  Brake orbits and homoclinics of autonomous dynamical systems correspond, via
Maupertuis principle, to geodesics in Riemannian manifolds endowed with a
metric which is singular on the boundary (Jacobi metric). Motivated by the
classical, yet still intriguing in many aspects, problem of establishing
multiplicity results for brake orbits and homoclinics, as done in [6, 7, 10],
and by the development of a Morse theory in [8] for geodesics in such kind of
metric, in this paper we study the related normal exponential map from a global
perspective.
