Abstract:
I will present a synthetic-Lorentzian analog of the generalized Cartan-Hadamard theorem for locally convex metric spaces established by S. Alexander and R. Bishop. This entails introducing an appropriate definition of local concavity for Lorentzian pre-length spaces as well as proving the existence and uniqueness of timelike geodesics between any pair of timelike-related points under the additional assumptions of global hyperbolicity and future one-connectedness. As a consequence we get globalization results for this notion of concavity and for upper timelike curvature bounds by κ ≥ 0.