The purpose of this article is to study the two homogeneous structures of the unit bundle UH^n of a real hyperbolic space. In both the cases, we determine the -invariant Riemannian metrics. In passing, we examine whether the geodesic flow is an isometry of UH^n when equipped with its Levi-Civita metric. Finally, we study the manifold of geodesics of H^n
seen as a homogeneous space.

hyperbolic space, isotropy representations, unit bundle