  We present a geometric proof of the averaging theorem for perturbed dynamical
systems on a Riemannian manifold, in the case where the flow of the unperturbed
vector field is periodic and the $\mathbb{S}^{1}$-action associated to this
vector field is not necessarily trivial. We generalize the averaging procedure
\cite{Arno-63,ArKN-88} defining a global averaging method based on a free
coordinate approach which allow us to formulate our results on any open domain
with compact closure.
