Natural operators in the view of Cartan geometries

Martin Panak

Address. Mathematical Institute of Academy of Science, branch Brno, Zizkova 22, Brno, Czech Republic


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.