Abstract
LetE 1, ...,E k andE be natural vector bundles defined over the categoryMf + m of smooth orientedm-dimensional manifolds and orientation preserving local diffeomorphisms, withm≥2. LetM be an object ofMf + m which is connected. We give a complete classification of all separately continuousk-linear operatorsD : Γc(E 1 M) × ... × Γc(E k M) → Γ(EM) defined on sections with compact supports, which commute whith Lie derivatives, i.e., which satisfy
for all vector fieldsX onM and sectionss j ε Γ c (E j M), 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.
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Communicated by P.W. Michor
First author supported by project P 77724 PHY of ‘Fonds zur Förderung der wissenschaftlichen Forschung’. Second author supported by grant No. 201/93/2125 of the GAČR.
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Čap, A., Slovák, J. On multilinear operators commuting with Lie derivatives. Ann Glob Anal Geom 13, 251–279 (1995). https://doi.org/10.1007/BF00773659
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DOI: https://doi.org/10.1007/BF00773659