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Concept
Poincaré metric
Summary
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.
metric tensor
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 used to construct the volume form. 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 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. 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,
The Poincaré metric tensor in the Poincaré half-plane model is given on the upper half-plane H as
where we write and .
Official source
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