Homogeneous and flow-invariant geometry on the unit tangent bundle of hyperbolic space

We construct the Sasaki metric on the unit tangent bundle $U_{g}M$ of a Riemannian manifold $(M,g)$ and describe the unit tangent bundle $U{\mathbb{H}}^n$ of real hyperbolic space as a homogeneous space, both under $S{O}_0(1,n)$ and under the larger group $S{O}_0(1,n)\times S{O}_0(1,1)$, yielding explicit of $G$-invariant metrics. Using Hopf coordinates and Busemann functions, we then construct a Riemannian metric $g_{Hopf}$ on $U{\mathbb{H}}^n$ that is invariant under the geodesic flow, and we identify the horospherical cylinders as totally geodesic leaves of a natural foliation associated to a Busemann function, with respect to an explicit metric connection with torsion.