Advertisement

Systems of vector valued forms on a fibred manifold and applications to gauge theories

  • Marco Modugno
IV. Differential Geometric Iechniques
Part of the Lecture Notes in Mathematics book series (LNM, volume 1251)

Abstract

The new concept of "system" over double fibred manifolds is introduced and systems of vector valued forms and connections are investigated.

A graded universal differential calculus for involutive systems induced by the Frölicher-Nijenhuis bracket is shown. The system of overconnections, which projects on a given system of connections of a fibred manifold and on the system of linear connections of the base space, is also presented.

A direct formulation of gauge theories and a re-formulation of the lagrangian approach are obtained by means of the graded universal calculus.

In the particular case of principal bundles, the standard differential techniques are recovered and new results are shown as well. The present approach, which is based on differential and functorial methods, can provide new hints for field theory.

Each notion and result is expressed both in an intrinsic way and by explicit formulas in local coordinates.

Keywords

Gauge Theory Vector Field Vector Bundle Principal Bundle Differential Calculus 
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]
    D. Bleecker: Gauge theories and variational principles, Addison-Wesley, MA, 1981.zbMATHGoogle Scholar
  2. [2]
    D.Canarutto:Bundle splittings, connections and locally principal fibred manifolds, Bollett. U.M.I., (1986), to appear.Google Scholar
  3. [3]
    D. Canarutto, C.T.J. Dodson: On the bundle of principal connections and the stability of b-incompleteness of manifolds, Math. Proc. Cambridge, Phil. Soc., 98, 1985, p.51–59.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    D.Canarutto, M.Modugno: On the graded Lie algebras of vector valued forms, Sem. 1st. Mat. Applic. "G. Sansone", Firenze, 1985, p.1–26.Google Scholar
  5. [5]
    L. Corwin, Y. Ne'eman, S. Sternberg: Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Mod. Phys., 47,3, (1975), p.573–603.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Crampin:Generalized Bianchi identities for horizontal distributions, Math. Proc. Cambridge Phil. Soc. (1983), 94, p.125–132.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M.Crampin, L.A.Ibort:Graded Lie algebras of derivations and Ehresmann connections, preprint.Google Scholar
  8. [8]
    M.Dubois Violette: The theory of overdetermined linear systems and its application to non-linear field equations, J. Geom. Phys., 1, 2, 1984, p.139–172.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    C.Ehresmann: Les connexions infinitésimales dans un espace fibré différentiable, Coll. Topologie (Bruxelles, 1950), Liège 1951, p.29–55.Google Scholar
  10. [10]
    M.Ferraris, M.Francaviglia: The theory of formal connections and fibred connections in fibred manifolds, in Differential geometry, L.A.Cordero editor, Pittman, 1985, p.297–317.Google Scholar
  11. [11]
    A. Frölicher, A. Nijenhuis: Theory of vector valued differential forms.Part I:Derivations in the graded ring of differential forms., Indag.Math., 18,(1956), p.338–385.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    P.L. Garcia: Connections and 1-jet fibre bundle, Rendic. Sem. Mat. Univ. Padova, 47, 1972, p.227–242.Google Scholar
  13. [13]
    P.L. Garcia: Gauge algebras, curvature and symplectic structure, J. Diff. Geom., 12,(1977), p.209–227.MathSciNetzbMATHGoogle Scholar
  14. [14]
    P.L. Garcia, A. Peréz-Rendón: Reducibility of the symplectic structure of minimal interactions, Lecture Notes in Mathematica, N.676, Springer-Verlag, Berlin, 1978.zbMATHGoogle Scholar
  15. [15]
    H. Goldschmidt: Integrability criteria for systems of non linear partial differential equations, J. Diff. Geom., 1,(1967), p.269–307.MathSciNetzbMATHGoogle Scholar
  16. [16]
    J. Grifone: Structure presque tangente et connexions, I, Ann. Inst. Fourier., 22, 1, (1972), p.287–334.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    R. Hermann: Gauge fields and Cartan-Ehresmann connections, Part A, Math. Sci. Press, Brookline, 1975.zbMATHGoogle Scholar
  18. [18]
    D. Kastler, R. Stora: Lie-Cartan pairs, J. Geom. Phys., 2, 3, 1985, p.1–31.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. Kobayashi, K. Nomizu: Foundations of differential geometry, Intersc. Publish., New York, 1963.zbMATHGoogle Scholar
  20. [20]
    I. Kolář: Higher order torsion of spaces with Cartan connection, Cahiers de Topologie et Géometrie Differentielle, 12,2 (1981), p.29–34.Google Scholar
  21. [21]
    I. Kolář:On generalized connections, Beiträge zur Algebra und Geometrie, II, (1981), p.29–34.MathSciNetzbMATHGoogle Scholar
  22. [22]
    J.L.Koszul: Lecture on fibre bundles and differential geometry, Tata Inst., 1960.Google Scholar
  23. [23]
    A. Kumpera, D. Spencer: Lie equations, vol.1: General theory, Ann. of Math. Studies, 73, Princeton University Press, Princeton, 1972.Google Scholar
  24. [24]
    P.Libermann: Sur les prolongements des fibrés principaux et grupoides différentiables, Sem. Anal. Glob., Montréal, 1969, p.7–108.Google Scholar
  25. [25]
    P. Libermann: Parallélismes, J. Diff. Geom., 8, 1973, p.511–539.MathSciNetzbMATHGoogle Scholar
  26. [26]
    P. Libermann: Remarques sur les systèmes diffèrentiels, Cahiers de Top. et Géom. Diff., 23,1, (1982), p.55–72.MathSciNetzbMATHGoogle Scholar
  27. [27]
    A. Lichnerowicz: Théorie globale des connexions et de groupes d'holonomie, Ediz. Cremonese, Roma, 1962.zbMATHGoogle Scholar
  28. [28]
    L. Mangiarotti, M. Modugno: New operators on jet spaces, Ann. Fac. Scie. Toulouse, 5,(1983), p.171–198.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    L. Mangiarotti, M. Modugno: Fibered spaces, jet spaces and connections for field theories, in Proceed. of Internat. Meet. "Geometry and Physics", Florence, 1982, Pitagora Editrice, Bologna, 1983, p.135–165.zbMATHGoogle Scholar
  30. [30]
    L. Mangiarotti, M. Modugno: Some results on the calculus of variations on jet spaces, Ann.Inst.H.Poinc.39,1,(1983), p.29–43.MathSciNetzbMATHGoogle Scholar
  31. [31]
    L. Mangiarotti, M. Modugno: Graded Lie algebras and connections on a fibred spaces, J.Math.Pur. et appl.63,(1984),p.111–120.MathSciNetzbMATHGoogle Scholar
  32. [32]
    L. Mangiarotti, M. Modugno: On the geometric structure of gauge theories, J.Math.Phys., 26,6,(1985), p.1373, 1379.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    P.W. Michor: A generalization of Hamiltonian mechanics, J. Geom. Phys., 2, 2, 1985, p. 67–82.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    P.W.Michor:Differential geometry and graded Lie algebras of derivations, preprint.Google Scholar
  35. [35]
    M.Modugno, R.Ragionieri, Fibred manifolds: a new context for field theories, Sem. 1st. Mat. Appl. "G.Sansone", Firenze, 1985, p.1–55.Google Scholar
  36. [36]
    M.Modugno:On structuring categories and systems, Semin. Istit. Mat. Applic. "G.Sansone", 1986.Google Scholar
  37. [37]
    H.K. Nickerson: On differential operators and connections, Trans. Amer. Math. Soc., 99, (1961), p.509–539.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    A. Nijenhuis: Jacoby-type identities for bilinear differential concomitants of certain tensor fields. I, Indag. Math., 17, 3, (1955), p.390–403.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    A. Peréz-Rendón: Lagrangiennes dans les théories jauge par rapport au groupe de Poincaré,Rend.Sem.Mat.Univ.Polit.Torino,40,3,(1982),p. 21–34.zbMATHGoogle Scholar
  40. [40]
    Peréz-Rendón: Principles of minimal interaction, in Proceed. of Internat. Meet. "Geometry and Physics", Florence, 1982, Pitagora Editrice, Bologna, 1983, p.185–216.zbMATHGoogle Scholar
  41. [41]
    J.F. Pommaret:Systems of partial differential equations and Lie pseudogroups, Gordon and Brach, New York, 1978.zbMATHGoogle Scholar
  42. [42]
    Y. Ne'eman, T. Regge: Gauge theory of gravity and supergravity on a group manifold, La Rivista del Nuovo Cimento, 5, 1978.Google Scholar
  43. [43]
    Tong Van Duc:Sur la géometrie differentielle des fibrés vectoriels, Kōdai Math., Sem. Rep.,26,4,1975,p.349–408.CrossRefzbMATHGoogle Scholar
  44. [44]
    A.Trautman: Fibre bundles, gauge fields and gravitation, in General relativity and gravitation, ed.by A.Held, I, Plenum Press, 1980, p.287–308.Google Scholar
  45. [45]
    M.W. Tulczyjew: The Euler-Lagrange resolution, Lecture Notes in Mathematics, N.836, Springer-Verlag, Berlin, 1980.zbMATHGoogle Scholar
  46. [46]
    T. Utiyama: Invariant theoretical interpretation of interaction, Phys. Rev., 101, 1956, p.1597–1607.MathSciNetCrossRefzbMATHGoogle Scholar

Further details can be found in the extended manuscript

  1. [47]
    M.Modugno: An introduction to systems of connections, Seminario Istituto di Matematica Applicata "G.Sansone", 1986, p.1–63.Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Marco Modugno
    • 1
  1. 1.Istituto di Matematica Applicata "G. Sansone"Firenze

Personalised recommendations