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

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

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

${\displaystyle l(\gamma )=\int _{\gamma }\lambda (z,{\overline {z}})\,|dz|}$

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

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

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

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

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

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

The Laplace–Beltrami operator is given by

${\displaystyle \Delta ={\frac {4}{\lambda ^{2}}}{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\overline {z}}}}={\frac {1}{\lambda ^{2}}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right).}$

The Gaussian curvature of the metric is given by

${\displaystyle K=-\Delta \log \lambda .\,}$

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

${\displaystyle f:S\to T\,}$

with ${\displaystyle f=w(z)}$ is an isometry if and only if it is conformal and if

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

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

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

that is,

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

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

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

where we write ${\displaystyle dz=dx+i\,dy}$ and ${\displaystyle d{\overline {z}}=dx-i\,dy}$. This metric tensor is invariant under the action of SL(2,R). That is, if we write

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

for ${\displaystyle ad-bc=1}$ then we can work out that

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

and

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

The infinitesimal transforms as

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

and so

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

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

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

The invariant volume element is given by

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

The metric is given by

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

${\displaystyle \rho (z_{1},z_{2})=\log {\frac {|z_{1}-{\overline {z_{2}}}|+|z_{1}-z_{2}|}{|z_{1}-{\overline {z_{2}}}|-|z_{1}-z_{2}|}}}$

for ${\displaystyle z_{1},z_{2}\in \mathbb {H} .}$

Another interesting form of the metric can be given in terms of the cross-ratio. Given any four points ${\displaystyle z_{1},z_{2},z_{3}}$ and ${\displaystyle z_{4}}$ in the compactified complex plane ${\displaystyle {\hat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \},}$ the cross-ratio is defined by

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

Then the metric is given by

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

Here, ${\displaystyle z_{1}^{\times }}$ and ${\displaystyle z_{2}^{\times }}$ are the endpoints, on the real number line, of the geodesic joining ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$. These are numbered so that ${\displaystyle z_{1}}$ lies in between ${\displaystyle z_{1}^{\times }}$ and ${\displaystyle z_{2}}$.

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

${\displaystyle w=e^{i\phi }{\frac {z-z_{0}}{z-{\overline {z_{0}}}}}}$

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

The canonical mapping is

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

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

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

by

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

The volume element is given by

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

The Poincaré metric is given by

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

for ${\displaystyle z_{1},z_{2}\in U.}$

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[]

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

${\displaystyle q=\exp(i\pi \tau )}$

where q is the nome and τ is the half-period ratio:

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

In the notation of the previous sections, τ is the coordinate in the upper half-plane ${\displaystyle \Im \tau >0}$. 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

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

The potential of the metric is

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

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.

See also[]

• Fuchsian group
• Fuchsian model
• Kleinian group
• Kleinian model
• Poincaré disk model
• Poincaré half-plane model
• Prime geodesic

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.)