Bundle functors of the jet type.

*(English)*Zbl 0951.58003
Kolář, Ivan (ed.) et al., Differential geometry and applications. Proceedings of the 7th international conference, DGA 98, and satellite conference of ICM in Berlin, Brno, Czech Republic, August 10-14, 1998. Brno: Masaryk University. 231-237 (1999).

Let \({\mathcal M}f\) be the category of all manifolds and all smooth maps, \({\mathcal M}f_m\) be the category of \(m\)-dimensional manifolds and their local diffeomorphims and \({\mathcal FM}_m\) be the category of fibered manifolds and their morphisms with the base map in \({\mathcal M}f_m\).

The aim of this article is to characterize all extensions of jet functors on \({\mathcal FM}_m \times {\mathcal M} F\).

The author starts by defining the general concept of jet functor, then, using as tools:

His previous work with W. M. Mikulski on product preserving bundle functors on \({\mathcal FM}_m\) [Differ. Geom. Appl. 11, No. 2, 105-115 (1999; Zbl 0935.58001)], from which he knows that every product preserving functor is a Weil functor; C. Ehresmann’s work of non-holonomic jets [C. R. Acad. Sci., Paris 239, 1762-1764 (1954; Zbl 0057.15603], and P. Libermann’s very readable survey on semi-holonomic jets [Arch. Math., Brno 33, No. 3, 173-189 (1996; Zbl 0915.58004)].

He proves that there are only: two extensions, horizontal and vertical, in the case of holonomic \(r\)-jets with \(r \geq 2\); he then gives a complete description of all extensions of \({\tilde J}^r(M,N) \supset J^r(M,N) \) the bundle of non-holonomic \(r\)-jets of \(M\) into \(N\). As a final application, he determines that in the case of a one-semi-holonomic jet functor, there are only two possible extensions.

See also I. Kolář, Czechosl. Math. J. 24(99), 311-330 (1974; Zbl 0313.53013).

For the entire collection see [Zbl 0923.00021].

The aim of this article is to characterize all extensions of jet functors on \({\mathcal FM}_m \times {\mathcal M} F\).

The author starts by defining the general concept of jet functor, then, using as tools:

His previous work with W. M. Mikulski on product preserving bundle functors on \({\mathcal FM}_m\) [Differ. Geom. Appl. 11, No. 2, 105-115 (1999; Zbl 0935.58001)], from which he knows that every product preserving functor is a Weil functor; C. Ehresmann’s work of non-holonomic jets [C. R. Acad. Sci., Paris 239, 1762-1764 (1954; Zbl 0057.15603], and P. Libermann’s very readable survey on semi-holonomic jets [Arch. Math., Brno 33, No. 3, 173-189 (1996; Zbl 0915.58004)].

He proves that there are only: two extensions, horizontal and vertical, in the case of holonomic \(r\)-jets with \(r \geq 2\); he then gives a complete description of all extensions of \({\tilde J}^r(M,N) \supset J^r(M,N) \) the bundle of non-holonomic \(r\)-jets of \(M\) into \(N\). As a final application, he determines that in the case of a one-semi-holonomic jet functor, there are only two possible extensions.

See also I. Kolář, Czechosl. Math. J. 24(99), 311-330 (1974; Zbl 0313.53013).

For the entire collection see [Zbl 0923.00021].

Reviewer: Lilia Del Riego (San Luis Potosí)

##### MSC:

58A20 | Jets in global analysis |

55R65 | Generalizations of fiber spaces and bundles in algebraic topology |

18F15 | Abstract manifolds and fiber bundles (category-theoretic aspects) |

17B65 | Infinite-dimensional Lie (super)algebras |

##### Keywords:

jet bundle functors; non-holonomic jet; Weil bundle; product preserving jet bundles; semi-holonomic jet functors
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\textit{I. Kolář}, in: Differential geometry and applications. Proceedings of the 7th international conference, DGA 98, and satellite conference of ICM in Berlin, Brno, Czech Republic, August 10--14, 1998. Brno: Masaryk University. 231--237 (1999; Zbl 0951.58003)

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