  B. Fedosov has given a simple and very natural construction of a deformation
quantization for any symplectic manifold, using a flat connection on the bundle
of formal Weyl algebras associated to the tangent bundle of a symplectic
manifold. The connection is obtained by affinizing, nonlinearizing, and
iteratively flattening a given torsion free symplectic connection. In this
paper, a classical analog of Fedosov's operations on connections is analyzed
and shown to produce the usual exponential mapping of a linear connection on an
ordinary manifold. A symplectic version is also analyzed. Finally, some remarks
are made on the implications for deformation quantization of Fedosov's index
theorem on general symplectic manifolds.
