Trajectories connecting two submanifolds on a non-complete Lorentzian manifold

This article presents existence and multiplicity results for orthogonal trajectories joining two submanifolds $\Sigma_1$ and $\Sigma_2$ of a static space-time manifold $M$ under the action of gravitational and electromagnetic vector potential. The main technical difficulties are because $M$ may not be complete and $\Sigma_1$, $\Sigma_2$ may be 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.