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 Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below.

Overview of metrics on Riemann surfaces[edit]

A metric on the complex plane may be generally expressed in the form

ds^2=\lambda^2(z,\overline{z})\, dz\,d\overline{z}

where λ is a real, positive function of z and \overline{z}. The length of a curve γ in the complex plane is thus given by

l(\gamma)=\int_\gamma \lambda(z,\overline{z})\, |dz|

The area of a subset of the complex plane is given by

\text{Area}(M)=\int_M \lambda^2 (z,\overline{z})\,\frac{i}{2}\,dz \wedge d\overline{z}

where \wedge is the exterior product used to construct the volume form. The determinant of the metric is equal to \lambda^4, so the square root of the determinant is \lambda^2. The Euclidean volume form on the plane is dx\wedge dy and so one has

dz \wedge d\overline{z}=(dx+i\,dy)\wedge (dx-i \, dy)= -2i\,dx\wedge dy.

A function \Phi(z,\overline{z}) is said to be the potential of the metric if

4\frac{\partial}{\partial z} 
\frac{\partial}{\partial \overline{z}} \Phi(z,\overline{z})=\lambda^2(z,\overline{z}).

The Laplace–Beltrami operator is given by

\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).

The Gaussian curvature of the metric is given by

K=-\Delta \log \lambda.\,

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 \lambda^2(z,\overline{z})\, dz \, d\overline{z} and T be a Riemann surface with metric \mu^2(w,\overline{w})\, dw\,d\overline{w}. Then a map

f:S\to T\,

with f=w(z) is an isometry if and only if it is conformal and if

\mu^2(w,\overline{w}) \;
\frac {\partial w}{\partial z}
\frac {\partial \overline {w}} {\partial \overline {z}} = 
\lambda^2 (z, \overline {z})
.

Here, the requirement that the map is conformal is nothing more than the statement

w(z,\overline{z})=w(z),

that is,

\frac{\partial}{\partial \overline{z}} w(z) = 0.

Metric and volume element on the Poincaré plane[edit]

The Poincaré metric tensor in the Poincaré half-plane model is given on the upper half-plane H as

ds^2=\frac{dx^2+dy^2}{y^2}=\frac{dz \, d\overline{z}}{y^2}

where we write dz=dx+i\,dy. This metric tensor is invariant under the action of SL(2,R). That is, if we write

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

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

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

and

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

The infinitesimal transforms as

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

and so

dz'd\overline{z}' = \frac{dz\,d\overline{z}}{|cz+d|^4}

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

The invariant volume element is given by

d\mu=\frac{dx\,dy}{y^2}.

The metric is given by

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

for z_1,z_2 \in \mathbb{H}.

Another interesting form of the metric can be given in terms of the cross-ratio. Given any four points z_1,z_2,z_3 and z_4 in the compactified complex plane \hat{\mathbb{C}} = \mathbb{C} \cup \infty, the cross-ratio is defined by

(z_1,z_3; z_2,z_4) = 
\frac{(z_1-z_2)(z_3-z_4)}{(z_3-z_2)(z_1-z_4)}.

Then the metric is given by

 \rho(z_1,z_2)= \log (z_1,z_2^\times ; z_2, z_1^\times).

Here, z_1^\times and z_2^\times are the endpoints, on the real number line, of the geodesic joining z_1 and z_2. These are numbered so that z_1 lies in between z_1^\times and z_2.

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.

Conformal map of plane to disk[edit]

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

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

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 \Im z =0 maps to the edge of the unit disk |w|=1. The constant real number \phi can be used to rotate the disk by an arbitrary fixed amount.

The canonical mapping is

w=\frac{iz+1}{z+i}

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

Metric and volume element on the Poincaré disk[edit]

The Poincaré metric tensor in the Poincaré disk model is given on the open unit disk U=\{z=x+iy:|z|=\sqrt{x^2+y^2} < 1 \} by

ds^2=\frac{4(dx^2+dy^2)}{(1-(x^2+y^2))^2}=\frac{4 dz\,d\overline{z}}{(1-|z|^2)^2}.

The volume element is given by

d\mu=\frac{4 dx\,dy}{(1-(x^2+y^2))^2}=\frac{4 dx\,dy}{(1-|z|^2)^2}.

The Poincaré metric is given by

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

for z_1,z_2 \in U.

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

The punctured disk model[edit]

J-invariant in punctured disk coordinates; that is, as a function of the nome.
J-invariant in Poincare disk coordinates; note this disk is rotated by 90 degrees from canonical coordinates given in this article

A second common mapping of the upper half-plane to a disk is the q-mapping

q=\exp(i\pi\tau)

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 \Im \tau >0. 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

ds^2=\frac{4}{|q|^2 (\log |q|^2)^2} dq \, d\overline{q}

The potential of the metric is

\Phi(q,\overline{q})=4 \log \log |q|^{-2}

Schwarz lemma[edit]

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

See also[edit]

References[edit]

  • 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.)