Let $G$ be a Lie group with Lie algebra $\g$. On the trivial principal $G$-bundle
over $\g$ there is a natural connection whose curvature is the Lie bracket of
$\g$. The exponential map of $G$ is given by parallel transport of this
connection. If $G$ is the diffeomorphism group of a manifold $M$, the curvature
of the natural connection is the Lie bracket of vectorfields on $M$. In the case
that $G=\SO(3)$ the motion of a sphere rolling on a plane is given by parallel
transport of a pullback of the natural connection by a map from the plane to
$\so(3)$. The motion of a sphere rolling on an oriented surface in $\R^3$ can be
described by a similar connection.
Publié le : 2009-12-15
Classification:
Differential geometry,
Connections,
Issues of holonomy,
53C05,
53C29
@article{1281106598,
author = {Morrison, Kent E.},
title = {A connection whose curvature is the Lie bracket},
journal = {J. Gen. Lie Theory Appl.},
volume = {3},
number = {3},
year = {2009},
pages = { 311-319},
language = {en},
url = {http://dml.mathdoc.fr/item/1281106598}
}
Morrison, Kent E. A connection whose curvature is the Lie bracket. J. Gen. Lie Theory Appl., Tome 3 (2009) no. 3, pp. 311-319. http://gdmltest.u-ga.fr/item/1281106598/