
That is
:
,
where
. That is to say, the starting point
goes to the tangent plane from
through the inverse of the exponential map. Then travels from one tangent plane to the other through the isometry
, and then down to the surface
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 th ...
,
and
, around
and
respectively. The requirement will be that the axis are mapped to each other, that is
goes to
. Then
preserves the first fundamental form.
In a geodesic polar system, the
Gaussian curvature can be expressed as
:
.
In addition K is constant and fulfills the following differential equation
:
Since
and
have the same constant Gaussian curvature, then they are locally isometric (
Minding's Theorem). That means that
is a local isometry between
and
. Furthermore, from the Hadamard's theorem it follows that
is also a covering map.
Since
is simply connected,
is a homeomorphism, and hence, a (global) isometry. Therefore,
and
are globally isometric, and because
has an infinite area, then
has an infinite area, as well.
Lemma 2: For each
exists a parametrization
, such that the
coordinate curves of
are asymptotic curves of
and form a Tchebyshef net.
Lemma 3: Let
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
such that the coordinate curves are asymptotic curves in
. Then the area A of any quadrilateral formed by the coordinate curves is smaller than
.
The next goal is to show that
is a parametrization of
.
Lemma 4: For a fixed
, the curve
, is an asymptotic curve with
as arc length.
The following 2 lemmas together with lemma 8 will demonstrate the existence of a
parametrization
Lemma 5:
is a local diffeomorphism.
Lemma 6:
is
surjective.
Lemma 7: On
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 asympto ...
s of
.
Lemma 8:
is
injective.
''Proof of Hilbert's Theorem:''
First, it will be assumed that an isometric immersion from a
complete surface with negative curvature exists:
As stated in the observations, the tangent plane
is endowed with the metric induced by the exponential map
. Moreover,
is an isometric immersion and Lemmas 5,6, and 8 show the existence of a parametrization
of the whole
, such that the coordinate curves of
are the asymptotic curves of
. This result was provided by Lemma 4. Therefore,
can be covered by a union of "coordinate" quadrilaterals
with
. By Lemma 3, the area of each quadrilateral is smaller than
. On the other hand, by Lemma 1, the area of
is infinite, therefore has no bounds. This is a contradiction and the proof is concluded.
See also
*
Nash embedding theorem, 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.
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Hyperbolic geometry
Theorems in differential geometry
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