Journal of Generalized Lie Theory and Applications
- J. Gen. Lie Theory Appl.
- Volume 3, Number 4 (2009), 311-319.
A connection whose curvature is the Lie bracket
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.
J. Gen. Lie Theory Appl., Volume 3, Number 4 (2009), 311-319.
First available in Project Euclid: 6 August 2010
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Morrison, Kent E. A connection whose curvature is the Lie bracket. J. Gen. Lie Theory Appl. 3 (2009), no. 4, 311--319. doi:10.4303/jglta/S090404. https://projecteuclid.org/euclid.jglta/1281106598