# Hyperbolic space

Last updated A perspective projection of a dodecahedral tessellation in H .Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of $\mathbb {R} ^{n}$ with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane.

## Contents

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. 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 $n$ -dimensional hyperbolic space or Hyperbolic $n$ -space, usually denoted $\mathbb {H} ^{n}$ , is the unique simply connected, $n$ -dimensional complete 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.

### 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 $\mathbb {R} ^{n}$ 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: this is the upper-half space $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}>0\}$ with the metric ${\tfrac {dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{2}}}$ • Poincaré disc model: this is the unit ball of $\mathbb {R} ^{n}$ with the metric $4{\tfrac {dx_{1}^{2}+\cdots +dx_{n}^{2}}{(1-(x_{1}^{2}+\cdots +x_{n}^{2}))^{2}}}$ . The isometry to the half-space model can be realised by a homography sending a point of the unit sphere to infinity.
• Hyperboloid model: In contrast with the previous two models this realises hyperbolic $n$ -space as isometrically embedded inside the $(n+1)$ -dimensional Minkowski space (which is not a Riemannian but rather a Lorentzian manifold). More precisely, looking at the quadratic form $q(x)=x_{1}^{2}+\cdots +x_{n}^{2}-x_{n+1}^{2}$ on $\mathbb {R} ^{n+1}$ , its restriction to the tangent spaces of the upper sheet of the hyperboloid given by $q(x)=-1$ 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 from the hyperboloid to the plane $\{x_{n+1}=0\}$ , taking the vertex from which to project to be $(0,\ldots ,0,1)$ for the ball and a point at infinity in the cone $q(x)=0$ inside projective space for the half-space.
• Klein model: This is another model realised on the unit ball of $\mathbb {R} ^{n}$ ; 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, $x_{n+1}=1$ ) from the origin $(0,\ldots ,0)$ .
• Symmetric space: Hyperbolic $n$ -space can be realised as the symmetric space of the simple Lie group $\mathrm {SO} (n,1)$ (the group of isometries of the quadratic form $q$ with positive determinant); as a set the latter is the coset space $\mathrm {SO} (n,1)/\mathrm {O} (n)$ . The isometry to the hyperboloid model is immediate through the action of the connected component of $\mathrm {SO} (n,1)$ on the hyperboloid.

## Geometric properties

### Parallel lines

Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometrical space analogous to Euclidean space, 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; 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.

### 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 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.

### Volume growth and isoperimetric inequality

The volume of balls in hyperbolic space increases exponentially with respect to the radius of the ball rather than polynomially as in Euclidean space. Namely, if $B(r)$ is any ball of radius $r$ in $\mathbb {H} ^{n}$ then:

$\mathrm {Vol} (B(r))=\mathrm {Vol} (S^{n-1})\int _{0}^{r}\sinh ^{n-1}(t)dt$ where $S^{n-1}$ is the total volume of the Euclidean $(n-1)$ -sphere of radius 1.

The hyperbolic space also satisfies a linear isoperimetric inequality, that is there exists a constant $i$ such that any embedded disk whose boundary has length $r$ has area at most $i\cdot r$ . 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, 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 M of constant negative curvature 1, which is to say, a hyperbolic manifold, is Hn. Thus, every such M can be written as Hn/Γ where Γ is a torsion-free discrete group of isometries on Hn. 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 surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group π1=Γ; the groups that arise this way are known as Fuchsian groups. The quotient space H²/Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model 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.

## Related Research Articles

In differential geometry, a Riemannian manifold or Riemannian space(M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure".

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature Kp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold. In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry, diffeomorphism, or homeomorphism (topology). Some authors insist that the action of G be faithful, although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. 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, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

In mathematics, the Teichmüller space of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm. Besson, Courtois & Gallot (1996) gave the simplest available proof.

In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature. 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 hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S+ or by wedge products of m vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the n-sphere is embedded in (n+1)-dimensional Euclidean space.

In differential geometry, the Margulis lemma is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold. Roughly, it states that within a fixed radius, usually called the Margulis constant, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

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 manifolds with constant curvature. In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it.

In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In the mathematical subject of geometric group theory, the Švarc–Milnor lemma is a statement which says that a group , equipped with a "nice" discrete isometric action on a metric space , is quasi-isometric to .

In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric.

In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler manifold, and it is characterised by being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between -1 and -1/4 : in particular, it is a CAT(-1/4) space.

• 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 100:442–455.
• Wolf, Joseph A. Spaces of constant curvature, 1967. See page 67.