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 di eomorphism group of a manifold M, the curvature of the natural connection is the Lie bracket of vector elds 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 R3 can be described by a similar connection.
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