COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 7, Number 1, 157-197, 1999 An Intrinsic Approach to the Geodesical Connectedness of Stationary Lorentzian Manifolds FABIO GIANNONI 1 AND PAOLO PICCIONE 2 We prove a variational principle for geodesies on a Lorentzian ma- nifold M. admitting a timelike Killing vector field. Using this prin- ciple and standard techniques of global nonlinear analysis we es- tablish the existence of geodesies that join two fixed points of .M, under a suitable coercivity assumption on M. Whenever M is non contractible, we also get a multiplicity result for geodesies in M. joining two fixed points. 1. Introduction. In this paper we consider the problem of the existence of geodesies that join two fixed point in a Lorentzian manifold. We will make a symmetry assumption on the metric of our Lorentzian manifold. Namely, we will assume that our spacetime possesses a 1- parameter group of (local) isometries, whose infinitesimal generator is a timelike vector field. Heuristically, this amounts to say that the coeffi- cients of our Lorentzian metric tensor are invariant by time translation (see Lemma C.l), and so our manifold is stationary with respect to a given ob- server field. Such a vector field is used to prove an alternative variational principle for geodesies, and this principle allows to reduce the search of geodesies to the study of the critical points of a smooth functional which is bounded from below. The class of stationary Lorentzian manifolds is reasonably large, and it contains examples that can be considered interesting both from a physicist's and from a mathematician's point of view. Among others, we would like to recall here the Schwarzsehild space-time, the Reissner-Nordstrom space- time and the Kerr spacertime. We refer to [9] for a detailed description of the mentioned examples and their physical interpretation. Martially supported by MURST Supported by CAPES, Processo AEX1697/96-0 157