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Causal Trajectories between Submanifolds in Lorentzian
Geometry

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*Paolo Piccione*

**E-mail:** piccione@ime.usp.br
**Abstract.** We set up an infinite dimensional differential structure
for the set of causal curves joining two submanifolds $P$ and $\Gamma$
of a Lorentzian manifold $\M$, and we prove a variational principle that
characterizes the timelike geodesics. Such principle, which is obtained
using a sort of {\em arrival time\/} functional, is given by a general-relativistic
version of the Fermat's principle for light rays in Classical Optics. The
lightlike geodesics between $P$ and $\Gamma$ are obtained with a limit
process, which is briefly discussed in the last section.

**AMSclassification.** 53C22, 53C50, 58E30

**Keywords.** Lorentzian geometry, causal geodesics, Fermat's Principle