# Hyperbolic space

hyperbolic 3-spacehyperbolic planehyperbolic 4-spacehyperbolichyperbolic 5-spacehyperbolic geometryhyperbolic ''n''-spacehyperbolic 3-space \mathbb H^3hyperbolic manifoldhyperbolic n-space
In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.wikipedia
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### Homogeneous space

homogeneoushomogeneous spacescoset space
In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.
The same is true of the models found of non-Euclidean geometry of constant curvature, such as hyperbolic space.

### Hyperbolic geometry

hyperbolic planehyperbolichyperbolic surface
It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.
Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases.

### Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature
Hyperbolic n-space, denoted H n, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with a constant negative sectional curvature.

### Constant curvature

curvatureconstantconstant curve
In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

### Hyperbolic manifold

hyperbolic planehyperbolic metrichyperbolic
As a result, the universal cover of any closed manifold M of constant negative curvature −1, which is to say, a hyperbolic manifold, is H n.
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension.

### Minkowski space

Minkowski spacetimeMinkowski metricflat spacetime
The hyperboloid model is closely related to the geometry of Minkowski space.
It is one of the model spaces of Riemannian geometry, the hyperboloid model of hyperbolic space.

### Hyperbolic 3-manifold

Hyperbolic 3-Manifoldshyperbolic structurescusped hyperbolic
It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group).

### Fuchsian group

Fuchsian symmetry groups
Most hyperbolic surfaces have a non-trivial fundamental group π 1 =Γ; the groups that arise this way are known as Fuchsian groups.
Then H is a model of the hyperbolic plane when endowed with the metric

### Lorentz group

orthochronousorthochronous Lorentz groupSO(3,1)
It is preserved by the action of the Lorentz group on R n,1.
But the homogeneous space SO + (1,3)/SO(3) is homeomorphic to hyperbolic 3-space H 3, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H 3.

### Poincaré half-plane model

Poincaré half-space modelPoincaré hyperbolic diskPoincaré half-plane
The Poincaré half plane is also hyperbolic, but is simply connected and noncompact.
This model can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.

### Mostow rigidity theorem

Mostow rigidityMostow's rigidity theoremstrong rigidity
Let \mathbb H^n be the n-dimensional hyperbolic space.

### Pseudosphere

pseudospherical surfaceTractricoidpseudo-sphere
It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane.

### Uniformization theorem

uniformizationuniformisationUniformisation Theorem
According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic.

### Discrete group

discrete subgroupdiscretediscrete subgroups
Thus, every such M can be written as H n /Γ where Γ is a torsion-free discrete group of isometries on H n.

### Mathematics

mathematicalmathmathematician
In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

### Constant function

constantconstant mapconstant mapping
In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

### Dimension

dimensionsdimensionalone-dimensional
It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

### Euclidean space

EuclideanspaceEuclidean vector space
It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

### 0

zerozero function0 (number)
It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

### Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

### Elliptic geometry

ellipticelliptic spaceelliptical geometry
It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point.

### Volume form

Riemannian volume formamount of spacesurface element
Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially.

### Ball (mathematics)

ballopen ballballs
Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially.

### Exponential growth

exponentiallyexponentialgrow exponentially
Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially.