Natural maps on the iterated jet prolongation of a fibred manifold

Using an analytic procedure by the first author [5, 6], we first determine all natural transformations of the second iterated jet prolongation J1J1Y of a fibred manifold Y→X into itself depending on a linear connection Λ on the base manifold X. We obtain two 3-parameter families and we interpret them geometrically. Our results clarify the distinguished role of the involution on J1 J1 Y depending on Λ introduced by the second author [11]. Then we discuss the role of our transformations in the theory of the natural operators transforming a connection Γ on a fibred manifold Y→X and a linear connection Λ on X into a connection on the first jet prolongation J1Y→X of Y. In the final remark we determine all natural transformations of the second sesquiholonomic and holomonic jet prolongations of Y into themselves. Our attention to second order jet spaces is due to the role they play in fundamental geometric and physical theories (cf. curvature of connections and lagrangian theories).