Define the hyperbolic metric $g$ on $\mathbb{H}^2$ by
$$
g = \frac{1}{2y^2} (d x^2 + d y^2).
$$
The Iwasawa decomposition yields an isomorpism $SL(2)/SO(2)=\mathbb{H}^2$ which is given by
$$
\pmatrix{a_{11} & a_{12}\\ a_{21} & a_{22}} \mapsto \frac{1}{(a_{21}^2 + a_{22}^2)}(a_{11} a_{21} + a_{12} a_{22}, 1).
$$
Moreover, if you define a metric $h$ on $SL(2)$ by
$$
h_A (V, W) = Tr(A^{-1} V A^{-1} W),
$$
then the projection $SL(2) \to \mathbb{H}^2$ is a Riemannian submersion. Some time ago, I've written some notes on this (and related things concerning the Poincare upper half plane). They are not in a very good shape but may be helpful nonetheless (they were meant as a first experiment with sage).