Annals of Global Analysis and Geometry

, Volume 13, Issue 3, pp 251–279 | Cite as

On multilinear operators commuting with Lie derivatives

  • Andreas Čap
  • Jan Slovák
Article
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Abstract

LetE1, ...,Ek 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(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 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.

Key words

Natural operators natural bundles Lie differentiation 

MSC 1991

53 A 55 58 A 20 
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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Andreas Čap
    • 1
    • 3
  • Jan Slovák
    • 2
  1. 1.Institut für MathematikUniversität WienWienAustria
  2. 2.Department of Algebra and GeometryMasaryk UniversityBrnoCzech Republic
  3. 3.Erwin Schrödinger International Institute for Mathematical PhysicsWienAustria

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