If I have a function $f(z) = \frac{z-i}{z + i}$, with $f: \Bbb{H} \to \Bbb{D}$ and the First Fundamental Form coefficients of each space ($\Bbb H$ and $\Bbb D$) given, how can I show that f is an isometry between $\Bbb H$ and $\Bbb D$?
Here z is standard, i.e z = u + iv.
$ \Bbb D := \{(x,y) : x^2 + y^2 < 1\}$
$ \Bbb H$ is the hyperbolic plane with $(u,v) \in \Bbb R \times (0,\infty)$