Lorentzian geometry is the privileged framework for the theory of general relativity. It is a variation on the usual riemannian geometry where one of the directions, called timelike, is negative whereas all the others, called spacelike, are positive. One of the main works of physical geometry in the 20th century has been to give geometric conditions on lorentzian spaces for them to serve as reasonable models for our spacetime ; conditions of time orientability, of causality... the strongest of those conditions is called global hyperbolicity. It is both a condition about the causality of the spacetime and the non-existence of points on its boundary.
One very natural way to generalize lorentzian geometry is, instead of only taking one timelike direction, to take q timelike directions and p spacelike directions. The resulting geometry is called a pseudo-riemannian geometry of signature (p,q), the case where q=1 being the lorentzian case. While pseudo-riemannian spaces for q bigger than two do not have any physical interpretation as of now, recent developments in Lie theory have given us reasons to study those spaces in more depth. Our goal will be to give a way to generalize the notion of global hyperbolicity in higher signature and to give a few uses of this generalization.