A connection whose curvature is the Lie bracket

TL;DR

This paper explores a natural connection on trivial principal G-bundles over Lie algebras, where the curvature equals the Lie bracket, illustrating this with examples including diffeomorphism groups and rolling ball motions.

Contribution

It introduces a natural connection whose curvature equals the Lie bracket and demonstrates its applications in geometric contexts like diffeomorphism groups and rolling motions.

Findings

The exponential map corresponds to parallel transport of this connection.

Curvature in specific cases is proportional to the Lie bracket.

Applications include the motion of a rolling ball on surfaces.

Abstract

Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie bracket of vectorfields on the manifold. The motion of a ball rolling on an oriented surface is the parallel transport of a similar connection on the trivial SO(3)-bundle over the surface. If the surface is a plane or a sphere, then the curvature of the connection is a scalar multiple of the Lie bracket in the Lie algebra of SO(3).