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 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

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

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:

\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.

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.)

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