A general criterion for jump set slicing and applications

In this paper a novel criterion for the slicing of the jump set of a function u: \R^{n} \to \R^{m} is provided, 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 [3] and more recently adapted to the case of functions of bounded \mathcal{A}-variation [5]. Our method instead, builds upon a recent rectifiability result of integral geometric measures and extends to Riemannian manifolds, where slicing is conducted along geodesics. In particular, we are able to deal with one dimensional slices of the form u(\gamma) \cdot g(\gamma, \dot{\gamma}) for a continuous function g \colon \R^{n} \times \R^{n} \to \R^{m} and for curves \gamma \colon \R \to \R^{n} solutions of a suitable second order ODE. This covers all the above mentioned spaces. As a particular application, we study the structure of the jump set of functions with (generalized) bounded deformation in a Riemannian setting.