Natural operations with projectable tangent valued forms on a fibred manifold

Summary Let p: EB be a fibred manifold. Then, we consider the sheaf (E)=(B)P(E) of (local) projectable tangent valued forms on E, where (B) is the sheaf of (local) differential forms on B andP(E) is the sheaf of (local) projectable vector fields on E. The Frölicher-Nijenhuis bracket makes (E) to be a sheaf of graded Lie algebras 18]. In this paper we study all natural R -bilinear operations on (E) which are of Frölicher-Nijenhuis type. By using the analytical method of 16], we prove that there is a three-parameter family of such operators on (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.This paper has been written during the author's visit at the Institute of Applied Mathematics «Giovanni Sansone». Florence, Italy. The author would like to thank ProfessorMarco Modugno for his kind hospitality and for stimulating discussions.