  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).
