Poincaré metric

In mathematics, the Poincaré metric is the natural metric tensor for Poincaré half-plane model of hyperbolic geometry. Two equivalent representations exist for hyperbolic geometry, one for the unit disc and one for the upper half-plane. These two are related by a conformal map. We give both forms below.

1 Conformal map
2 Metric and volume element on the Poincaré plane
3 Metric and volume element on the Poincaré disk
4 Schwarz lemma
5 See also
6 References

Conformal map

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 canonical mapping is

which takes i to the center of the disk, and 0 to the bottom of the disk.

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

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

and

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

The infinitesimal transforms as

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

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>.

The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis.

Metric and volume element on the Poincaré disk

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

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 geodesics for this metric tensor are circular arcs whose endpoints are orthogonal to the boundary of the disk.

Schwarz lemma

The Poincaré metric is distance-decreasing on harmonic functions. This is an extension of the Schwarz lemma, called the Schwarz-Alhfors-Pick theorem.

References

Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0–387–90465–4.
Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3–540–43299-X (See Section 2.3).

Categories: Conformal geometry | Hyperbolic geometry | Riemannian geometry | Riemann surfaces