Electron. J. Diff. Eqns., Vol. 2004(2004), No. 10, pp. 1-20.

Trajectories connecting two submanifolds on a non-complete Lorentzian manifold

Rossella Bartolo, Anna Germinario, & Miguel Sanchez

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 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.

Submitted November 21, 2003. Published January 14, 2004.
Math Subject Classifications: 58E30, 53C50, 83C10, 83C50.
Key Words: Lorentzian manifolds, gravitational and electromagnetic fields, convex boundary, critical point theory.

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Rossella Bartolo
Dipartimento di Matematica
Politecnico di Bari
Via G. Amendola, 126/B
70126 Bari Italy
email: rossella@poliba.it
Anna Germinario
Dipartimento di Matematica
Universita degli Studi di Bari
Via E. Orabona, 4
70125 Bari Italy
email: germinar@dm.uniba.it
Miguel Sanchez
Departamento de Geometria y Topologia
Fac. Ciencias, Univ. Granada
Avenida Fuentenueva s/n
18071 Granada Spain
email: sanchezm@ugr.es

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