An intrinsic approach to the geodesical connectedness of stationary Lorentzian manifolds

We prove a variational principle for geodesics on a Lorentzian manifold M admitting a timelike Killing vector field. Using this principle and standard techniques of global nonlinear analysis we establish the existence of geodesics 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 geodesics in M joining two fixed points.