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
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
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 (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis.
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.
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.
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
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