A connection whose curvature is the Lie bracket
Differential Geometry
2010-01-02 v2
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).
Cite
@article{arxiv.0803.3321,
title = {A connection whose curvature is the Lie bracket},
author = {Kent E. Morrison},
journal= {arXiv preprint arXiv:0803.3321},
year = {2010}
}
Comments
13 pages, revised and expanded