Natural transformations of separated jets. (English) Zbl 1049.58008

Doupovec, Miroslav; Kolář, Ivan

Arch. Math., Brno 36, No. 4, 297-303 (2000).

Let \(M,N,Q\) be manifolds. Given a map \(f:M\times N \to Q\), the concept of jets \(J^{r;s}(M,N,Q)\) of separated orders \(r\) and \(s\) was introduced by M. Kawaguchi [Proc. Japan Acad. 37, 18–22 (1961; Zbl 0171.42102)]. The authors prove that \(J^{r;s}\) is a functor on the product category \({\mathcal M}f_m \times {\mathcal M}f_n \times {\mathcal M}f\) and show that the exchange diffeomorphism \(\kappa _{M,N,Q}: J^{r;s}(M,N,Q) \to J^{s;r}(N,M,Q)\) introduced by I. Kolář [Math. Nachr. 69, 297–306 (1975; Zbl 0318.53034)] is a natural equivalence \(J^{r;s} \to J^{s;r}\). Finally it is proved that for \(r\geq 2\), \(s\geq 2\) the exchange diffeomorphism \(\kappa \) is the only natural equivalence. For \(r=1\) or \(s=1\) there are others natural equivalences which are completely classified and described geometrically.

Reviewer: Josef Janyška (Brno)

MSC:

58A20

Jets in global analysis

Keywords:

separated jet; canonical exchange; natural transformation

Citations:

Zbl 0171.42102; Zbl 0318.53034