  We prove existence of global solutions for differential equations driven by a
geometric rough path under the condition that the vector fields have linear
growth. We show by an explicit counter-example that the linear growth condition
is not sufficient if the driving rough path is not geometric. This settle a
long-standing open question in the theory of rough paths. So in the geometric
setting we recover the usual sufficient condition for differential equation.
The proof rely on a simple mapping of the differential equation from the
Euclidean space to a manifold to obtain a rough differential equation with
bounded coefficients.
