In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
,
hyperbolic n-space, denoted
Hn, is the maximally symmetric, simply connected,
n-dimensional
Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor...
with constant
sectional curvatureIn 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 geometryIn 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-
sphereA 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 planeIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
Hyperbolic space, developed independently by Lobachevsky and Bolyai, is a geometrical space analogous to
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
, but such that
Euclid's parallel postulateIn 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.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
,
hyperbolic n-space, denoted
Hn, is the maximally symmetric, simply connected,
n-dimensional
Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor...
with constant
sectional curvatureIn 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 geometryIn 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-
sphereA 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 planeIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
Models of hyperbolic space
Hyperbolic space, developed independently by Lobachevsky and Bolyai, is a geometrical space analogous to
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
, but such that
Euclid's parallel postulateIn 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
isometryIn mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent....
; 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
isometricThe term isometric comes from the Greek for "having equal measurement".isometric may mean:* Isometric projection , a method for the visual representation of three-dimensional objects in two dimensions; a form of orthographic projection, or more specifically, an axonometric projection.* Isometry and...
.
The hyperboloid model
The first model realizes hyperbolic space as a hyperboloid in
Rn+1 = {(
x0,...,
xn)|
xi∈
R, i=0,1,...,n}. The hyperboloid is the locus
Hn of points whose coordinates satisfy
In this model a "line" (or
geodesicIn mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".In the presence of a metric, geodesics are defined to be the shortest path between points on 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 spaceIn physics and mathematics, Minkowski space is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for...
. The
quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,is a quadratic form in the variables x and y....
which defines the hyperboloid
polarizesIn mathematics, the polarization identity is any one of a family of formulas that express the dot product of two vectors in terms of the Euclidean norm...
to give the
bilinear formIn mathematics, a bilinear form on a vector space V is a bilinear mapping V × V → F, where F is the field of scalars...
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 spaceIn 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 groupIn physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena.The mathematical form of*the kinematical laws of special relativity,...
on
Rn,1. Hence the Lorentz group acts as a transformation group of
isometriesIn mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent....
on
Hn.
The Klein model
An alternative model of hyperbolic geometry is on a certain
domainIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
in
projective spaceIn 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=R
2 or V=R
3 are the projective line and the projective plane, respectively.The idea of a projective space relates to perspective,...
. The Minkowski quadratic form
Q defines a subset
Un ⊂
RPn given as the locus of points for which
Q(
x) > 0 in the
homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, make calculations possible in projective space just as Cartesian coordinates do in Euclidean space...
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
x ∈
Un 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
bijectionIn mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y and no unmapped element remains in both X and Y.Alternatively, f is bijective if it is a one-to-one correspondence...
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
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 projectionIn geometry, the stereographic projection 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
semicircleIn 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°...
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 pointIn Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*...
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
completeIn mathematical analysis, a metric space M is said to be 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....
,
connectedIn topology and related branches of mathematics, a connected space is a topological space which 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
isometricThe term isometric comes from the Greek for "having equal measurement".isometric may mean:* Isometric projection , a method for the visual representation of three-dimensional objects in two dimensions; a form of orthographic projection, or more specifically, an axonometric projection.* Isometry and...
to the real hyperbolic space
Hn. As a result, the universal cover of any
closed manifoldIn 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 manifoldIn mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends which are the product of a Euclidean n-1-manifold and the closed half-ray...
, is
Hn. Thus, every such
M can be written as
Hn/Γ where Γ is a torsion-free
discrete groupIn 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
isometriesIn mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent....
on
Hn. That is, Γ is a
latticeIn 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 surfaceIn 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 theoremIn mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. In fact, one can find a metric with constant Gaussian curvature in any given conformal class....
, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial
fundamental groupIn mathematics, more specifically algebraic topology, the fundamental group or Poincaré 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...
; the groups that arise this way are known as
Fuchsian groupIn mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. A Fuchsian group is always a discrete group contained in the semisimple Lie group PSL. The name is given in honour of Immanuel Lazarus Fuchs....
s. The
quotient spaceIn 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...
H/Γ of the upper half-plane
moduloThe word modulo, in the mathematical community, is often used informally, in many imprecise ways. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"...
the fundamental group is known as the
Fuchsian modelIn mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group...
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 modelIn mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where Γ is a discrete subgroup of PSL. Here, the subgroup Γ, a Kleinian group, is defined so that it is isomorphic to the fundamental group of the surface N...
.