On multilinear operators commuting with Lie derivatives

Abstract

LetE1, ...,Ek andE be natural vector bundles defined over the categoryMfm+ of smooth orientedm-dimensional manifolds and orientation preserving local diffeomorphisms, withm≥2. LetM be an object ofMfm+ which is connected. We give a complete classification of all separately continuousk-linear operatorsD : Γc(E1M) × ... × Γc(EkM) → Γ(EM) defined on sections with compact supports, which commute whith Lie derivatives, i.e., which satisfy $$\mathcal{L}_X (D(s_1 , \ldots ,s_k )) = \sum\limits_{i = 1}^k D (s_1 , \ldots ,\mathcal{L}_X s_i , \ldots ,s_k ),$$ for all vector fieldsX onM and sectionssj e Γc(EjM), in terms of local natural operators and absolutely invariant sections. In special cases we do not need the continuity assumption. We also present several applications in concrete geometrical situations, in particular we give a completely algebraic characterization of some well-known Lie brackets.