##

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