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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 surface]]s.
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| 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 [[isometry|isometries]] are given by [[Möbius transformation]]s. A third representation is on the [[punctured disk]], where relations for [[q-analog]]ues are sometimes expressed. These various forms are reviewed below.
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| ==Overview of metrics on Riemann surfaces==
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| A metric on the complex plane may be generally expressed in the form
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| :<math>ds^2=\lambda^2(z,\overline{z})\, dz\,d\overline{z}</math> | |
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| where λ is a real, positive function of <math>z</math> and <math>\overline{z}</math>. The length of a curve γ in the complex plane is thus given by
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| :<math>l(\gamma)=\int_\gamma \lambda(z,\overline{z})\, |dz|</math>
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| The area of a subset of the complex plane is given by
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| :<math>\text{Area}(M)=\int_M \lambda^2 (z,\overline{z})\,\frac{i}{2}\,dz \wedge d\overline{z}</math>
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| 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
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| :<math>dz \wedge d\overline{z}=(dx+i\,dy)\wedge (dx-i \, dy)= -2i\,dx\wedge dy.</math>
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| A function <math>\Phi(z,\overline{z})</math> is said to be the '''potential of the metric''' if
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| :<math>4\frac{\partial}{\partial z}
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| \frac{\partial}{\partial \overline{z}} \Phi(z,\overline{z})=\lambda^2(z,\overline{z}).</math>
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| The [[Laplace–Beltrami operator]] is given by
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| :<math>\Delta = \frac{4}{\lambda^2}
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| \frac {\partial}{\partial z}
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| \frac {\partial}{\partial \overline{z}}
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| = \frac{1}{\lambda^2} \left(
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| \frac {\partial^2}{\partial x^2} +
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| \frac {\partial^2}{\partial y^2}
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| \right).</math>
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| The Gaussian [[curvature]] of the metric is given by
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| :<math>K=-\Delta \log \lambda.\,</math>
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| This curvature is one-half of the [[Ricci scalar curvature]].
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| 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})\, dz \, d\overline{z}</math> and ''T'' be a Riemann surface with metric <math>\mu^2(w,\overline{w})\, dw\,d\overline{w}</math>. Then a map
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| :<math>f:S\to T\,</math>
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| with <math>f=w(z)</math> is an isometry if and only if it is conformal and if
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| :<math>\mu^2(w,\overline{w}) \;
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| \frac {\partial w}{\partial z}
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| \frac {\partial \overline {w}} {\partial \overline {z}} =
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| \lambda^2 (z, \overline {z})
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| </math>.
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| Here, the requirement that the map is conformal is nothing more than the statement
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| :<math>w(z,\overline{z})=w(z),</math>
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| that is,
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| :<math>\frac{\partial}{\partial \overline{z}} w(z) = 0.</math>
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| ==Metric and volume element on the Poincaré plane==
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| The '''Poincaré metric tensor''' in the [[Poincaré half-plane model]] is given on the [[upper half-plane]] '''H''' as
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| :<math>ds^2=\frac{dx^2+dy^2}{y^2}=\frac{dz \, d\overline{z}}{y^2}</math> | |
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| where we write <math>dz=dx+i\,dy.</math>
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| This metric tensor is invariant under the action of [[SL2(R)|SL(2,'''R''')]]. That is, if we write
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| :<math>z'=x'+iy'=\frac{az+b}{cz+d}</math>
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| for <math>ad-bc=1</math> then we can work out that
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| :<math>x'=\frac{ac(x^2+y^2)+x(ad+bc)+bd}{|cz+d|^2}</math>
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| and
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| :<math>y'=\frac{y}{|cz+d|^2}.</math>
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| The infinitesimal transforms as
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| :<math>dz'=\frac{dz}{(cz+d)^2}</math>
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| and so
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| :<math>dz'd\overline{z}' = \frac{dz\,d\overline{z}}{|cz+d|^4}</math>
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| thus making it clear that the metric tensor is invariant under SL(2,'''R''').
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| The invariant [[volume element]] is given by
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| :<math>d\mu=\frac{dx\,dy}{y^2}.</math>
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| The metric is given by
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| :<math>\rho(z_1,z_2)=2\tanh^{-1}\frac{|z_1-z_2|}{|z_1-\overline{z_2}|}</math>
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| :<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>
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| for <math>z_1,z_2 \in \mathbb{H}</math>.
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| 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 [[Riemann sphere|compactified complex plane]] <math>\hat{\mathbb{C}} = \mathbb{C} \cup \infty</math>, the cross-ratio is defined by
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| :<math>(z_1,z_3; z_2,z_4) =
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| \frac{(z_1-z_2)(z_3-z_4)}{(z_3-z_2)(z_1-z_4)}.</math>
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| Then the metric is given by
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| :<math> \rho(z_1,z_2)= \log (z_1,z_2^\times ; z_2, z_1^\times).</math>
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| 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>.
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| 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.
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| ==Conformal map of plane to disk==
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| The upper half plane can be [[conformal map|mapped conformally]] to the [[unit disc|unit disk]] with the [[Möbius transformation]]
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| :<math>w=e^{i\phi}\frac{z-z_0}{z-\overline {z_0}}</math> | |
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| 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''<sub>0</sub> 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.
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| The canonical mapping is
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| :<math>w=\frac{iz+1}{z+i}</math>
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| which takes ''i'' to the center of the disk, and ''0'' to the bottom of the disk.
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| ==Metric and volume element on the Poincaré disk==
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| The '''Poincaré metric tensor''' in the [[Poincaré disk model]] is given on the open [[unit disk]] <math>U=\{z=x+iy:|z|=\sqrt{x^2+y^2} < 1 \}</math> by
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| :<math>ds^2=\frac{4(dx^2+dy^2)}{(1-(x^2+y^2))^2}=\frac{4 dz\,d\overline{z}}{(1-|z|^2)^2}.</math>
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| The volume element is given by
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| :<math>d\mu=\frac{4 dx\,dy}{(1-(x^2+y^2))^2}=\frac{4 dx\,dy}{(1-|z|^2)^2}.</math>
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| The Poincaré metric is given by
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| :<math>\rho(z_1,z_2)=\tanh^{-1}\left|\frac{z_1-z_2}{1-z_1\overline{z_2}}\right|</math>
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| for <math>z_1,z_2 \in U.</math>
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| The geodesics for this metric tensor are circular arcs whose endpoints are orthogonal to the boundary of the disk.
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| ==The punctured disk model==
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| [[File:J-inv-modulus.jpeg|thumb|J-invariant in punctured disk coordinates; that is, as a function of the nome.]]
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| [[File:J-inv-poincare.jpeg|thumb|J-invariant in Poincare disk coordinates; note this disk is rotated by 90 degrees from canonical coordinates given in this article]]
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| A second common mapping of the [[upper half-plane]] to a disk is the [[q-analog|q-mapping]]
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| :<math>q=\exp(i\pi\tau)</math> | |
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| where ''q'' is the [[nome (mathematics)|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 [[codomain|image]] of the map.
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| The Poincaré metric on the upper half-plane induces a metric on the q-disk
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| :<math>ds^2=\frac{4}{|q|^2 (\log |q|^2)^2} dq \, d\overline{q}</math>
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| The potential of the metric is
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| :<math>\Phi(q,\overline{q})=4 \log \log |q|^{-2}</math>
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| ==Schwarz lemma==
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| The Poincaré metric is [[contraction mapping|distance-decreasing]] on [[laplace operator|harmonic]] functions. This is an extension of the [[Schwarz lemma]], called the [[Schwarz-Alhfors-Pick theorem]].
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| ==See also==
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| *[[Fuchsian group]]
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| *[[Fuchsian model]]
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| *[[Kleinian group]]
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| *[[Kleinian model]]
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| *[[Poincaré disk model]]
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| *[[Poincaré half-plane model]]
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| *[[Prime geodesic]]
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| ==References==
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| * Hershel M. Farkas and Irwin Kra, ''Riemann Surfaces'' (1980), Springer-Verlag, New York. ISBN 0-387-90465-4.
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| * Jurgen Jost, ''Compact Riemann Surfaces'' (2002), Springer-Verlag, New York. ISBN 3-540-43299-X ''(See Section 2.3)''.
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| * [[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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| {{DEFAULTSORT:Poincare metric}}
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| [[Category:Conformal geometry]]
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| [[Category:Hyperbolic geometry]]
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| [[Category:Riemannian geometry]]
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| [[Category:Riemann surfaces]]
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