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.

Hyperbolic 2-space, H2, is also called the hyperbolic plane.

Definition

Hyperbolic space is most commonly defined 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. Minkowski space Rn,1 is identical to Rn+1 except that the metric is given by the quadratic form

Note that the Minkowski metric is not positive-definite, but rather has signature (−, +, +, …, +). This gives it rather different properties than Euclidean space.

Hyperbolic space, Hn, is then given as a hyperboloid of revolution in Rn,1:

The condition x0 > 0 selects only the top sheet of the two-sheeted hyperboloid so that Hn is connected. The space Hn is formally called the real hyperbolic space of dimension n .

The metric on Hn is induced from the metric on Rn,1. Explicitly, the tangent space to a point xHn can be identified with the orthogonal complement of x in Rn,1. The metric on the tangent space is obtained by simply restricting the metric on Rn,1. It is important to note that the metric on Hn is positive-definite even through the metric on Rn,1 is not. This means that Hn is a true Riemannian manifold (as opposed to a pseudo-Riemannian manifold).

Although hyperbolic space Hn is diffeomorphic to Rn its negative curvature metric gives it very different geometric properties.

Symmetry

The group O(n,1) is the Lie group of real matrices that preserve the bilinear form

That is, O(n,1) is the group of isometries of Minkowski space Rn,1 fixing the origin. This group is sometimes called the (n+1)-dimensional Lorentz group. The subgroup which preserves the orientation of x0 is called the orthochronous Lorentz group, denoted O+(n,1).

The action of O+(n,1) on Rn,1 restricts to an action on Hn. This group clearly preserves the hyperbolic metric on Hn. In fact, O+(n,1) is the full isometry group of Hn. This isometry group has dimension n(n+1)/2, the maximal number of isometries for a Riemannian manifold. Therefore, hyperbolic space is said to be maximally symmetric. The group of orientation preserving isometries of Hn is the group SO+(n,1), which is the identity component of the full Lorentz group.

The orientation preserving isometry group SO+(n,1) acts transitively and faithfully on Hn. Which is to say that Hn is a homogeneous space for the action of SO+(n,1). The isotropy group of the vector is a matrix of the form

where A is a matrix in the rotation group SO(n); that is, A is an orthogonal matrix with determinant +1. Hyperbolic space Hn is therefore isomorphic to the quotient space SO+(n,1)/SO(n).

The bilinear form is the Cartan-Killing form, the unique SO+(n,1)-invariant quadratic form on SO+(n,1).

General case

Every complete, 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 is Hn. Thus, every such M can be written as Hn/Γ where Γ is a discrete group of isometries on Hn. That is, Γ is a lattice in SO+(n,1).

See also