By a natural vector bundle (or a vector bundle functor) the authors mean a functorial assignment to each oriented \(m\)-dimensional manifold \(M\), \(m > 1\), of a vector bundle \(FM \to M\), and to every orientation preserving local diffeomorphism \(h : M_1 \to M_2\) of a fibrewise invertible vector bundle morphism \(Fh : FM_1 \to FM_2\) so that for every inclusion \(i : U \to M\) of an open submanifold \(U \subset M\) the morphism \(F_i : FU \to FM\) is the pullback along \(i\) of vector bundles. By a natural operator between vector bundle functors \(E\), \(F\) the authors mean a family \(D_M\), indexed by the manifolds \(M\), of local smooth operators \(\Gamma(EM) \to \Gamma(FM)\), commuting with the actions of smooth diffeomorphisms.
By general theory, the action of \(F\) on morphisms induces an action on vector fields via their flows, and, moreover, implies a well-defined Lie derivative \({\mathcal L}_X s\) of sections \(s \in \Gamma (FM)\) by vector fields \(X\) on the base manifold \(M\). It is known that any \(k\)-linear natural operator \(D: E_1 \times \dots \times E_k \to F\) commutes with Lie derivatives, this is to say, satisfies \({\mathcal L}_X (D_M (s_1, \dots, s_k)) = \sum_i D_M (s_1, \dots, {\mathcal L}_X s_i, \dots, s_k)\). According to a recent result by the same authors, each \(k\)-linear local operator defined on a fixed manifold \(M\) and commuting with Lie derivatives extends to a unique natural operator [Differ. Geom. Appl. 2, No. 1, 45-55 (1992; Zbl 0744.53010)]. In the reviewed paper the authors aim at generalizing the last result to nonlocal operators, and, in particular, at finding the general form of a nonlocal operator commuting with Lie derivatives.
The main result of the paper is a theorem, according to which, under a weak condition of separate continuity, every \(k\)-linear operator acting on sections with a compact support of a fixed connected oriented \(m\)- dimensional manifold \(M\) and commuting with Lie derivatives is identical to what the authors call an “almost natural operator”, i.e., an operator composed in a natural way of local natural operators \(D_M\), and of operators of taking integrals of the form \(\int_{M} \langle D_M(s_1, \dots, s_n),s\rangle\), where \(\langle \cdot, \cdot\rangle\) is the natural pairing \(E^* M \otimes \Lambda^m T^* M \times EM \to \Lambda^m T^* M\). The continuity assumption may be omitted, e.g., when the operator is bilinear and the target vector bundle has no absolutely invariant sections (sections invariant with respect to every vector field on \(M\)).
Finally the authors show how some strong results on a uniqueness of several brackets (Lie, Schouten, Schouten-Nijenhuis, and a “compression” of the Frölicher-Nijenhuis bracket) follow rather easily from their main theorem.