Abstract
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.
Citation
Kent E. Morrison. "A connection whose curvature is the Lie bracket." J. Gen. Lie Theory Appl. 3 (4) 311 - 319, December 2009. https://doi.org/10.4303/jglta/S090404
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