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Bundle Functors on Manifolds

In: Natural Operations in Differential Geometry

Author

Listed:
  • Ivan Kolář

    (Masaryk University, Department of Algebra and Geometry, Faculty of Science)

  • Jan Slovák

    (Masaryk University, Department of Algebra and Geometry, Faculty of Science)

  • Peter W. Michor

    (Universität Wien, Institut für Mathematik)

Abstract

The description of the product preserving bundle functors on Mf in terms of Weil algebras reflects their general properties in a rather complete way. In the present chapter we use some other procedures to deduce the basic geometric properties of arbitrary bundle functors on Mf. Hence the basic subject of this theory is a bundle functor on Mf that does not preserve products. Sometimes we also contrast certain properties of the product-preserving and non-productpreserving bundle functors on Mf. First we study the bundle functors with the so-called point property, i.e. the image of a one-point set is a one-point set. In particular, we deduce that their fibers are numerical spaces and that they preserve products if and only if the dimensions of their values behave well. Then we show that an arbitrary bundle functor on manifolds is, in a certain sense, a ‘bundle’ of functors with the point property. For an arbitrary vector bundle functor F on Mf with the point property we also derive a canonical Lie group structure on the prolongation FG of a Lie group G.

Suggested Citation

  • Ivan Kolář & Jan Slovák & Peter W. Michor, 1993. "Bundle Functors on Manifolds," Springer Books, in: Natural Operations in Differential Geometry, chapter 0, pages 329-349, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-02950-3_9
    DOI: 10.1007/978-3-662-02950-3_9
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