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 This metric tensor is invariant under the action of SL. 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. 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. 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 for this metric tensor are circular arcs perpendicular to 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 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. 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 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. Geodesic flows 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 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 of the map. The Poincaré metric on the upper half-plane induces a metric on the q-disk The potential of the metric is
