  This paper presents an averaging method for nonlinear systems defined on
Riemannian manifolds. We extend closeness of solutions results for ordinary
differential equations on $R^{n}$ to dynamical systems defined on Riemannian
manifolds by employing differential geometry. A generalization of closeness of
solutions for periodic dynamical systems on compact time intervals is derived
for dynamical systems evolving on compact Riemannian manifolds. Under local
asymptotic (exponential) stability of the average vector field, we further
relax the compactness of the ambient Riemannian manifold and obtain the
closeness of solutions on the infinite time interval by employing the notion of
uniform normal neighborhoods of an equilibrium point of a vector field. These
results are also presented for time-varying dynamical systems where their
averaged systems are almost globally asymptotically or exponentially stable on
compact manifolds. The main results of the paper are illustrated by several
examples.
