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 [ edit ]
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 [ edit ]
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 [ edit ]
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 z 0 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 [ edit ]
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\}}
by
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 [ edit ]
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 [ edit ]
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 [ edit ]
References [ edit ]
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.)