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

## TL;DR

This paper explores the algebraic and geometric properties of invariant connections on manifolds and algebroids, providing a foundation for advanced numerical integration methods like Runge-Kutta on complex geometric spaces.

## Contribution

It generalizes classical results to invariant connections on algebroids, linking algebraic properties to geometric structures relevant for numerical integrators.

## Key findings

- Characterization of spaces suitable for Lie-Butcher series methods
- Extension of Cartan and Nomizu results to algebroids
- Fundamental insights for numerical integration on manifolds

## Abstract

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.