On multilinear operators commuting with Lie derivatives

TL;DR

This paper classifies all multilinear operators on sections of natural vector bundles over manifolds that commute with Lie derivatives, revealing their structure in terms of local natural operators and invariant sections, with applications to Lie brackets.

Contribution

It provides a complete classification of continuous multilinear operators commuting with Lie derivatives on natural vector bundles over manifolds, including explicit descriptions and applications.

Findings

Classification of all such multilinear operators.

Representation of operators via local natural operators and invariant sections.

Applications to algebraic characterizations of Lie brackets.

Abstract

Let $E_1,\dots ,E_k$ and $E$ be natural vector bundles defined over the category $\Cal Mf_m^+$ of smooth oriented $m$--dimensional manifolds and orientation preserving local diffeomorphisms, with $m\geq 2$. Let $M$ be an object of $\Cal Mf_m^+$ which is connected. We give a complete classification of all separately continuous $k$--linear operators $D\:\Ga _c(E_1M)\x\dots\x\Ga_c(E_kM)\to \Ga (EM)$ defined on sections with compact supports, which commute with Lie derivatives, i\.e\. which satisfy $$ \Cal L_X(D(s_1,\dots ,s_k))=\sum _{i=1}^kD(s_1,\dots ,\Cal L_Xs_i,\dots,s_k), $$ for all vector fields $X$ on $M$ and sections $s_j\in\Ga_c(E_jM)$, 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.