Acta Mathematica Sinica, English Series

, Volume 26, Issue 1, pp 169–184 | Cite as

Reduction theorems for principal and classical connections



We prove general reduction theorems for gauge natural operators transforming principal connections and classical linear connections on the base manifold into sections of an arbitrary gauge natural bundle. Then we apply our results to the principal prolongation of connections. Finally we describe all such gauge natural operators for some special cases of a Lie group G.


Gauge natural operator reduction theorem principal prolongation 

MR(2000) Subject Classification

53C05 53C80 58A20 58A32 


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  1. [1]
    Utiyama, R.: Invariant theoretical interpretation of interaction. Phys. Rev., 101, 1597–1607 (1956)MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Eck, D. E.: Gauge-natural Bundles and Generalized Gauge Theories, American Mathematical Society, Provindence, RI, 1981Google Scholar
  3. [3]
    Kolář, I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry. Springer-Verlag, New York, 1993MATHGoogle Scholar
  4. [4]
    Janyška, J.: Higher order Utiyama-like theorem. Rep. Math. Phys., 58, 93–118 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Janyška, J.: Reduction theorems for general linear connections. Differential Geom. Appl., 20, 177–196 (2004)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Janyška, J.: Higher order valued reduction theorems for general linear connections. Note di Matematica, 23, 75–97 (2004)MATHMathSciNetGoogle Scholar
  7. [7]
    Kolář, I., Virsik, G.: Connections in first principal prolongations. Rend. Circ. Mat. Palermo, Series II, Suppl., 43, 163–171 (1996)Google Scholar
  8. [8]
    Fatibene, L., Francaviglia, M.: Natural and Gauge Natural Formalism for Classical Field Theories, Kluwer, 2003Google Scholar
  9. [9]
    Fatibene, L., Francaviglia, M., Palese M.: Conservation laws and variational sequences in gauge-natural theories. Math. Proc. Cambridge Philos. Soc., 130(3), 559–569 (2001)CrossRefMathSciNetGoogle Scholar
  10. [10]
    Kureš, M.: Weil modules and gauge bundles. Acta Mathematica Sinica, English Series, 22(1), 271–278 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Thomas, T. Y.: Differential Invariants of Generalized Spaces, Cambridge University Press, Cambridge, 1934MATHGoogle Scholar
  12. [12]
    Horndeski, G. W.: Replacement theorems for concomitants of gauge fields. Utilitas Math., 19, 215–246 (1981)MATHMathSciNetGoogle Scholar
  13. [13]
    Mikulski, W. M.: Higher order linear connections from first order ones. Arch. Math. (Brno), 43, 285–288 (2007)MATHMathSciNetGoogle Scholar
  14. [14]
    Doupovec, M., Mikulski, W. M.: Holonomic extension of connections and symmetrization of jets. Rep. Math. Phys., 60, 299–316 (2007)MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Kolář, I.: On some operations with connections. Math. Nachr., 69, 297–306 (1973)Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsBrno University of TechnologyBrnoCzech Republic
  2. 2.Institute of MathematicsJagiellonian UniversityKrakówPoland

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