  Smooth manifolds are not the suitable context for trying to generalize the
concept of rough paths on a manifold. Indeed, when one is working with smooth
maps instead of Lipschitz maps and trying to solve a rough differential
equation, one loses the quantitative estimates controlling the convergence of
the Picard sequence. Moreover, even with a definition of rough paths in smooth
manifolds, ordinary and rough differential equations can only be solved locally
in such case. In this paper, we first recall the foundations of the Lipschitz
geometry, introduced in "Rough Paths on Manifolds" (Cass, T., Litterer, C. &
Lyons, T.), along with the main findings that encompass the classical theory of
rough paths in Banach spaces. Then we give what we believe to be a minimal
framework for defining rough paths on a manifold that is both less rigid than
the classical one and emphasized on the local behaviour of rough paths. We end
by explaining how this same idea can be used to define any notion of coloured
paths on a manifold.
