Global hyperbolicity in higher signatures

Global hyperbolicity is the strongest causality condition in Lorentzian geometry. It encapsulates both the non-existence of causal loops and of causal paths going to the edge of the spacetime in finite time and is of interest to both physicists and mathematicians. In non-Lorentzian pseudo-Riemannian geometry of signature (p,q), i.e such that p and q are greater than 1, there does not exist any obvious generalization of global hyperbolicity as there does not exist any causality for non-positive paths. This stems for the connectedness of the space of negative vectors in $R^{p,q}$ when q is greater than 1. My goal in this talk will be to introduce a way to generalize global hyperbolicity in higher signatures as well as other concepts in Lorentzian geometry.