Hyperbolic space

A perspective projection of a dodecahedral tessellation in H³.
Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E³

In mathematics, 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.

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):

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.

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).

The hyperboloid model

Main article: Hyperboloid model

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

x_{0}^{2}-x_{1}^{2}-\cdots -x_{n}^{2}=1,\quad x_{0}>0.

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

Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-\cdots -x_{n}^{2},

which defines the hyperboloid, polarizes to give the bilinear form

B(x,y)=(Q(x+y)-Q(x)-Q(y))/2=x_{0}y_{0}-x_{1}y_{1}-\cdots -x_{n}y_{n}.

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

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

d(x,y)=\operatorname {arcosh}B(x,y).

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

The Klein model

Main article: Klein model

An alternative model of hyperbolic geometry is on a certain domain in projective space. The Minkowski quadratic form Q defines a subset Uⁿ ⊂ RPⁿ 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

d(x, y) = \operatorname{arcosh}\left(\frac{B(x,y)}{\sqrt{Q(x)Q(y)}}\right).

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 L_x 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.

The Poincaré ball model

Main article: Poincaré disc 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 R^n,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

B^{n}=\{(x_{1},\ldots ,x_{n})|x_{1}^{2}+\cdots +x_{n}^{2}<1\}

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.

The Poincaré half space model

Main article: Poincaré half-plane 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.

References

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

A'Campo, Norbert and Papadopoulos, Athanase, (2012) Notes on hyperbolic geometry, in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, SBN ISBN 978-3-03719-105-7, DOI 10.4171/105.
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.
Hyperbolic Voronoi diagrams made easy, Frank Nielsen

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