Geometric interpretations of the symmetric product in affine   differential geometry
  The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpretation of the symmetric product is used to provide an intrinsic proof of the fact that the distributions closed under the symmetric product are exactly those distributions invariant under the geodesic flow. 
M. Barbero-Li\~n\'an, A. D. Lewis
math.DG