“Your space-time could have corners, edges and folds, and it doesn’t matter. This approach doesn’t care about non-smoothness.”
Einstein’s general relativity equations break down when spacetime isn’t smooth, which is exactly what happens at black hole singularities and possibly at quantum scales. Mathematicians at the University of Vienna figured out how to measure curvature without requiring smoothness by adapting triangle-comparison geometry to work with “time separation” instead of distance. They proved versions of Hawking’s Big Bang theorem and Penrose’s black hole theorem without assuming smooth spacetime. That means these fundamental results are even more universal than the people who discovered them realized.