Graded derivations of the algebra of differential forms associated with a connection

Abstract

The central part of calculus on manifolds is usually the calculus of differential forms and the best known operators are exterior derivative, Lie derivatives, pullback and insertion operators. Differential forms are a graded commutative algebra and one may ask for the space of graded derivations of it. It was described by Frolicher and Nijenhuis in [1], who found that any such derivation is the sum of a Lie derivation Θ(K) and an insertion operator i(L) for tangent bundle valued differential forms K,L ∈ Ω(M ;TM). The Lie derivations give rise to the famous Frolicher-Nijenhuis bracket, an extension of the Lie bracket for vector fields to a graded Lie algebra structure on the space Ω(M ;TM) of vector valued differential forms. The space of graded derivations is a graded Lie algebra with the graded commutator as bracket, and this is the natural living ground for even the usual formulas of calculus of differential forms. In [8] derivations of even degree were integrated to local flows of automorphisms of the algebra of differential forms. In [6] we have investigated the space of all graded derivations of the graded Ω(M)module Ω(M ;E) of all vector bundle valued differential forms. We found that any such derivation, if a covariant derivative ∇ is fixed, may uniquely be written as Θ∇(K) + i(L) + μ(Ξ) and that this space of derivations is a very convenient setup for covariant derivatives, curvature etc. and that one can get the characteristic classes of the vector bundle in a very straightforward and simple manner. But the question arose of how all these nice formulas may be lifted to the linear frame bundle of the vector bundle. This paper gives an answer. In [7] we have shown that differential geometry of principal bundles carries over nicely to principal bundles with structure group the diffeomorphism group of a fixed manifold S, and that it may be expressed completely in terms of finite dimensional manifolds, namely as (generalized) connections on fiber bundles with standard fiber S, where the structure group is the whole diffeomorphism group. But some of the properties of connections remain true for still more general situations: in the main part of this paper a connection will be just a fiber projection onto a (not necessarily integrable) distribution or sub vector