Geodesics in static Lorentzian manifolds with critical quadratic behavior

The aim of this paper is to study the geodesic connectedness of a complete static Lorentzian manifold (M, <(.),(.)> L) such that, if K is its irrotational timelike Killing vector field, the growth of beta = -<K, K>(L) is (at most) quadratic. This is shown to be equivalent to a critical variational problem on a Riemannian manifold, where some geometrical interpretations yield the optimal results. Multiplicity of connecting geodesics is studied, especially in the timelike case, where the restrictions for the number of causal connecting geodesics are stressed. Extensions to the non-complete case, including discussions on pseudoconvexity, are also given.