In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Poincaré metric, named after
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
, is the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
describing a two-dimensional surface of constant negative
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
. It is the natural metric commonly used in a variety of calculations in
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
or
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s.
There are three equivalent representations commonly used in two-dimensional hyperbolic
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. One is the
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincar� ...
, defining a model of hyperbolic space on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
. The
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk ...
defines a model for hyperbolic space on the
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
. The disk and the upper half plane are related by a
conformal map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
, and
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
are given by
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
s. A third representation is on the
punctured disk
In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' mea ...
, 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
:
where λ is a real, positive function of
and
. The length of a curve γ in the complex plane is thus given by
:
The area of a subset of the complex plane is given by
:
where
is the
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
used to construct the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
. The determinant of the metric is equal to
, so the square root of the determinant is
. The Euclidean volume form on the plane is
and so one has
:
A function
is said to be the potential of the metric if
:
The
Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named ...
is given by
:
The Gaussian
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
of the metric is given by
:
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
and ''T'' be a Riemann surface with metric
. Then a map
:
with
is an isometry if and only if it is conformal and if
:
.
Here, the requirement that the map is conformal is nothing more than the statement
:
that is,
:
Metric and volume element on the Poincaré plane
The Poincaré metric tensor in the
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincar� ...
is given on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
H as
:
where we write
and
.
This metric tensor is invariant under the action of
SL(2,R). That is, if we write
:
for
then we can work out that
:
and
:
The infinitesimal transforms as
:
and so
:
thus making it clear that the metric tensor is invariant under SL(2,R). Indeed,
:
The invariant
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV ...
is given by
:
The metric is given by
:
:
for
Another interesting form of the metric can be given in terms of the ''
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
''. Given any four points
and
in the
compactified complex plane the cross-ratio is defined by
:
Then the metric is given by
:
Here,
and
are the endpoints, on the real number line, of the geodesic joining
and
. These are numbered so that
lies in between
and
.
The
geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
with the
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
:
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
maps to the edge of the unit disk
The constant real number
can be used to rotate the disk by an arbitrary fixed amount.
The canonical mapping is
:
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
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk ...
is given on the open
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
:
by
:
The volume element is given by
:
The Poincaré metric is given by
:
for
The geodesics for this metric tensor are circular arcs whose endpoints are orthogonal to the boundary of the disk.
Geodesic flow
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s on the Poincaré disk are
Anosov flows; that article develops the notation for such flows.
The punctured disk model
A second common mapping of the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
to a disk is the
q-mapping
:
where ''q'' is the
nome and τ is the
half-period ratio:
:
.
In the notation of the previous sections, τ is the coordinate in the upper half-plane
. The mapping is to the punctured disk, because the value ''q''=0 is not in the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of the map.
The Poincaré metric on the upper half-plane induces a metric on the q-disk
:
The potential of the metric is
:
Schwarz lemma
The Poincaré metric is
distance-decreasing on
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
functions. This is an extension of the
Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapp ...
, called the
Schwarz–Ahlfors–Pick theorem
In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model.
The Schwarz–Pick lemma states that every holomorphic function from the unit disk ''U ...
.
See also
*
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
*
Fuchsian model
*
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...
*
Kleinian model
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold ''N'' by the quotient space \mathbb^3 / \Gamma where \Gamma is a discrete subgroup of PSL(2,C). Here, the subgroup \Gamma, a Kleinian group, is defined so ...
*
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk ...
*
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincar� ...
*
Prime geodesic
In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they ob ...
References
* Hershel M. Farkas and Irwin Kra, ''Riemann Surfaces'' (1980), Springer-Verlag, New York. .
* Jurgen Jost, ''Compact Riemann Surfaces'' (2002), Springer-Verlag, New York. ''(See Section 2.3)''.
*
Svetlana Katok, ''Fuchsian Groups'' (1992), University of Chicago Press, Chicago ''(Provides a simple, easily readable introduction.)''
{{DEFAULTSORT:Poincare metric
Conformal geometry
Hyperbolic geometry
Riemannian geometry
Riemann surfaces
Metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...