![]() |
|
|
| |
|
||||
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 symmetryThere is a group action of the special linear group SL(2,R) on H defined by
The action is transitive and the stabilizer of i is the rotation group
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 planeThe Poincaré metric tensor on H is given by
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
for <math>ad-bc=1<math> then we can work out that
and
The infinitesimal transforms as
and so
thus making it clear that the metric tensor is invariant under SL(2,R). The invariant volume element is given by
The metric is given by
for <math>z_1,z_2 \in \mathbb{H}.<math> Metric and volume element on the Poincaré diskThe upper half plane can be mapped conformally to the unit disk with the Möbius transformation
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
The volume element is given by
The Poincaré metric is given by
for <math>z_1,z_2 \in U.<math> The Poincaré metric is distance-decreasing on harmonic functions, stated as follows. Schwarz lemmaAn 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
|
||
|
|
|
|
|
|
Copyright 2008 WordIQ.com - Privacy Policy
::
Terms of Use
:: Contact Us
:: About Us This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Upper half plane". |