Trajectories connecting two submanifolds on non--complete Lorentzian manifolds

This article presents existence and multiplicity results for orthog- onal trajectories joining two submanifolds Σ1 and Σ2 of a static space-time manifold M under the action of gravitational and electromagnetic vector po- tential. The main technical difficulties are because M may not be complete and Σ1 , Σ2 may not be compact. Hence, a suitable convexity assumption and hypotheses at infinity are needed. These assumptions are widely discussed in terms of the electric and magnetic vector fields naturally associated. Then, these vector fields become relevant from both their physical interpretation and the mathematical gauge invariance of the equation of the trajectories.