Some remarks on Causality Theory and Variational Methods in Lorentzian manifolds

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?