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.

A metric on the complex plane may be generally expressed in the form where λ is a real, positive function of

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 used to construct the volume form.

The Euclidean volume form on the plane is

is said to be the potential of the metric if The Laplace–Beltrami operator is given by The Gaussian curvature 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.

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, The Poincaré metric tensor in the Poincaré half-plane model is given on the upper half-plane H as where we write

This metric tensor is invariant under the action of SL(2,R).

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 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 the compactified complex plane

are the endpoints, on the real number line, of the geodesic joining

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.

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

maps to the edge of the unit disk

can be used to rotate the disk by an arbitrary fixed amount.

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

A second common mapping of the upper half-plane 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

This is an extension of the Schwarz lemma, called the Schwarz–Ahlfors–Pick theorem.