Characterization of Exact Lumpability for Vector Fields on Smooth Manifolds

TL;DR

This paper provides a comprehensive characterization of exact lumpability for smooth vector fields on manifolds, offering necessary and sufficient conditions from multiple perspectives, and generalizing prior Euclidean results.

Contribution

It introduces a new partial connection related to the Bott connection, unifies various lumpability conditions, and extends the theory to smooth manifolds.

Findings

Derived necessary and sufficient conditions for lumpability.

Introduced a partial connection related to the Bott connection.

Generalized existing Euclidean results to smooth manifolds.

Abstract

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.