In this talk a novel criterion for the slicing of the jump set of functions is presented, which overcomes the limitation of the classical approach based on a combination of codimension-one slices and the parallelogram law. This latter technique was originally developed for functions of bounded deformation and more recently adapted to the case of functions of bounded A-variation . Our method instead, builds upon a recent rectifiability result of integral geometric measures and extends to Riemannian manifolds, where slicing is conducted along geodesics. As a particular application, we derive the structure of the jump set of functions with (generalized) bounded deformation in a Riemannian setting. Furthermore, if time permits, I will show the critical role these structural results play in deriving certain non-local to local free-discontinuity problems.