  Billiard trajectories (broken generalised geodesics) are considered in the
exterior of an obstacle $K$ with smooth boundary on an arbitrary Riemannian
manifold. We prove a generalisation of the well-known Santalo's formula. As a
consequence, it is established that if the set of trapped points has positive
measure, then for all sufficiently small smooth perturbations of the boundary
of $K$ the set of trapped points for the new obstacle obtained in this way also
has positive measure. More generally the measure of the set of trapped points
depends continuously on perturbations of the obstacle $K$. Some consequences of
the generalised Santalo's formula are derived in the case of scattering by an
obstacle $K$ in an Euclidean space. For example, it is shown that, for a large
class of obstacles $K$, the volume of $K$ is uniquely determined by the average
travelling times of scattering rays in the exterior of $K$.
