Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie-Butcher series methods, which generalize Runge-Kutta methods, may be applied.