Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds
This work provides foundational theoretical insights for numerical integration on manifolds, but the results are incremental and primarily of interest to researchers in geometric numerical integration.
The paper establishes a connection between invariant connections and Lie algebra actions to generalize classical results, providing a characterization of spaces where Butcher and Lie-Butcher series methods (generalizations of Runge-Kutta) can be applied for 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.