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 Hyperbolic orthogonal dodecahedral honeycomb.png
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, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. 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.

Contents

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

Formal definition

Hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with a constant negative sectional curvature. Hyperbolic space is a space exhibiting hyperbolic geometry. It is the negative-curvature analogue of the n-sphere. 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 plane.

Models of hyperbolic space

Hyperbolic space, developed independently by Nikolai Lobachevsky and János Bolyai, 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):

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.

Models of hyperbolic spaces that can be embedded in a flat (e.g. Euclidean) spaces may be constructed. 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, the Poincaré ball model and the Poincaré half space model. These all model the same geometry in the sense that any two of them can be related by a transformation that preserves all the geometrical properties of the space, including isometry (though not with respect to the metric of a Euclidean embedding).

Hyperboloid model

The hyperboloid model realizes hyperbolic space as a hyperboloid in Rn+1 = {(x0,...,xn)|xiR, i=0,1,...,n}. The hyperboloid is the locus Hn of points whose coordinates satisfy

In this model a line (or geodesic) is the curve formed by the intersection of Hn with a plane through the origin in Rn+1.

The hyperboloid model is closely related to the geometry of Minkowski space. The quadratic form

which defines the hyperboloid, polarizes to give the bilinear form

The space Rn+1, equipped with the bilinear form B, is an (n+1)-dimensional Minkowski space Rn,1.

One can associate a distance on the hyperboloid model by defining [1] the distance between two points x and y on Hn to be

This function satisfies the axioms of a metric space. It is preserved by the action of the Lorentz group on Rn,1. Hence the Lorentz group acts as a transformation group preserving isometry on Hn.

Klein model

An alternative model of hyperbolic geometry is on a certain domain in projective space. The Minkowski quadratic form Q defines a subset UnRPn given as the locus of points for which Q(x) > 0 in the homogeneous coordinates 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

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 xUn 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 bijection 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.

Poincaré ball model

A closely related pair of models of hyperbolic geometry are the Poincaré ball and Poincaré half-space models.

The ball model comes from a stereographic projection 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 semicircles that are perpendicular to the boundary sphere of Bn. Isometries of the ball are generated by spherical inversion in hyperspheres perpendicular to the boundary.

Poincaré half-space model

The half-space model results from applying inversion in a circle with centre a boundary point of the Poincaré ball model Bn above and a radius of twice the radius.

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 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.

See also

Related Research Articles

In geometry, a pseudosphere is a surface with constant negative Gaussian curvature. Hilbert's theorem says that no pseudosphere can be immersed into three-dimensional space.

Riemannian geometry branch of differential geometry dealing with (generalized) Riemannian manifolds

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Hyperbolic geometry Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

Gaussian curvature product of the principal curvatures of a surface

In differential geometry, the Gaussian curvature or Gauss curvatureΚ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point:

In mathematics, the upper half-planeH is the set of points (x, y) in the Cartesian plane with y > 0.

Homogeneous space Topological space in group theory

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.

Poincaré half-plane model Upper-half plane model of hyperbolic non-Euclidean geometry

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

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

Affine connection Construct allowing differentiation of tangent vector fields of manifolds

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan and Hermann Weyl. The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.

Low-dimensional topology branch of topology that studies topological spaces of four or fewer dimensions

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed particularly by Mario Pieri in his reduction of the primitive notions of geometry to merely point and motion.

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-)hyperbolic groups.

In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds. They are named after Herbert Busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "The geometry of geodesics".

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.

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 hyperboloid in (n+1)-dimensional Minkowski space and m-planes are represented by the intersections of the (m+1)-planes in Minkowski space with S+. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the n-dimensional hyperbolic space is closely related to the Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group.

Beltrami–Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

Differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric

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.

Poincaré disk model model of hyperbolic geometry

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.

References

  1. Note the similarity with the chordal metric on a sphere, which uses trigonometric instead of hyperbolic functions.