We know that every hyperbolic manifold $M$ is locally isometric to hyperbolic plane $\mathbb{H}_2$. Can we consider the inverse image, under the isometry, of 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.e ideal points do not belong to $\mathbb{H}_2$, I doubt whether this is meaningful.
1 Answer
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Mind you, the manifold $M$ may be compact and have no boundary at all. Still, of course, one can talk about geodesics on it. The right way to consider their behaviour at infinity is to pass to the unit tangent bundle. Then one can define the stable foliation of the geodesic flow: its leaves consist of tangent vectors with the property that at infinity the corresponding geodesics are getting closer and closer to each other. If you pass to the universal cover, the latter property means precisely that those geodesics go to the same boundary point. Alternatively, one can first define the stable foliation for the universal cover, and then quotient by the fundamental group. In this way it is obvious that all (except for countably many ones that correspond to the boundary points with non-trivial stabilizers in the fundamental group) leaves of the stable foliation are isometric to the hyperbolic plane.
