Natural Transformations of Connections on the First Principal Prolongation

We consider a vector bundle E →M and the principal bundle PE of frames of E. We determine all natural transformations of the connection bundle of the first order principal prolongation of principal bundle PE into itself. Unless otherwise specified, we use the terminology and notation from the book [7]. All manifolds and maps are assumed to be infinitely differentiable. In the monograph [7], the following assertion was deduced. Proposition 1. All natural operators transforming principal connection Λ on P 1M into principal connection Λ1 on P 1M are of zero order and form a 3-parameter family Λ1 = Λ + Φ1(Λ) where Φ1(Λ) is a natural (1, 2)-tensor field of the form Φ1 = a1T + a2 IdTM ⊗c1(T ) + a3c1(T )⊗ IdTM ai ∈ R , where T denotes the torsion tensor of Λ and c1 denotes the contraction to the first subscript. We discuss the generalization of this problem, i.e. the case of the principal connections on the first order prolongation W 1P of a principal bundle P . 1. Principal connections We consider a principal bundle P = (P,M, π;G) with a structure group G. We denote by (x, z) fibered coordinates on P , λ = 1, . . . ,dimM, a = 1, . . . ,dimG. A principal connection on P is defined as lifting linear mapping