Starting from the Amann-Conley-Zehnder finite reduction framework in the non-compact Viterbo's version we discuss the existence of global generating function with a finite number of auxiliary parameters describing the two-points Characteristic Relation related to the geodesic problem in the Hamiltonian formalism. This applies both to Analytical Mechanics and to General Relativity - we construct a global object generalizing the World Function introduced by Synge, which is well-defined only locally. Whenever the auxiliary parameters can be fully removed, Synge's World Function is restored.
Franco Cardin. Antonio Marigonda. "Global World Functions." J. Geom. Symmetry Phys. 2 1 - 17, 2004. https://doi.org/10.7546/jgsp-2-2004-1-17