Annals of Global Analysis and Geometry

, Volume 6, Issue 2, pp 109–117 | Cite as

On the natural operators on vector fields

  • Ivan Kolář

We determine all natural operators transforming every vector field on a manifold M into a vector field on FM, where F is any natural bundle corresponding to a product preserving functor.

This research was finished during the author's stay at the University of Vienna. The author acknowledges its kind hospitality and is grateful to Peter Michor, Jan Slovák and Jiří Vanžura for several useful comments.

All manifolds and maps are assumed to be infinitely differentiable.


Vector Field Group Theory Natural Operator Preserve Functor Kind Hospitality 
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  1. [1]
    Doupovec, M.: Natural operators transforming vector fields to the r-th order tangent bundle. Časopis Pěst. Mat. (to appeear).Google Scholar
  2. [2]
    Eck, D. J.: Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra 42 (1986), 133–140.Google Scholar
  3. [3]
    Ehresmann, C.: Introduction á la théorie des structures infinitésimales et des pseudo-groupes de Lie. Colloque du C.N.R.S., Strasbourg 1953, 97–110.Google Scholar
  4. [4]
    Gancarzewicz, J.: Liftings of functions and vector fields to natural bundles. Dissertationes Mathematicae, CCXII, Warszawa 1983.Google Scholar
  5. [5]
    Kainz, G. and Michor, P.: Natural transformation in differential geometry. Czechoslovak Math.J. 37 (112) (1987), 584–607.Google Scholar
  6. [6]
    Kolář, I.: Covariant approach to natural transformations of Weil functors. Comment. Math. Univ. Carolin. 7 (1986), 723–729.Google Scholar
  7. [7]
    Kolář, I.: Some natural operations with connections. J. Nat. Acad. Math. India (to appear).Google Scholar
  8. [8]
    Krupka, D., and Janyška, J.: Lectures on differential invariants. To appear.Google Scholar
  9. [9]
    De León, M. and Rodrigues, P. R.: Almost tangent geometry and higher order mechanical systems. Differential Geometry and Its Applications, Proceedings, D. Reidel Publishing Company, 1987, 179–195.Google Scholar
  10. [10]
    Luciano, O. O.: Categories of multiplicative functors and Morimoto‘s conjecture. Preprint N° 46, Institut Fourier, Laboratoire de Mathématiques, Grenoble, 1986.Google Scholar
  11. [11]
    Morimoto, A.: Prolongation of connéctions to bundles of infinitely near points. J. Differential Geometry 11 (1976), 479–498.Google Scholar
  12. [12]
    Nijenhuis, A.: Natural bundles and their general properties. Differential Geometry in honor of K. Yano, Kinokuniya, Tokyo 1972, 317–334.Google Scholar
  13. [13]
    Okassa, E.: Prolongements des champs de vecteurs á des variétés de points proches. C.R.A.S. Paris 300, (1985), série I, 173–176.Google Scholar
  14. [14]
    Palais, R. S. and Terng, C. J.: Natural bundles have finite order. Topology 16 (1977), 271–277.Google Scholar
  15. [15]
    Pohl, F. W.: Differential geometry of higher order. Topology 1 (1962), 169–211.Google Scholar
  16. [16]
    Sekizawa, M.: Natural transformations of vector fields on manifolds to vector fields on tangent bundles. To appear.Google Scholar
  17. [17]
    Weil, A.: Théorie des points proches sur les variétés différentiables. Colloque du C.N.R.S., Strasbourg 1953, 111–117.Google Scholar
  18. [18]
    Zajtz, A.: A method of estimation of the order of natural differential operators and bitings. To appear.Google Scholar

Copyright information

© VEB Deutscher Verlag der Wissenschaften 1988

Authors and Affiliations

  • Ivan Kolář
    • 1
  1. 1.Matematický ústav ČSAV pobočka BrnoBrnoCzechoslovakia

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