In this chapter, we discuss integration over the unit tangent bundle of a given Riemannian manifold. The geodesic flow of the Riemannian metric acts on the unit tangent bundle, and one of its salient features is the existence of a natural measure on the unit tangent bundle, called the kinematic density or the Liouville measure, which is invariant under the action of the geodesic flow. Furthermore, the integral of a function on the unit tangent bundle can be calculated by first integrating the function relative to the kinematic density over each of the fibers ((n – 1)–spheres) in the unit tangent bundle and then integrating the resulting function on the base manifold relative to its Riemannian measure. The measure on the fibers is the natural measure on spheres induced by Lebesgue measure on the tangent spaces.

We could present the kinematic density by simply writing it as the local product measure of the natural measure on tangent spheres and the Riemannian measure on the base manifold, and then verifying that it is invariant relative to the geodesic flow. However, we prefer a different route, one that detours through the formalism of classical analytical mechanics. This affords an opportunity to connect the discussion to an extremely important collection of ideas, important historically and in current research. We do not pursue this connection here to any extent, rather, we concentrate on Riemannian results that emerged from these notions.