Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
I'd recommend taking a look at the following monograph and surveys (and at the references cited therein):
Masiello, Antonio, Variational methods in Lorentzian geometry, Pitman Research Notes in Mathematics Series. 309. Harlow, Essex: Longman Scientific & Technical. New York, NY: Wiley. xix, 175 p. (1994). ZBL0816.58001.
Masiello, Antonio, Applications of calculus of variations to general relativity, Casciaro, B. (ed.) et al., Recent developments in general relativity. Proceedings of the 13th Italian conference on general relativity and gravitational physics, Cala Corvino-Monopoli, (Bari), Italy, September 21-25, 1998. Milan: Springer. 173-195 (2000). ZBL0995.83015.
Masiello, Antonio, Some variational problems in semi-Riemannian geometry, Frauendiener, Jörg (ed.) et al., Analytical and numerical approaches to mathematical relativity. Papers based on the presentation at the 319th WE-Heraeus seminar `Mathematical relativity: New ideas and developments’, Bad Honnef, Germany, March 1--5, 2004. With a foreword by Roger Penrose. Berlin: Springer (ISBN 3-540-31027-4/pbk). Lecture Notes in Physics 692, 51-77 (2006). ZBL1100.58006.
Candela, Anna Maria; Sánchez, Miguel, Geodesics in semi-Riemannian manifolds: geometric properties and variational tools, Alekseevsky, Dmitri V. (ed.) et al., Recent developments in pseudo-Riemannian geometry. Zürich: European Mathematical Society (ISBN 978-3-03719-051-7/pbk). ESI Lectures in Mathematics and Physics, 359-418 (2008). ZBL1204.53034.