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Natural transformations of semi-holonomic 3-jets. (English) Zbl 0852.58003

Let \({\mathcal M} f\) denote the category of all finite-dimensional smooth manifolds and smooth mappings between them, let \({\mathcal M} f_m\) denote its subcategory of \(m\)-dimensional manifolds and local diffeomorphisms between them. Let \(\overline J^3\) denote the functor \((M,N) \mapsto\) the manifold of semi-holonomic 3-jets of smooth mappings \(M \to N\), as defined on the product category \({\mathcal M} f_m \times {\mathcal M} f\).
The following two results are proved: (1) Any natural transformation \(\overline J^3 \to \overline J^3\) is either the identity, or the “contraction” (hence, is factorizable as \(\overline J^3 \to \text{Id} \to \overline J^3)\). These two transformations exist in any jet order. (2) Let \(\overline J^{3,2}\) denote the functor of semi-holonomic 3-jets which are holonomic in the second order. Then natural transformations \(\overline J^{3,2} \to \overline J^{3,2}\) form two 5-parametric families. Their geometric meaning is clarified. The first result contrasts with what is known for the 2nd and 4th order semi-holonomic jets for which nontrivial families of natural transformations exist.
Reviewer: M.Marvan (Opava)

MSC:

58A20 Jets in global analysis
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