On multilinear operators commuting with Lie derivatives

Let E 1, . . . , Ek and E be natural vector bundles defined over the category Mf m of smooth oriented m–dimensional manifolds and orientation preserving local diffeomorphisms, with m ≥ 2. Let M be an object of Mf m which is connected. We give a complete classification of all separately continuous k–linear operators D : Γc(E1M) × . . . × Γc(EkM) → Γ(EM) defined on sections with compact supports, which commute with Lie derivatives, i.e. which satisfy LX(D(s1, . . . , sk)) = k ∑ i=1 D(s1, . . . ,LXsi, . . . , sk), for all vector fields X on M and sections sj ∈ 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. 1. Natural operators In this section we give a brief survey over notions and results from the theory of natural bundles and operators which we will need in the sequel and we formulate our main result. Besides the references cited for the results they can all be found in [Kolář–Michor–Slovák]. 1.1. Definition. A bundle functor (or natural bundle) on the categoryMf+ m of mdimensional oriented manifolds and orientation preserving smooth locally invertible mappings (so called local diffeomorphisms) is a functor F : Mf+ m → FM with values in fibered manifolds which satisfies: (i) B ◦ F = idMf+ m where B : FM→Mf is the base functor (ii) for every inclusion i : U →M of an open submanifold, FU is the restriction p−1 M (U) of the value FM = (pM : FM → M) to U and Fi is the inclusion p−1 M (U)→ FM . A vector bundle functor (or natural vector bundle) is a bundle functor with values in the category of finite dimensional vector bundles and fiberwise invertible vector bundle homomorphisms. 1991 Mathematics Subject Classification. 53A55, 58A20.