Natural operations with projectable tangent valued forms on a fibred manifold

Abstract

Let p: E→B be a fibred manifold. Then, we consider the sheaf $$\mathfrak{B}$$ (E)=Ω(B)⊗P(E) of (local) projectable tangent valued forms on E, where Ω(B) is the sheaf of (local) differential forms on B andP(E) is the sheaf of (local) projectable vector fields on E. The Frolicher-Nijenhuis bracket makes $$\mathfrak{B}$$ (E) to be a sheaf of graded Lie algebras [18]. In this paper we study all natural R -bilinear operations on $$\mathfrak{B}$$ (E) which are of Frolicher-Nijenhuis type. By using the analytical method of [16], we prove that there is a three-parameter family of such operators on $$\mathfrak{B}$$ (E). As a consequence, we obtain a result on the unicity of the covariant differential of tangent valued forms and of the curvature associated with a given connection on E. All manifolds and mappings are assumed to be infinitely differentiable.