Hyperbolic space

Hyperbolic space

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, hyperbolic space is a type of non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

. Whereas spherical geometry
Spherical geometry
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....

has a constant positive curvature, hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

has a negative curvature: every point in hyperbolic space is a saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...

. Parallel lines are not uniquely paired: given a line and a point not on that line, any number of lines can be drawn through the point which are coplanar with the first and do not intersect it. This contrasts with Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, where parallel lines are a unique pair, and spherical geometry, where parallel lines do not exist, as all lines (which are great circles) cross each other.
Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, hyperbolic space is a type of non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

. Whereas spherical geometry
Spherical geometry
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....

has a constant positive curvature, hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

has a negative curvature: every point in hyperbolic space is a saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...

. Parallel lines are not uniquely paired: given a line and a point not on that line, any number of lines can be drawn through the point which are coplanar with the first and do not intersect it. This contrasts with Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, where parallel lines are a unique pair, and spherical geometry, where parallel lines do not exist, as all lines (which are great circles) cross each other. Another distinctive property is the amount of space covered by the n-ball
N-ball
N-ball is a two-dimensional physics game released by Rag Doll Software in 2005. The game is divided into four different campaigns with a total of 40 levels. The central idea of the game is move the glowing ball to the finish plate. This is done by holding or tapping the space bar to make the ball...

in hyperbolic n-space -- it increases exponentially with respect to the radius of the ball, rather than polynomially.

Formal definition

Hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

with constant sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

−1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

. It can be thought of as the negative-curvature analogue of the n-sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

.
Although hyperbolic space Hn is diffeomorphic to Rn its negative-curvature metric gives it very different geometric properties.

Hyperbolic 2-space, H², is also called the hyperbolic plane.

Models of hyperbolic space

Hyperbolic space, developed independently by Lobachevsky and Bolyai, is a geometrical space analogous to Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, but such that Euclid's parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

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 in fact infinitely many such lines through P. Note that this axiom still does not uniquely characterize the hyperbolic plane uniquely up to isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

; there is an extra constant, the curvature K<0, which must be specified. However, it does uniquely characterize it up to homothety, 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 K=-1.

Hyperbolic spaces are constructed in order to model such a modification of Euclidean geometry. In particular, the existence of model spaces implies that the parallel postulate is logically independent of the other axioms of Euclidean geometry.

There are several important models of hyperbolic space: the Klein model, the hyperboloid model, and the Poincaré model. These all model the same geometry in the sense that any two of them can be related by a transformation which preserves all the geometrical properties of the space. They are isometric
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

.

The hyperboloid model

The first model realizes hyperbolic space as a hyperboloid in Rn+1 = {(x0,...,xn)|xiR, i=0,1,...,n}. The hyperboloid is the locus Hn of points whose coordinates satisfy
In this model a "line" (or geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

) is the curve cut out by intersecting Hn with a plane through the origin in Rn+1.

The hyperboloid model is closely related to the geometry of Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

which defines the hyperboloid polarizes
Polarization identity
In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \|x\| \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...

to give the bilinear form B defined by
The space Rn+1, equipped with the bilinear form B is an (n+1)-dimensional Minkowski space Rn,1.

From this perspective, one can associate a notion of distance to the hyperboloid model, by defining the distance between two points x and y on H to be
This function satisfies the axioms of a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

. Moreover, it is preserved by the action of the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...

on Rn,1. Hence the Lorentz group acts as a transformation group of isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

on Hn.

The Klein model

An alternative model of hyperbolic geometry is on a certain domain
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

in projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

. The Minkowski quadratic form Q defines a subset UnRPn given as the locus of points for which Q(x) > 0 in the homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

x. The domain Un is the Klein model of hyperbolic space.

The lines of this model are the open line segments of the ambient projective space which lie in Un. The distance between two points x and y in Un is defined by
Note that this is well-defined on projective space, since the ratio under the inverse hyperbolic cosine is homogeneous of degree 0.

This model is related to the hyperboloid model as follows. Each point xUn corresponds to a line Lx through the origin in Rn+1, by the definition of projective space. This line intersects the hyperboloid Hn in a unique point. Conversely, through any point on Hn, there passes a unique line through the origin (which is a point in the projective space). This correspondence defines a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

between Un and Hn. It is an isometry since evaluating d(x,y) along Q(x) = Q(y) = 1 reproduces the definition of the distance given for the hyperboloid model.

The Poincaré models

Main articles: Poincaré disc model, Poincaré half-plane model
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....

Another closely related pair of models of hyperbolic geometry are the Poincaré ball and Poincaré half-space models. The ball model comes from a stereographic projection
Stereographic projection
The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...

of the hyperboloid in Rn+1 onto the hyperplane {x0 = 0}. In detail, let S be the point in Rn,1 with coordinates (-1,0,0,...,0): the South pole for the stereographic projection. For each point P on the hyperboloid Hn, let P* be the unique point of intersection of the line SP with the plane {x0 = 0}. This establishes a bijective mapping of Hn into the unit ball
in the plane {x0 = 0}.

The geodesics in this model are semicircle
Semicircle
In mathematics , a semicircle is a two-dimensional geometric shape that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle always measures 180° or a half turn...

s which are perpendicular to the boundary sphere of Bn. Isometries of the ball are generated by spherical inversion in hyperspheres perpendicular to the boundary.

The half-space model results from applying an inversion in a point of the boundary of Bn. This sends circles to circles and lines, and is moreover a conformal transformation. Consequently the geodesics of the half-space model are lines and circles perpendicular to the boundary hyperplane.

Hyperbolic manifolds

Every complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

, connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

, simply-connected manifold of constant negative curvature −1 is isometric
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

M of constant negative curvature −1, which is to say, a hyperbolic manifold
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...

, is Hn. Thus, every such M can be written as Hn/Γ where Γ is a torsion-free discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

of isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

on Hn. That is, Γ is a lattice
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...

in SO+(n,1).

Riemann surfaces

Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

s. According to the uniformization theorem
Uniformization theorem
In mathematics, the uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature...

, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

π1=Γ; the groups that arise this way are known as Fuchsian group
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...

s. The quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

H²/Γ of the upper half-plane modulo
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

the fundamental group is known as the Fuchsian model
Fuchsian model
In mathematics, a Fuchsian model is a construction of a hyperbolic Riemann surface R as a quotient of the upper half-plane H. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group \pi_1...

of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply connected and noncompact. It is the universal cover of the other hyperbolic surfaces.

The analogous construction for three-dimensional hyperbolic surfaces is the Kleinian model.

• Hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

• Mostow rigidity theorem
Mostow rigidity theorem
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique...

• Hyperbolic manifold
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...

• Hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...

• Pseudosphere
Pseudosphere
In geometry, the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid....

• Dini's surface
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:- Uses :...