Poincaré metric
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In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.
There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Mobius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below.
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[edit] Overview of metrics on Riemann surfaces
A metric on the complex plane may be generally expressed in the form
- <math>ds^2=\lambda^2(z,\overline{z})\, dzd\overline{z}</math>
where λ is a real, positive function of z and <math>\overline{z}</math>. The length of a curve γ in the complex plane is thus given by
- <math>l(\gamma)=\int_\gamma \lambda(z,\overline{z})\, |dz|</math>
The area of a subset of the complex plane is given by
- <math>\mbox{Area}(M)=\int_M \lambda^2 (z,\overline{z})\,\frac{i}{2}dz \wedge d\overline{z}</math>
where <math>\wedge</math> is the exterior product used to construct the volume form. The determinant of the metric is equal to <math>\lambda^4</math>, so the square root of the determinant is <math>\lambda^2</math>. The Euclidean volume form on the plane is <math>dx\wedge dy</math> and so one has
- <math>dz \wedge d\overline{z}=(dx+idy)\wedge (dx-idy)= -2idx\wedge dy</math>.
A function <math>\Phi(z,\overline{z})</math> is said to be the potential of the metric if
- <math>4\frac{\partial}{\partial z}
\frac{\partial}{\partial \overline{z}} \Phi(z,\overline{z})=\lambda^2(z,\overline{z})</math>.
The Laplace-Beltrami operator is given by
- <math>\Delta = \frac{4}{\lambda^2}
\frac {\partial}{\partial z} \frac {\partial}{\partial \overline{z}} = \frac{1}{\lambda^2} \left( \frac {\partial^2}{\partial x^2} + \frac {\partial^2}{\partial y^2} \right)</math>
The Gaussian curvature of the metric is given by
- <math>K=-\Delta \log \lambda</math>
This curvature is one-half of the Ricci scalar curvature.
Isometries preserve angles and arc-lengths. On Riemann surfaces, isometries are identical to changes of coordinate: that is, both the Laplace-Beltrami operator and the curvature are invariant under isometries. Thus, for example, let S be a Riemann surface with metric <math>\lambda^2(z,\overline{z})\, dzd\overline{z}</math> and T be a Riemann surface with metric <math>\mu^2(w,\overline{w})\, dwd\overline{w}</math>. Then a map
- <math>f:S\to T</math>
with <math>f=w(z)</math> is an isometry if and only if it is conformal and if
- <math>\mu^2(w,\overline{w}) \;
\frac {\partial w}{\partial z} \frac {\partial \overline {w}} {\partial \overline {z}} = \lambda^2 (z, \overline {z}) </math>.
Here, the requirement that the map is conformal is nothing more than the statement
- <math>w(z,\overline{z})=w(z)</math>,
that is,
- <math>\frac{\partial}{\partial \overline{z}} w(z) = 0</math>.
[edit] Metric and volume element on the Poincaré plane
The Poincaré metric tensor in the Poincaré half-plane model is given on the upper half-plane H as
- <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>.
Another interesting form of the metric can be given in terms of the cross ratio. Given any four points <math>z_1,z_2,z_3</math> and <math>z_4</math> in the compactified complex plane <math>\hat \mathbb{C} = \mathbb{C} \cup \infty</math>, the cross ratio is defined by
- <math>(z_1,z_2; z_3,z_4) =
\frac{(z_1-z_2)(z_3-z_4)}{(z_2-z_3)(z_4-z_1)}</math>
Then the metric is given by
- <math> \rho(z_1,z_2)= \ln (z_1,z_2^\times ; z_2, z_1^\times)</math>
Here, <math>z_1^\times</math> and <math>z_2^\times</math> are the endpoints, on the real number line, of the geodesic joining <math>z_1</math> and <math>z_2</math>. These are numbered so that <math>z_1</math> lies in between <math>z_1^\times</math> and <math>z_2</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.
[edit] Conformal map of plane to 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 canonical mapping is
- <math>w=\frac{iz+1}{z+i}</math>
which takes i to the center of the disk, and 0 to the bottom of the disk.
[edit] Metric and volume element on the Poincaré disk
The Poincaré metric tensor in the Poincaré disk model is given on the unit disk <math>U=\{z=x+iy:|z|=\sqrt{(x^2+y^2)} \leq 1 \}</math> 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 geodesics for this metric tensor are circular arcs whose endpoints are orthogonal to the boundary of the disk.
[edit] The punctured disk model
A second common mapping of the upper half-plane to a disk is the q-mapping
- <math>q=exp(i\pi\tau)</math>
where q is the nome and τ is the half-period ratio. In the notation of the previous sections, τ is the coordinate in the upper half-plane <math>\Im \tau >0</math>. The mapping is to the punctured disk, because the value q=0 is not in the image of the map.
The Poincaré metric on the upper half-plane induces a metric on the q-disk
- <math>ds^2=\frac{4}{|q|^2 (\log |q|^2)^2} dq d\overline{q}</math>
The potential of the metric is
- <math>\Phi(q,\overline{q})=4 \log \log |q|^{-2}</math>
[edit] 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.
[edit] See also
- Fuchsian group
- Fuchsian model
- Kleinian group
- Kleinian model
- Poincaré disk model
- Poincaré half-plane model
- Prime geodesic
- Schwarz-Alhfors-Pick theorem
[edit] 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).
- Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0-226-42583-5 (Provides a simple, easily readable introduction.)
