Generalized bounded deformation in non-Euclidean settings

We introduce a new space of generalized functions of bounded deformation GBDF, made of functions u whose one-dimensional slices u(γ) · ˙γ have bounded variation in a generalized sense for all curves γ solution of the second order ODE ¨γ = F(γ, ˙γ) for a fixed field F. For u ∈ GBDF we study the structure of the jump set in connection with its slices and prove the existence of a curvilinear approximate symmetric gradient. With a particular choice of F in terms of the Christoffel symbols of a Riemannian manifold M, we are able to define and recover similar properties for a space of 1-forms on M which have generalized bounded deformation in a suitable sense.