Submetries in non-positive signature and applications

We consider a generalization of the notion of submetry tailored to the setting of spacetimes. We show that, under suitable completeness assumptions, Lorentzian submetries between smooth spacetimes correspond to locally C^{1,1} Semi-Riemannian submersions. Moreover, we establish some applications regarding timelike curvature bounds, isometric group actions, acausal foliations and Lorentz-Wasserstein geometry.