## Hyperbolic metrics

Let denote the positive function defined on the upper half plane. One can define a metric on using this function as follows: for a piecewise smooth curve , we define the length of by the integral

The distance between two points in , called the Poincaré distance, is defined as the infimum of the lengths of (piecewise smooth) curves joining them. We called the Poincaré metric on .

Theorem 1.18   The group acts by isometries on with respect to the Poincaré metric.

By the Riemann Mapping Theorem, if is a proper, simply connected subset of , then there exists a biholomorphic function . We can then use the Poincaré metric on the upper half plane to get a metric on . In particular, in the unit disc we have that the expression of the metric is given by the function .

It is easy to show that the Poincaré metric has constant negative curvature equal to ; or some other negative constant, if the metric is a positive multiple of the expression given above, as some textbooks do.

Pablo Ares Gastesi 2005-08-31