  We introduce a new space of generalized functions of bounded deformation
$GBD_{F}$, made of functions u whose one-dimensional slice $u(\gamma) \cdot
\dot{\gamma}$ has bounded variation in a generalized sense for all curves
$\gamma$ solution of the second order ODE $\ddot{\gamma} = F(\gamma,
\dot{\gamma})$ for a fixed field F. For $u \in GBD_{F}$ we study the structure
of the jump set in connection 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.
