Advertisement

Torsion-free connections on higher order frame bundles

  • Ivan Kolář
Part of the Mathematics and Its Applications book series (MAIA, volume 350)

Abstract

We deduce that torsion-free connections on the r-th order frame bundle P r M of a manifold M can be identified with certain reductions of P r+1 M. They are also interpreted as splittings of T*M into the bundle of all (1,r+l)-covelocities on M. Finally we determine all natural operators transforming torsion-free connections on P 1 M into torsion-free connections on P 2 M.

Keywords

Principal Bundle Natural Operator Frame Bundle Exponential Operator Canonical Bijection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aguirre-Dabán E., Sánchez-Rodriquez I.: On structure equations for second order connections, Differential Geometry and Its Applications, Proceedings Opava (1993), pp. 257–264.Google Scholar
  2. 2.
    Ferraris M., Francaviglia M., Tubiello F.: Connections over the bundle of second-order frames, Proceedings of Conference on Differential Geometry and Applications, Brno 1989, World-Scientific, Singapure, 1990, pp. 33-46.Google Scholar
  3. 3.
    Janyška J.: On natural operations with linear connections, Czechoslovak Math. J. 35, (1985), pp. 106–115.MathSciNetGoogle Scholar
  4. 4.
    Janyška J., Kolář I.: On the connections naturally induced on the second order frame bundle, Arch. Math. (Brno) 22 (1986), pp. 21–28.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kobayashi S.: Canonical forms on frame bundles of higher order contact, Proc. of symposia in pure math., vol III, A.M.S., 1961, pp. 186–193Google Scholar
  6. 6.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry, Vol.I., J. Wiley-Interscience, 1963.Google Scholar
  7. 7.
    Kolář I.: A geometrical version of the higher order Hamilton formalism in fibered manifolds, J. Geometry and Physics 1 (1984), pp. 127–137.zbMATHCrossRefGoogle Scholar
  8. 8.
    Kolář I.: Some natural operators in differential geometry, Proc. Conf. Diff. Geom. and Its Applications, Brno 1986, Dodrecht, 1987, pp. 91-110.Google Scholar
  9. 9.
    Kolář I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry, Springer-Verlag, 1993.Google Scholar
  10. 10.
    Yuen P.C.: Higher order frames and linear connections, Cahiers Topol. Geom. Diff. 12 (1971), pp. 333–371.MathSciNetzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Ivan Kolář
    • 1
  1. 1.Department of Algebra and Geometry Faculty of ScienceMasaryk UniversityBrnoCzech Republic

Personalised recommendations