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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.
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
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