Poincaré metric

Metric tensor describing constant negative (hyperbolic) curvature

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

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Main article: metric tensor

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

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

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

l ( γ ) = ∫ γ λ ( z , z ¯ ) | d z | {\displaystyle l(\gamma )=\int _{\gamma }\lambda (z,{\overline {z}})\,|dz|}

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

Area ( M ) = ∫ M λ 2 ( z , z ¯ ) i 2 d z ∧ d z ¯ {\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  λ 4 {\displaystyle \lambda ^{4}} , so the square root of the determinant is  λ 2 {\displaystyle \lambda ^{2}} . The Euclidean volume form on the plane is  d x ∧ d y {\displaystyle dx\wedge dy} and so one has

d z ∧ d z ¯ = ( d x + i d y ) ∧ ( d x − i d y ) = − 2 i d x ∧ d y . {\displaystyle dz\wedge d{\overline {z}}=(dx+i\,dy)\wedge (dx-i\,dy)=-2i\,dx\wedge dy.}

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

4 ∂ ∂ z ∂ ∂ z ¯ Φ ( z , z ¯ ) = λ 2 ( z , z ¯ ) . {\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

Δ = 4 λ 2 ∂ ∂ z ∂ ∂ z ¯ = 1 λ 2 ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) . {\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

K = − Δ log ⁡ λ . {\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  λ 2 ( z , z ¯ ) d z d z ¯ {\displaystyle \lambda ^{2}(z,{\overline {z}})\,dz\,d{\overline {z}}} and T be a Riemann surface with metric  μ 2 ( w , w ¯ ) d w d w ¯ {\displaystyle \mu ^{2}(w,{\overline {w}})\,dw\,d{\overline {w}}} . Then a map

f : S -> T {\displaystyle f:S\to T\,}

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

μ 2 ( w , w ¯ ) ∂ w ∂ z ∂ w ¯ ∂ z ¯ = λ 2 ( z , z ¯ ) {\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

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

that is,

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

## 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

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

where we write  d z = d x + i d y {\displaystyle dz=dx+i\,dy} and  d z ¯ = d x − i d y {\displaystyle 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 ′ + i y ′ = a z + b c z + d {\displaystyle z'=x'+iy'={\frac {az+b}{cz+d}}}

for  a d − b c = 1 {\displaystyle ad-bc=1} then we can work out that

x ′ = a c ( x 2 + y 2 ) + x ( a d + b c ) + b d | c z + d | 2 {\displaystyle x'={\frac {ac(x^{2}+y^{2})+x(ad+bc)+bd}{|cz+d|^{2}}}}

and

y ′ = y | c z + d | 2 . {\displaystyle y'={\frac {y}{|cz+d|^{2}}}.}

The infinitesimal transforms as

d z ′ = ∂ ∂ z ( a z + b c z + d ) d z = a ( c z + d ) − c ( a z + b ) ( c z + d ) 2 d z = a c z + a d − c a z − c b ( c z + d ) 2 d z = a d − c b ( c z + d ) 2 d z = a d − c b = 1 1 ( c z + d ) 2 d z = d z ( c z + d ) 2 {\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

d z ′ d z ¯ ′ = d z d z ¯ | c z + d | 4 {\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,

d z ′ d z ¯ ′ y ′ 2 = d z d z ¯ | c z + d | 4 y 2 | c z + d | 4 = d z d z ¯ y 2 . {\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

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

The metric is given by

ρ ( z 1 , z 2 ) = 2 tanh − 1 ⁡ | z 1 − z 2 | | z 1 − z 2 ¯ | {\displaystyle \rho (z_{1},z_{2})=2\tanh ^{-1}{\frac {|z_{1}-z_{2}|}{|z_{1}-{\overline {z_{2}}}|}}}

ρ ( z 1 , z 2 ) = log ⁡ | z 1 − z 2 ¯ | + | z 1 − z 2 | | z 1 − z 2 ¯ | − | z 1 − 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  z 1 , z 2 ∈ H . {\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  z 1 , z 2 , z 3 {\displaystyle z_{1},z_{2},z_{3}} and  z 4 {\displaystyle z_{4}} in the compactified complex plane C ^ = C ∪ { ∞ } , {\displaystyle {\hat {\mathbb {C} }}=\mathbb {C} \cup \\{\infty \\},} the cross-ratio is defined by

( z 1 , z 2 ; z 3 , z 4 ) = ( z 1 − z 3 ) ( z 2 − z 4 ) ( z 1 − z 4 ) ( z 2 − z 3 ) . {\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

ρ ( z 1 , z 2 ) = log ⁡ ( z 1 , z 2 ; z 1 × , z 2 × ) . {\displaystyle \rho (z_{1},z_{2})=\log \left(z_{1},z_{2};z_{1}^{\times },z_{2}^{\times }\right).}

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

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

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

The canonical mapping is

w = i z + 1 z + i {\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

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

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

d s 2 = 4 ( d x 2 + d y 2 ) ( 1 − ( x 2 + y 2 ) ) 2 = 4 d z d z ¯ ( 1 − | z | 2 ) 2 . {\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

d μ = 4 d x d y ( 1 − ( x 2 + y 2 ) ) 2 = 4 d x d y ( 1 − | z | 2 ) 2 . {\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

ρ ( z 1 , z 2 ) = 2 tanh − 1 ⁡ | z 1 − z 2 1 − z 1 z 2 ¯ | {\displaystyle \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 ∈ U . {\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

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

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

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

In the notation of the previous sections, τ is the coordinate in the upper half-plane  ℑ τ > 0 {\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

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

The potential of the metric is

Φ ( q , q ¯ ) = 4 log ⁡ log ⁡ | q | − 2 {\displaystyle \Phi (q,{\overline {q}})=4\log \log |q|^{-2}}

## Schwarz lemma

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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

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

## References

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* 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.)