In geometry, the Half-plane model of hyperbolic geometry is described by hyperbolic motions.

In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. It is the domain of many functions of interest in complex analysis, especially modular forms. It is also a model of the hyperbolic plane.

This article will present mathematical theorems and formulas for both the plane and the Poincare disk. A less technical overview can be found in the articles on hyperbolic geometry and Poincaré half-plane model.

Contents

1 Group symmetry 2 Metric and volume element on the Poincaré plane 3 Metric and volume element on the Poincaré disk 4 Schwarz lemma 5 See also

Group symmetry

There is a group action of the special linear group SL(2,R) on H defined by

<math>\left(\begin{matrix}a&b\\ c&d\\ \end{matrix}\right) \cdot z = \frac{az+b}{cz+d} = {(ac|z|^2+bd+(ad+bc)\Re(z)) + i\Im(z)\over|cz+d|^2}.<math>

The action is transitive and the stabilizer of i is the rotation group

<math>{\rm SO}(2) = \left\{ \left(\begin{matrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{matrix}\right)\,:\,\theta\in{\mathbf R}\right\}.<math>

Therefore, H = SL(2,R)/SO(2). The upper half plane is tessellated by the modular group SL(2,Z).

Metric and volume element on the Poincaré plane

The Poincaré metric tensor on H is given by

<math>ds^2=\frac{dx^2+dy^2}{y^2}=\frac{dzd\overline{z}}{y^2}<math>

where we write <math>dz=dx+idy.<math> This metric tensor is invariant under the action of SL(2,R). That is, if we write

<math>z'=x'+iy'=\frac{az+b}{cz+d}<math>

for <math>ad-bc=1<math> then we can work out that

<math>x'=\frac{ac(x^2+y^2)+x(ad+bc)+bd}{|cz+d|^2}<math>

and

<math>y'=\frac{y}{|cz+d|^2}.<math>

The infinitesimal transforms as

<math>dz'=\frac{dz}{(cz+d)^2}<math>

and so

<math>dz'd\overline{z}' = \frac{dzd\overline{z}}{|cz+d|^4}<math>

thus making it clear that the metric tensor is invariant under SL(2,R).

The invariant volume element is given by

<math>d\mu=\frac{dxdy}{y^2}.<math>

The metric is given by

<math>\rho(z_1,z_2)=2\tanh^{-1}\frac{|z_1-z_2|}{|z_1-\overline{z_2}|}<math>

<math>\rho(z_1,z_2)=\log\frac{|z_1-\overline{z_2}|+|z_1-z_2|}{|z_1-\overline{z_2}|-|z_1-z_2|}<math>

for <math>z_1,z_2 \in \mathbb{H}.<math>

Metric and volume element on the Poincaré disk

The upper half plane can be mapped conformally to the unit disk with the Möbius transformation

<math>w=e^{i\phi}\frac{z-z_0}{z-\overline {z_0}}<math>

where w is the point on the unit disk that corresponds to the point z in the upper half plane. In this mapping, the constant z₀ can be any point in the upper half plane; it will be mapped to the center of the disk. The real axis <math>\Im z =0<math> maps to the edge of the unit disk <math>|w|=1.<math> The constant real number <math>\phi<math> can be used to rotate the disk by an arbitrary fixed amount.

The Poincaré metric tensor on the unit disk <math>U=\{z=x+iy:|z|=\sqrt{(x^2+y^2)} \leq 1 \}<math> is given by

<math>ds^2=\frac{dx^2+dy^2}{(1-(x^2+y^2))^2}=\frac{dz\,d\overline{z}}{(1-|z|^2)^2}.<math>

The volume element is given by

<math>d\mu=\frac{dx\,dy}{(1-(x^2+y^2))^2}=\frac{dx\,dy}{(1-|z|^2)^2}.<math>

The Poincaré metric is given by

<math>\rho(z_1,z_2)=\tanh^{-1}\left|\frac{z_1-z_2}{1-z_1\overline{z_2}}\right|<math>

for <math>z_1,z_2 \in U.<math>

The Poincaré metric is distance-decreasing on harmonic functions, stated as follows.

Schwarz lemma

An extension of the Schwarz lemma, called the Schwarz-Alhfors-Pick theorem, states that every holomorphic automorphism from the disk U to U or the half plane H to H, with distances defined by the Poincare metric, is a contraction mapping. That is, every such analytic mapping will not increase the distance between points. Stated more precisely:

Theorem: (Schwarz-Alhfors-Pick) For all holomorphic automorphisms <math>f:U\rightarrow U<math>, one has <math>\rho(f(z_1),f(z_2)) \leq \rho(z_1,z_2)<math> for points <math>z_1,z_2 \in U<math> and Poincaré distance <math>\rho.<math> For any tangent vector T, the hyperbolic length of the tangent vector does not increase: <math>|f^*(T)| \leq |T|.<math>

See also

• The article on the Poincaré half-plane model provides a non-mathematical introduction to this topic.