In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, hyperbolic space of dimension n is the unique
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
, n-dimensional
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
of constant
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a p ...
equal to -1. It is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
, and satisfies the stronger property of being a
symmetric space. There are many ways to construct it as an open subset of
with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H
2, which was the first instance studied, is also called 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 ' ...
.
It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of
hyperbolic geometry
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 ...
. Sometimes the qualificative "real" is added to differentiate it from
complex hyperbolic spaces,
quaternionic hyperbolic spaces and the
octononic hyperbolic plane which are the other symmetric spaces of negative curvature.
Hyperbolic space serves as the prototype of a
Gromov hyperbolic space which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature. Another generalisation is the notion of a
CAT(-1) space.
Formal definition and models
Definition
The
-dimensional hyperbolic space or Hyperbolic
-space, usually denoted
, is the unique simply connected,
-dimensional
complete
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 ...
Riemannian manifold with a constant negative sectional curvature equal to -1. The unicity means that any two Riemannian manifolds which satisfy these properties are isometric to each other. It is a consequence of the
Killing–Hopf theorem
In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuou ...
.
Models of hyperbolic space
To prove the existence of such a space as described above one can explicitly construct it, for example as an open subset of
with a Riemannian metric given by a simple formula. There are many such constructions or models of hyperbolic space, each suited to different aspects of its study. They are isometric to each other according to the previous paragraph, and in each case an explicit isometry can be explicitly given. Here is a list of the better-known models which are described in more detail in their namesake articles:
*
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincar� ...
: this is the upper-half space
with the metric
*
Poincaré disc model: this is the unit ball of
with the metric
. The isometry to the half-space model can be realised by a
homography
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In gener ...
sending a point of the unit sphere to infinity.
*
Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbo ...
: In contrast with the previous two models this realises hyperbolic
-space as isometrically embedded inside the
-dimensional
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
(which is not a Riemannian but rather a
Lorentzian manifold). More precisely, looking at the quadratic form
on
, its restriction to the tangent spaces of the upper sheet of the
hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
given by
are definite positive, hence they endow it with a Riemannian metric which turns out to be of constant curvature -1. The isometry to the previous models can be realised by
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
from the hyperboloid to the plane
, taking the vertex from which to project to be
for the ball and a point at infinity in the cone
inside
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
for the half-space.
*
Klein model: This is another model realised on the unit ball of
; rather than being given as an explicit metric it is usually presented as obtained by using stereographic projection from the hyperboloid model in Minkowski space to its horizontal tangent plane (that is,
) from the origin
.
*Symmetric space: Hyperbolic
-space can be realised as the symmetric space of the simple Lie group
(the group of isometries of the quadratic form
with positive determinant); as a set the latter is the
coset space . The isometry to the hyperboloid model is immediate through the action of the connected component of
on the hyperboloid.
Geometric properties
Parallel lines
Hyperbolic space, developed independently by
Nikolai Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, ...
,
János Bolyai
János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consis ...
and
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, is a geometrical space analogous to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, but such that
Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):
* Given any line ''L'' and point ''P'' not on ''L'', there are at least two distinct lines passing through ''P'' which do not intersect ''L''.
It is then a theorem that there are infinitely many such lines through ''P''. This axiom still does not uniquely characterize the hyperbolic plane up to
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
; there is an extra constant, the curvature , which must be specified. However, it does uniquely characterize it up to
homothety
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...
, meaning up to bijections which only change the notion of distance by an overall constant. By choosing an appropriate length scale, one can thus assume, without loss of generality, that .
Euclidean embeddings
The hyperbolic plane cannot be isometrically embedded into Euclidean 3-space by
Hilbert's theorem. On the other hand the
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 ...
implies that hyperbolic n-space can be isometrically embedded into some Euclidean space of larger dimension (4 for the hyperbolic plane).
When isometrically embedded to a Euclidean space every point of a hyperbolic space is a
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...
.
Volume growth and isoperimetric inequality
The volume of balls in hyperbolic space increases
exponentially with respect to the radius of the ball rather than
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
ly as in Euclidean space. Namely, if
is any ball of radius
in
then:
where
is the total volume of the Euclidean
-sphere of radius 1.
The hyperbolic space also satisfies a linear
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
, that is there exists a constant
such that any embedded disk whose boundary has length
has area at most
. This is to be contrasted with Euclidean space where the isoperimetric inequality is quadratic.
Other metric properties
There are many more metric properties of hyperbolic space which differentiate it from Euclidean space. Some can be generalised to the setting of Gromov-hyperbolic spaces which is a generalisation of the notion of negative curvature to general metric spaces using only the large-scale properties. A finer notion is that of a CAT(-1)-space.
Hyperbolic manifolds
Every
complete
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 ...
,
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
,
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
manifold of constant negative curvature −1 is
isometric to the real hyperbolic space H
''n''. As a result, the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of any
closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example i ...
''M'' of constant negative curvature −1, which is to say, a
hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, r ...
, is H
''n''. Thus, every such ''M'' can be written as H
''n''/Γ where Γ is a
torsion-free discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
of
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
on H
''n''. That is, Γ is a
lattice in
SO+(''n'',1).
Riemann surfaces
Two-dimensional hyperbolic surfaces can also be understood according to the language of
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s. According to the
uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
π
1=Γ; the groups that arise this way are known as
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
s. The
quotient space H²/Γ of the upper half-plane
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
the fundamental group is known as the
Fuchsian model of the hyperbolic surface. The
Poincaré half plane is also hyperbolic, but is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
and
noncompact. It is the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of the other hyperbolic surfaces.
The analogous construction for three-dimensional hyperbolic surfaces is the
Kleinian model
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold ''N'' by the quotient space \mathbb^3 / \Gamma where \Gamma is a discrete subgroup of PSL(2,C). Here, the subgroup \Gamma, a Kleinian group, is defined so ...
.
See also
*
Dini's surface
In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations:
: \begin
x&=a \cos u \sin v \\
y&=a \ ...
*
Hyperbolic 3-manifold
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...
*
Ideal polyhedron
*
Mostow rigidity theorem Mostow may refer to: People
* George Mostow (1923–2017), American mathematician
** Mostow rigidity theorem
* Jonathan Mostow
Jonathan Mostow (born November 28, 1961) is an American film director, screenwriter, and producer. He has directed f ...
*
Murakami–Yano formula
*
Pseudosphere
References
{{reflist
* Ratcliffe, John G., ''Foundations of hyperbolic manifolds'', New York, Berlin. Springer-Verlag, 1994.
* Reynolds, William F. (1993) "Hyperbolic Geometry on a Hyperboloid",
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an ...
100:442–455.
* Wolf, Joseph A. ''Spaces of constant curvature'', 1967. See page 67.
Homogeneous spaces
Hyperbolic geometry
Topological spaces