  A manifold with an arbitrary affine connection is considered and the geodesic
spray associated with the connection is studied in the presence of a Lie group
action. In particular, results are obtained that provide insight into the
structure of the reduced dynamics associated with the given invariant affine
connection. The geometry of the frame bundle of the given manifold is used to
provide an intrinsic description of the geodesic spray. A fundamental
relationship between the geodesic spray, the tangent lift and the vertical lift
of the symmetric product is obtained, which provides a key to understanding
reduction in this formulation.
