# 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**

### 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)**

### Euclidean geometry

**plane geometryEuclideanEuclidean plane geometry**

### Elliptic geometry

**ellipticelliptic spaceelliptical geometry**

### Saddle point

**saddle surfacesaddlesaddle points**

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.