In Peter Scott's "The geometry of 3-manifolds" he says that: "As $U\mathbb{H}^2$ is a circle bundle over $\mathbb{H}^2$, we see that $\widetilde{SL(2,\mathbb{R})}$ is naturally a line bundle over $\mathbb{H}^2$"
$U\mathbb{H}^2$ is the unit tangent bundle of $\mathbb{H}^2$ and previously he shows that $U\mathbb{H}^2$ induces a metric on $\widetilde{SL(2,\mathbb{R})}$.