Hyperbolic metrics

Let $\lambda_{\mathbb{H}}$ denote the positive function $\lambda_{\mathbb{H}}(z) = \frac{1}{Im(z)}$ defined on the upper half plane. One can define a metric on $\mathbb{H}$ using this function as follows: for a piecewise smooth curve $\gamma:[a,b] \to \mathbb{H}$, we define the length of $\gamma$ by the integral

$$\vert\gamma\vert = \int_\gamma \lambda_{\mathbb{H}} = \int_a^b \frac{1}{\mathrm{Im}(\gamma(t)}  \vert\gamma'(t)\vert dt.$$

The distance between two points in $\mathbb{H}$, called the Poincaré distance, is defined as the infimum of the lengths of (piecewise smooth) curves joining them. We called $\lambda_{\mathbb{H}} \vert dz\vert$ the Poincaré metric on $\mathbb{H}$.

Theorem 1.18   The group $\mathrm{PSL}(2,{\mathbb{R}})$ acts by isometries on $\mathbb{H}$ with respect to the Poincaré metric.

By the Riemann Mapping Theorem, if $D$ is a proper, simply connected subset of $\mathbb{C}$, then there exists a biholomorphic function $f:{\mathbb{H}} \to D$. We can then use the Poincaré metric on the upper half plane to get a metric on $D$. In particular, in the unit disc $\mathbb{D}$ we have that the expression of the metric is given by the function $\lambda_{\mathbb{D}}(z) = \frac{2}{1-\vert z\vert^2}$.

It is easy to show that the Poincaré metric has constant negative curvature equal to $-1$; or some other negative constant, if the metric is a positive multiple of the expression given above, as some textbooks do.