  In this conference published in 1997 some problems on the geodesics of a
Lorentzian manifold concerning causality and infinite-dimensional variational
methods, are pointed out. Even though a big progress on many of these questions
have been carried out since then, some computations in this paper may be useful
and have not been published elsewhere. Among them, for example, the following
one (Section 3). Consider a spacetime which can be written globally as a
product $R x M$, such that the natural vector field associated to the
coordinate $t$ in $R$ is timelike. When is this spacetime globally hyperbolic
with Cauchy hypersurfaces the slices $t=$ constant?
