Hyperbolic space

In mathematics, hyperbolic "n"-space, denoted "H"^"n", is the maximally symmetric, simply connected, "n"-dimensional Riemannian manifold with constant sectional curvature −1. Hyperbolic space is the principal example of a space exhibiting hyperbolic geometry. It can be thought of as the negative-curvature analogue of the "n"-sphere. Although hyperbolic space "H"^"n" is diffeomorphic to R^"n" its negative-curvature metric gives it very different geometric properties.

Hyperbolic 2-space, "H"², is also called the hyperbolic plane.

Models of hyperbolic space

Hyperbolic space, developed independently by Lobachevsky and 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". [Strictly speaking, a slightly more restrictive condition is necessary for uniqueness of the hyperbolic plane: There are "exactly two" such lines which are asymptotically parallel to "L".] 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 isometric.

The hyperboloid model

The first model realizes hyperbolic space as a hyperboloid in Rⁿ⁺¹ = {("x"₀,...,"x"_n)|"x"_i∈R, i=0,1,...,n}. The hyperboloid is the locus "H"ⁿ of points whose coordinates satisfy:$x\_0^2-x\_1^2-ldots-x\_n^2=1,quad\; x\_0>0.$In this model a "line" (or geodesic) is the curve cut out by intersecting "H"ⁿ 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" defined by:$B(x,y)\; =\; (Q(x+y)-Q(x)-Q(y))/2=x\_0y\_0\; -\; x\_1y\_1\; -\; cdots\; -\; x\_ny\_n.$The space Rⁿ⁺¹, equipped with the bilinear form "B" is an ("n"+1)-dimensional Minkowski space R^n,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:$d(x,\; y)\; =\; operatorname\{arccosh\},\; B(x,y).$This function satisfies the axioms of a metric space. [Note the similarity with the chordal metric on a sphere, which uses trigonometric instead of hyperbolic functions.] Moreover, it is preserved by the action of the Lorentz group on R^n,1. Hence the Lorentz group acts as a transformation group of isometries on "H"ⁿ.

The Klein model

An alternative model of hyperbolic geometry is on a certain domain in projective space. The Minkowski quadratic form "Q" defines a subset "U"ⁿ ⊂ RPⁿ given as the locus of points for which "Q"("x") > 0 in the homogeneous coordinates "x". The domain "U"ⁿ 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 "U"ⁿ. The distance between two points "x" and "y" in "U"ⁿ is defined by:$d(x,\; y)\; =\; operatorname\{arccosh\}left(frac\{B(x,y)\}\{sqrt\{Q(x)Q(y)\; ight).$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" ∈ "U"ⁿ corresponds to a line "L"_x through the origin in Rⁿ⁺¹, by the definition of projective space. This line intersects the hyperboloid "H"ⁿ in a unique point. Conversely, through any point on "H"ⁿ, there passes a unique line through the origin (which is a point in the projective space). This correspondence defines a bijection between "U"ⁿ and "H"ⁿ. 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"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 projection of the hyperboloid in Rⁿ⁺¹ onto the plane {"x"₀ = 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 "H"ⁿ, let "P"^* be the unique point of intersection of the line "SP" with the plane {"x"₀ = 0}. This establishes a bijective mapping of "H"ⁿ into the unit ball :in the plane {"x"₀ = 0}.

The geodesics in this model are semicircles which are perpendicular to the boundary sphere of "B"_n. 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 point of the boundary of "B"ⁿ. 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 "H"^"n". As a result, the universal cover of any closed manifold "M" of constant negative curvature −1, which is to say, a hyperbolic manifold, is "H"^"n". Thus, every such "M" can be written as "H"^"n"/Γ where Γ is a torsion-free discrete group of isometries 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 surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group $pi\_1=Gamma$; the groups that arise this way are known as Fuchsian groups. The quotient space H/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.

ee also

* Mostow rigidity theorem
* Hyperbolic manifold
* Hyperbolic 3-manifold

References

* 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.