On the gauge version of exponential map *

Let P(M, G) be a principal bundle, Г be a principal connection on P and Λ be a classical connection on M. For every u ∈ P, we construct a local map expuΓ,Λ: TuP → P. Then we clarify how these maps can be used for finding all natural operators transforming (Γ, Λ) into a principal connection of the r-th principal gauge prolongation Wr P of P. We also deduce that expΓ,Λ coincides with the exponential map of a classical connection on P.