  We characterize the exact lumpability of smooth vector fields on smooth
manifolds. We derive necessary and sufficient conditions for lumpability and
express them from four different perspectives, thus simplifying and
generalizing various results from the literature that exist for Euclidean
spaces. We introduce a partial connection on the pullback bundle that is
related to the Bott connection and behaves like a Lie derivative. The lumping
conditions are formulated in terms of the differential of the lumping map, its
covariant derivative with respect to the connection and their respective
kernels. Some examples are discussed to illustrate the theory.
