TheInfoList In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, hyperbolic space of dimension n is the unique
simply connected In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
, n-dimensional
Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...
of constant
sectional curvatureIn Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
equal to -1. It is
homogeneous space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, and satisfies the stronger property of being a
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...
. There are many ways to construct it as an open subset of $\mathbb R^n$ with an explicitely 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... . 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 János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For an ...
. 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 Complete may refer to: Logic * Completeness (logic) * Complete theory, 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, ...
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 of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geom ...
.

## 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 In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H \ , together with a metric (mathematics), metric, the Poincaré metric, that makes it a mathematical model, model of two-dimensional hyperboli ...
: this is the upper-half space $\$ with the metric $\tfrac$ *
Poincaré disc model Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Lucien Poin ...
: this is the unit ball of $\mathbb R^n$ with the metric $2\tfrac$. 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 line (geometry), lines to lines, and thus a collineatio ...
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 the points on the forward sheet ''S''+ of a two-sheeted hyperbol ...
: In contrast with the previous two models this realises hyperbolic $n$-space as isometrocally embedded inside the $\left(n+1\right)$-dimensional
Minkowski space In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
(which is not a Riemannian but rather a
Lorentzian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a co ...
). More precisely, looking at the quadratic form $q\left(x\right) = x_1^2 + \cdots + x_n^2 - x_^2$ on $\mathbb R^$, its restriction to the tangent spaces of the upper sheet of the
hyperboloid In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
given by $q\left(x\right) = -1$ 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ... from the hyperboloid to the plane $\$, taking the vertex from which to project to be $\left(0, \ldots, 0, 1\right)$ for the ball and a point at infinity in the cone $q\left(x\right)=0$ inside
projective space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
for the half-space. *
Klein model Klein may refer to: People *Klein (surname) *Klein (musician) Places *Klein (crater), a lunar feature *Klein, Montana, United States *Klein, Texas, United States *Klein (Ohm), a river of Hesse, Germany, tributary of the Ohm *Klein River, a river ... : 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, $x_=1$) from the origin $\left(0, \ldots, 0\right)$. *Symmetric space: Hyperbolic $n$-space can be realised as the symmetric space of the simple Lie group $\mathrm\left(n, 1\right)$ (the group of isometries of the quadratic form $q$ with positive determinant); as a set the latter is the
coset space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a Group (mathematics), group ''G'' is a Empty set, non-empty manifold or topological space ''X'' on which ''G'' Group act ...
$\mathrm\left(n, 1\right)/\mathrm\left(n\right)$. The isometry to the hyperbolid model is immediate through the action of the connected component of $\mathrm\left(n, 1\right)$ 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, kn ...
,
János Bolyai János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 1946 * Hungarians, ethn ...
and
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ... , is a geometrical space analogous to
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
; there is an extra constant, the curvature , which must be specified. However, it does uniquely characterize it up to
homothety In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... , 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. John Forbes Nash Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory Game theory ...
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 (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ... .

## Volume growth and isoperimetric inequality

The volume of balls in hyperbolic space increases
exponentially Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ... with respect to the radius of the ball rather than
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... ly as in Euclidean space. Namely, if $B\left(r\right)$ is any ball of radius $r$ in $\mathbb H^n$ then:$\mathrm(B(r)) = c_n \int_0^r t\sinh(t) dt$ where $c_n$ is a constant depending only on the dimension, namely the total volume of the $\left(n-1\right)$-sphere. The hyperbolic space also satisfies a linear
isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with pr ...
, 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 Complete may refer to: Logic * Completeness (logic) * Complete theory, 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, ...
,
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
manifold of constant negative curvature −1 is
isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from ...
to the real hyperbolic space H''n''. As a result, the
universal cover In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space C to a topological space X such that each point in X has an Neighborhood_(mathematics)#Definitions, ...
of any
closed manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...
''M'' of constant negative curvature −1, which is to say, a
hyperbolic manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, is H''n''. Thus, every such ''M'' can be written as H''n''/Γ where Γ is a torsion-free
discrete group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of
isometries In mathematics, an isometry (or congruence (geometry), congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be Bijection, bijective. "We shall find it convenient to use the w ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. According to the
uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is Conformal equivalence, conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theore ...
, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial
fundamental group In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... π1=Γ; the groups that arise this way are known as
Fuchsian group In mathematics, a Fuchsian group is a Discrete group, discrete subgroup of PSL(2,R), PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group (mathematics), group of isometry, isometries of the Hyperbolic geometry, hyperbolic plane, or ...
s. The quotient space H²/Γ of the upper half-plane modulo the fundamental group is known as the
Fuchsian model In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface ''R'' as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazaru ...
of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is
simply connected In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
and noncompact. It is the
universal cover In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space C to a topological space X such that each point in X has an Neighborhood_(mathematics)#Definitions, ...
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 (topology), quotient space \mathbb^3 / \Gamma where \Gamma is a discrete group, discrete subgroup of PSL(2,C). Here, the subgroup \ ...
.

*
Dini's surface In geometry, Dini's surface is a surface (mathematics), 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 ... *
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 is ...
*
Ideal polyhedron In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose Vertex (geometry), vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as ...
*
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 complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fun ...
* Murakami–Yano formula *
Pseudosphere In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of fi ... # 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are co ...
100:442–455. * Wolf, Joseph A. ''Spaces of constant curvature'', 1967. See page 67. Homogeneous spaces Hyperbolic geometry Topological spaces