We know that any $2$-dimensional manifold $M$ with constant negative curvature is locally isometric to hyperbolic plane $\mathbb{H}_2$. Can this be extended to a local isometry to its closure as well, i.e $\overline{\mathbb{H}_2} = \mathbb{H}_2\cup \partial\mathbb{H}_2$? If so, when is this possible? 

The motivation behind this question is to pull-back, under the isometry, the geodesic $\gamma$ that connects a point $p\in \mathbb{H}_2$ and another point $q\in \partial \mathbb{H}_2$ at infinity. But since $q\notin \mathbb{H}_2$, I doubted whether this is meaningful.