Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 10, pp. 1-20.
Title: Trajectories connecting two submanifolds on a non-complete
Lorentzian manifold
Authors: Rossella Bartolo (Politecnico di Bari, Italy)
Anna Germinario (Univ. degli Studi di Bari, Italy)
Miguel Sanchez (Univ. Granada, Spain)
Abstract:
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.