During the past years there has been a considerable amount of research
related to the problem of geodesic connectedness of Lorentzian manifolds. This
topic has wide applications in Physics, but for mathematicians its interest is
essentially due to the peculiar di culty of this natural problem, which makes it
challenging from both an analytical and a geometrical point of view.
In this talk I discuss the geodesic connectedness problem on globally hyper-
bolic spacetimes endowed with a complete, timelike or lightlike, Killing vector
eld and a complete Cauchy hypersurface.
Then I introduce the notion of open subset with convex boundary and
present some applications of previous results.
