Characterization of Exact Lumpability for Vector Fields on Smooth Manifolds

摘要

We characterize the exact lumpability of smooth vector fields on smoothmanifolds. We derive necessary and sufficient conditions for lumpability andexpress them from four different perspectives, thus simplifying andgeneralizing various results from the literature that exist for Euclideanspaces. We introduce a partial connection on the pullback bundle that isrelated to the Bott connection and behaves like a Lie derivative. The lumpingconditions are formulated in terms of the differential of the lumping map, itscovariant derivative with respect to the connection and their respectivekernels. Some examples are discussed to illustrate the theory.