Poincaré metric
# Poincaré metric

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

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

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 $dz=dx+i\,dy$ and $d{\overline {z}}=dx-i\,dy$. 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

    $dz'={\frac {\partial }{\partial z}}{\Big{\frac {az+b}{cz+d}}{\Big )}\,dz={\frac {a(cz+d)-c(az+b)}{(cz+d)^{2}}}\,dz={\frac {acz+ad-caz-cb}{(cz+d)^{2}}}\,dz={\frac {ad-cb}{(cz+d)^{2}}}\,dz\,\,{\overset {ad-cb=1}{=}}\,\,{\frac {1}{(cz+d)^{2}}}\,dz={\frac {dz}{(cz+d)^{2}}}$

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). Indeed,

    ${\frac {dz'\,d{\overline {z}}'}{y'^{2}}}={\frac {\frac {dzd{\overline {z}}}{|cz+d|^{4}}}{\frac {y^{2}}{|cz+d|^{4}}}}={\frac {dz\,d{\overline {z}}}{y^{2}}}.$

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

    $(z_{1},z_{2};z_{3},z_{4})={\frac {(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{1}-z_{4})(z_{2}-z_{3})}}.$

Then the metric is given by

    $\rho (z_{1},z_{2})=\log \left(z_{1},z_{2};z_{1}^{\times },z_{2}^{\times }\right).$

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

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 $z_0$ 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

The Poincaré metric tensor in the Poincaré disk model is given on the open unit disk

    $U=\left\{z=x+iy:|z|={\sqrt {x^{2}+y^{2}}}<1\right\}$

by

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

The volume element is given by

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

The Poincaré metric is given by

    $\rho (z_{1},z_{2})=2\tanh ^{-1}\left|{\frac {z_{1}-z_{2}}{1-z_{1}{\overline {z_{2}}}}}\right|$

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. Geodesic flows on the Poincaré disk are Anosov flows; that article develops the notation for such flows.

## The punctured disk model

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:

    $\tau ={\frac {\omega _{2}}{\omega _{1}}}$ .

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

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

