Addenda and corrections to work done on the path-integral approach to classical mechanics

We continue the study of the path-integral approach to classical mechanics and in particular we correct and better clarify, with respect to previous papers, the geometrical meaning of the variables entering this formulation. We show that the space spanned by the whole set of variables (φ,c,λ,c̅ ) of our path integral is the cotangent bundle to the reversed-parity tangent bundle of the phase space M of our system and it is indicated as T^⋆(ΠTM). We also show that it is possible to build a different path integral made only of bosonic variables. These turn out to be the coordinates of T^⋆(T^⋆M) which is the double cotangent bundle to phase space.