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Natural operators in the view of Cartan geometries

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*Martin Panak*

**Address.** Mathematical Institute of Academy of Science, branch Brno,
Zizkova 22, Brno, Czech Republic
**E-mail:** naca@math.muni.cz

**Abstract.**

We prove, that $r$-th order gauge natural operators on the bundle of
Cartan connections with a target

in the gauge natural bundles of the order $(1,0)$ (``tensor bundles")
factorize through the curvature and

its invariant derivatives up to order $r-1$. On the course to this
result we also prove that the invariant

derivations (a generalization of the covariant derivation for Cartan
geometries) of the curvature function

of a Cartan connection have the tensor character. A modification of
the theorem is given for the reductive

and torsion free geometries.

**AMSclassification.** 58A20, 53A55.

**Keywords.** Cartan geometry, gauge natural bundle, natural operator,
natural sheaf, reductive Cartan geometry.