Hyperbolic space

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In mathematics, hyperbolic n-space, denoted Hⁿ, 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 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.

Contents 1 Minkowski space and hyperbolic space 2 Hyperbolic manifolds 3 See also 4 References

Minkowski space and hyperbolic space

Hyperbolic spaces may be regarded as models, over the real numbers, of hyperbolic geometry, that is, they satisfy the axioms of hyperbolic geometry. The principal models of hyperbolic geometry can all be related to Minkowski space.

The Minkowski quadratic form is the form

<math>Q(x) = x_0^2 - x_1^2 - x_2^2 - \cdots - x_n^2.</math>

In terms of Q, we may define the Minkowski bilinear form B by

<math>B(x,y) = (Q(x+y)-Q(x)-Q(y))/2 =

x_0y_0 - x_1y_1 - \cdots - x_ny_n.</math> In terms of the bilinear form, the quadratic form in turn may be defined by Q(x) = B(x,x). Rⁿ⁺¹ equipped with the Minkowski forms defines n+1 dimensional Minkowski space, R^n,1.

From the Minkowski quadratic form Q we may define a projective semialgebraic variety as the set Uⁿ defined by all x in R^n,1 such that Q(x)>0. Given two points x and y in Uⁿ, we may define a distance function by

<math>d(x, y) = \operatorname{arccosh}\left(\frac{B(x,y)}{\sqrt{Q(x)Q(y)}}\right).</math>

This is a homogenous function in each coordinate, d(λx, μy) = d(x, y), for λ, μ other than zero, and so defines a function on the projective semialgebraic variety Uⁿ. This function satisfies the axioms of a metric space, and makes Uⁿ into a model of hyperbolic space, which we may consider to be a representative form of n-dimensional hyperbolic space Hⁿ.

From this model we may derive the closely related Klein and hyperboloid models. If we choose points x in Uⁿ such that Q(x)=1, x₀ > 1, then we obtain the hyperboloid model. We may normalize to the hyperboloid model by changing the sign of x if x₀ < 0 and then dividing by <math>\sqrt{Q(x)}</math>. Similarly, we normalize to the Klein model by dividing x by x₀, which since Q(x)>0, cannot be zero. The Poincaré disk model can then be obtained by a slightly more elaborate mapping; see the Klein and hyperbolic model articles for mappings from these to the Poincaré disk model. The Poincaré disk model, in turn, is closely related via conformal mapping to the Poincaré half-plane model.

Another point of view on the hyperboloid model is to view it as a submanifold of (n+1)-dimensional Minkowski space, in much the same manner as the n-sphere is defined as a submanifold of (n+1)-dimensional Euclidean space. Note that the Minkowski metric tensor is not positive-definite, but rather has signature (+, −, −, …, −). This gives it rather different properties than Euclidean space. The condition x₀ > 0 selects only the top sheet of the two-sheeted hyperboloid so that Hⁿ is connected. The metric on Hⁿ is induced from the metric on R^n,1. Explicitly, the tangent space to a point x ∈ Hⁿ can be identified with the orthogonal complement of x in R^n,1. The metric on the tangent space is obtained by simply restricting the metric on R^n,1. It is important to note that the metric on Hⁿ is positive-definite even through the metric on R^n,1 is not. This means that Hⁿ is a true Riemannian manifold (as opposed to a pseudo-Riemannian manifold).

Hyperbolic manifolds

Every complete, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Uⁿ. 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).

See also

• Mostow rigidity theorem
• Hyperboloid model
• Klein model
• Poincaré disk model
• Hyperbolic manifold
• Hyperbolic 3-manifold
• Hyperbolic geometry

References

Ratcliffe, John G., Foundations of hyperbolic manifolds, New York, Berlin. Springer-Verlag, 1994.