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Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially relevant the systematic introduction of (infinite-dimensional) variational methods.
Our purpose is to give an overview, from refinements of classical results to updated variational settings. First, several properties (and especially completeness) of geodesics in some ambient spaces are studied. This includes heuristic constructions of compact incomplete examples, geodesics in warped, GRW or stationary spacetimes, properties in surfaces and spaceforms, or problems on stability of completeness.
Then we study the variational framework, and focus on two fundamental problems of this approach, which regards geodesic connectedness. The first one deals with a variational principle for stationary manifolds, and its recent implementation inside causality theory. The second one concerns orthogonal splitting manifolds, and a reasonably self-contained development is provided, collecting some steps spread in the literature.