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Functorial prolongations of some functional bundles. (English) Zbl 1133.58301

Let \(E_1\) and \(E_2\) be two fiber bundles over the same base \(M\) and denote by \(\mathcal F(E_1,E_2)\to M\) the functional bundle of all smooth maps from a fiber of \(E_1\) into the fiber of \(E_2\) over the same base point.
Given a Weil functor \(T^A\) determined by a Weil algebra \(A\), the authors first discuss the functorial prolongation \(T^A\mathcal F(E_1,E_2)\to T^AM\), which generalizes the tangent bundle \(T\mathcal F(E_1,E_2)\to TM\). Then they construct another functorial prolongation \(G\mathcal F(E_1,E_2)\to M\), where \(G\) is a product preserving bundle functor on the category \(\mathcal F\mathcal M_m\) of all fibered manifolds with \(m\)-dimensional bases and all fibered manifold morphisms covering local diffeomorphisms. The main geometric problem of the paper is the prolongation of vector fields on \(\mathcal F(E_1,E_2)\) into vector fields on \(T^A\mathcal F(E_1,E_2)\) and on \(G\mathcal F(E_1,E_2)\), respectively. It is proved that in both cases the bracket is preserved.

MSC:

58A20 Jets in global analysis
58A32 Natural bundles
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Full Text: arXiv