The Frölicher-Nijenhuis bracket on some functional spaces

Two fiber bundles E₁ and E₂ over the same base space M yield the fibered set ℱ(E₁,E₂) → M, whose fibers are defined as ${\displaystyle C^{\infty }(E₁ₓ,E₂ₓ)}$, for each x ∈ M. This fibered set can be regarded as a smooth space in the sense of Frölicher and we construct its tangent prolongation. Then we extend the Frölicher-Nijenhuis bracket to projectable tangent valued forms on ℱ(E₁,E₂). These forms turn out to be a kind of differential operators. In particular, we consider a general connection on ℱ(E₁,E₂) and study the associated covariant differential and curvature.