Skip to main content
SpringerLink
Log in

Fiber spaces, connections and Yang-Mills fields

  • Conference paper
  • First Online:
Geometric Techniques in Gauge Theories

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 926))

  • 504 Accesses

Abstract

From the point of view of a differential geometer, Yang-Mills Fields are connections on principal fiber bundles whose curvature satisfies certain first-order differential equations. These lecture notes assume a knowledge of the formalism of calculus on manifolds, i.e., the theory of differential forms and vector fields, and are based on the theory of connections in fiber spaces, developed primarily by E. Cartan and C. Ehresmann in the period 1920–1955. To make the material more readily accessible to someone familiar with classical physics, the emphasis will be on Maxwell electromagnetic theory, considered as a Yang-Mills with an abelian structure group. Some of the material is from Interdisciplinary Mathematics, some is new.

The Association for Physical and System Mathematics Research supported by Ames Research Center (NASA), NSG-2402; U.S. Army Research Office, #LILG1102RHN7-05 MATH; and The National Science Foundation MCS 8003227.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. A. Barut, Electrodynamics, Macmillan, New York, 1964.

    Google Scholar 

  2. A. Belavin, A. Polyakov, A. Schwartz, and Y. Tyupkin, Phys. Lett. B 59, 85 (1975).

    Article  MathSciNet  Google Scholar 

  3. M. Born and H. Wolf, Principles of Optics, Bergaman Press, 1975.

    Google Scholar 

  4. R. Bott, “Topological Obstruction to Integrability,” Proc. Symp. Pure Math, Vol. 16, 1975, Amer. Math. Soc., Providence, R.I.

    MATH  Google Scholar 

  5. E. Cartan, “Sur certaines expressions differentielles et le problème de Pfaff,” Ann. Ec. Norm., 16, 239–332 (1899).

    MathSciNet  MATH  Google Scholar 

  6. E. Cartan, Les éspaces a connexion projective, Conforme, Ann. Soc. Pol. Math., 2, 1923, 171–221.

    MATH  Google Scholar 

  7. E. Cartan, Les éspaces a connexion projective, Bull. Soc. Math., France, 52, 205–241, 1929.

    MathSciNet  MATH  Google Scholar 

  8. E. Cartan, “Sur les varietés a connexion affine et la theorie de la Relativité Generalisée,” Ann. Ec. Norm., 4D, 32F 412, 1940.

    MATH  Google Scholar 

  9. E. Cartan, “Sue l’equivalence absolue de certains systemes d’equations differentielles,” Oevres Complètes, Gauthier-Villars, Paris, 1953.

    Google Scholar 

  10. E. Cartan, “Sur l’integration de certains systèmes de pfaff de caractere deux,” Oevres Complètes, Gauthier-Villars, Paris, 1953.

    Google Scholar 

  11. E. Cartan, Les Sustèmes Differentiels Exterieurs et leurs Applications Geometriques, Herman, Paris, 1945.

    Google Scholar 

  12. E. Cartan, “Les systèmes de Pfaff a cinq variables,” Oevres Complètes, Pt. II, Gauthier-Villars, Paris, 1953.

    Google Scholar 

  13. E. Corrigan and D.B. Fairlie, Phys. Lett. B 67, 69 (1977).

    Article  MathSciNet  Google Scholar 

  14. Y. Choquet-Bruhat, Geometrie Differentielle et Systèmes Exterieurs, Dunod, Paris, 1963.

    MATH  Google Scholar 

  15. J. Dieudonne, Treatise on Analysis, Vol. 4, Academic Press, New York, 1974.

    MATH  Google Scholar 

  16. C. Ehresmann, “Sur la topologie des certains éspaces homogenes,” Annals of Math 35, 395–443 (1934).

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Ehresmann “Les connexions infinitesimales dans un éspace fibré,” Colloque de Topologie, Bruxelles, 1950.

    Google Scholar 

  18. F. Estabrook and H. Wahlquist, “The geometric approach to sets of ordinary differential equations and Hamiltonian mechanics,” SIAM Rev. 17, 201–220 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Estabrook and H. Wahlquist, “Prolongation structures of nonlinear evolution equations,” I. J. Math. Phys. 16, 1–7 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Estabrook and H. Wahlquist, “Prolongation structures of nonlinear evolution equations,” II; J. Math. Phys. 17, 1293–1297 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Estabrook and H. Wahlquist and R. Hermann “Differential-geometric prolongations and Backlund transformations,” in R. Hermann (ed.) The Ames Research Center (NASA) 1976 Conference on the Geometric Theory of Non-Linear Waves, Math Sci Press, Brookline, MA 1977.

    Google Scholar 

  22. F. Estabrook and H. Wahlquist, “Prolongation structures, connection theory and Backlund transformation,” in F. Calogero (ed.), Nonlinear Evolution Equations Solvable by the Spectral Transform, Research Notes in Mathematics, No. 26, Pittman, London, 1978.

    Google Scholar 

  23. F. Estabrook, “Some old and new techniques for the practical use of differential forms,” in R. Miura (ed.) Backlund Transformation, the Inverse Scattering Method, Solitons and their Application, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin, New York, 1976.

    Google Scholar 

  24. F. Estabrook and B.K. Harrison, “Geometric approach to inverse groups and solution of partial differential systems,” J. Math. Phys. 12, 653–666 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  25. R.B. Gardner, “Invariants of Pfaffian systems,” Trans. Am. Math. Soc. 126, 514–543 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  26. E. Goursat, Lecons sur le Problème de Pfaff, Herman, Paris, 1922.

    MATH  Google Scholar 

  27. V. Guillemin and S. Sternberg, Geometric Asymptotics, American Math. Soc., 1977.

    Google Scholar 

  28. R. Hermann, “Sur les isométries infinitésimaux et le groupe d’holonomie d’un espace de Riemann,” C.R. Acad. Sci., Paris, 239, 1178–1180 (1954).

    MathSciNet  MATH  Google Scholar 

  29. R. Hermann, “Sur les automorphismes infinitesimaux d’une G. Structure,” C.R. Acad. Sci., Paris 239, 1760–1761 (1954).

    MathSciNet  MATH  Google Scholar 

  30. R. Hermann, “A sufficient condition that a map of Riemannian manifolds be a fiber bundle,” Proc. Am. Math. Soc. 11, 236 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Hermann, “The differential geometry of foliations,” Ann. Math. 72, 445–457, (1960).

    Article  MathSciNet  MATH  Google Scholar 

  32. R. Hermann, Lie Groups for Physicists, W.A. Benjamin, Reading, MA, 1966.

    MATH  Google Scholar 

  33. R. Hermann, “Equivalence of submanifolds of homogeneous spaces,” Math. Ann. 158, 284–289 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Hermann, “Cartan connections and the equivalence problems for geometric structures,” Contributions to Differential Equations 3, 199–248 (1964).

    MathSciNet  Google Scholar 

  35. R. Hermann, “The differential geometry of foliations, II,” J. Math. Mech. 11, 303–316 (1962).

    MathSciNet  MATH  Google Scholar 

  36. R. Hermann, “Existence in the large of parallelism homomorphisms,” Trans. Am. Math. Soc. 161, 170–183 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Hermann “Formal linearization of Lie algebras of vector fields near an invariant submanifold,” J. Differential Geom., 1973.

    Google Scholar 

  38. R. Hermann, “A geometric formula for current algebra commutation relations,” Phys. Rev. 177, 2449 (1969).

    Article  MATH  Google Scholar 

  39. R. Hermann, “Quantum field theories with degenerate Lagrangians,” Phys. Rev. 177, 2453 (1969).

    Article  MATH  Google Scholar 

  40. R. Hermann “Formal linearization of Lie algebras of vector fields near an invariant submanifold,” Trans. Am. Math. Soc., 1973.

    Google Scholar 

  41. R. Hermann, “Spectrum-generating algebras in classical mechanics, I and II,” J. Math. Phys. 13, 833, 878 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  42. R. Hermann, “Currents in classical field theories,” J. Math. Phys. 13, 97 (1972).

    Article  MathSciNet  Google Scholar 

  43. R. Hermann, “Current algebra, Sugawara model and differential geometry,” J. Math. Phys. 11, 1825–1829 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Hermann, Fourier Analysis on Groups and Partial Wave Analysis, W.A. Benjamin, New York, 1969.

    MATH  Google Scholar 

  45. R. Hermann, “Infinite dimensional Lie algebra and current algebra,” Proc. 1969 Battele-Seattle Rencontres on Math. Phys., Springer-Verlag, Berlin, 1970.

    Book  MATH  Google Scholar 

  46. R. Hermann, Lie Algebras and Quantum Mechanics, W.A. Benjamin, New York, 1970.

    MATH  Google Scholar 

  47. R. Hermann, Vector Bundles in Mathematical Physics, I and II, W.A. Benjamin, New York, 1970.

    MATH  Google Scholar 

  48. R. Hermann, Geometry, Physics and Systems, M. Dekker, New York, 1973.

    MATH  Google Scholar 

  49. R. Hermann, Gauge Fields and Cartan-Ehresmann Connections, Part A, Interdisciplinary Mathematics, Vol. 10, Math. Sci. Press, Brookline, MA, 1975.

    MATH  Google Scholar 

  50. R. Hermann, “Modern differential geometry in elementary particle physics,” VII GIFT Conf. on Theoretical Physics, Salamanca, Spain, 1977, A. Azcarraga (ed.), Springer-Verlag, Berlin.

    Google Scholar 

  51. R. Hermann, Quantum and Fermion Differential Geometry, Part A, Math Sci Press, Brookline, MA, 1977.

    MATH  Google Scholar 

  52. R. Hermann, Geometry of Non-Linear Differential Equations, Backlünd Transformations and Solitons, Part A. Interdisciplinary Mathematics, Vol. 12, Math. Sci. Press, Brookline, MA, 1976.

    MATH  Google Scholar 

  53. R. Hermann (ed.) Sophus Lie’s 1880 Transformation Group Paper, comments and additional material by R. Hermann, Math Sci Press, Brookline, MA, 1975.

    Google Scholar 

  54. R. Hermann (ed.), Ricci and Levi-Civita’s Tensor Analysis Paper, translation, comments, and additional material by R. Hermann, Math Sci Press, 1975.

    Google Scholar 

  55. R. Hermann (ed.), Sophus Lie’s 1884 Differential Invariants Paper, comments and additional material by R. Hermann, Math Sci Press, 1975.

    Google Scholar 

  56. R. Hermann (ed.), Proc. Ames (NASA) 1976 Conf. on the Geometry of Solitons, Math Sci Press, Brookline, MA, 1977.

    Google Scholar 

  57. R. Hermann, “The Lie-Cartan geometric theory of differential equations and scattering theory,” Proc. 1977 Park City, Utah, Conf. on Differential Equations, C. Byrnes (ed.).

    Google Scholar 

  58. R. Hermann, “Backlund transformations, and Lie theory as algorithms for solving and understanding nonlinear differential equations,” in Solitons in Action, K. Lonngren and A. Scott (eds.), Academic Press, New York, 1978.

    Google Scholar 

  59. R. Hermann, Toda Lattices, Cosymplectic Manifolds, Backlünd Transformations and Kinds, Parts A and B, Math Sci Press, Brookline, MA, 1977.

    MATH  Google Scholar 

  60. R. Hermann, Quantum and Fermion Differential Geometry, Part A, Math Sci Press, Brookline, MA, 1977.

    MATH  Google Scholar 

  61. R. Hermann, Fourier Analysis on Groups and Partial Wave Analysis, W.A. Benjamin, Reading, MA, 1969.

    MATH  Google Scholar 

  62. R. Hermann, Energy-Momentum Tensors, Interdisciplinary Mathematics, Vol. 4, Math Sci Press, Brookline, MA, 1973.

    MATH  Google Scholar 

  63. R. Hermann, Topics in General Relativity, Interdisciplinary Mathematics, Vol. 5, Math Sci Press, Brookline, MA, 1973.

    MATH  Google Scholar 

  64. R. Hermann, Topics in the Mathematics of Quantum Mechanics, Interdisciplinary Mathematics, Vol. 6, Math Sci Press, Brookline, MA 1973.

    MATH  Google Scholar 

  65. R. Hermann, Lectures on Mathematical Physics, II, W.A. Benjamin, Reading, MA 1972.

    Google Scholar 

  66. R. Hermann, Yang-Mills, Naluza-Klein, and the Einstein Program, Interdisciplinary Mathematics, Vol. 19, Math Sci Press, Brookline, MA, 1978.

    Google Scholar 

  67. R. Hermann, Differential Geometry and the Calculus of Variations, 2nd Ed., Interdisciplinary Mathematics, Vol. 17, Math Sci Press, Brookline, MA, 1977.

    MATH  Google Scholar 

  68. R. Hermann, “E. Cartan’s geometric theory of partial differential equations,” Advances in Math. 1, 265–317 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  69. W.V. Hodge, Theory and Application of Harmonic Integrals, Cambridge Univ. Press, 1941.

    Google Scholar 

  70. G. Hooft, Phys. Rev. D 14, 3432 (1976).

    Article  Google Scholar 

  71. J.D. Jackson, Classical Electrodynamics, Wiley, 1962.

    Google Scholar 

  72. M. Jimbo, T. Miwa, K. Vend, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients,” Physica 2D 306–352 (1981).

    MathSciNet  MATH  Google Scholar 

  73. K. Kodaira and D.C. Spencer, “On deformation of complex analytic structures,” Am. M. Math. 67, 328–446 (1958).

    MathSciNet  MATH  Google Scholar 

  74. S. Kobayaski, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin, 1972.

    Book  Google Scholar 

  75. S. Kobayaski and K. Nomizu, Foundation of Differential Geometry, Vols. I and II, Interscience, New York, 1963.

    Google Scholar 

  76. K. Kondo, The RAAG Memoirs, Tokyo, 1955.

    Google Scholar 

  77. G. Kron, “Non-Riemannian geometry of rotating electrical machinery,” J. Math. Mech. 13, 103–194 (1934).

    MATH  Google Scholar 

  78. J. Milnor and S. Stasheff, Characteristic Classes, Princeton Univ. Press, 1965.

    Google Scholar 

  79. B. O’Neill, “Submersions and geodesics,” Duke Math. J. 34, 363–373 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  80. B. Reinhart, “Foliated manifolds with bundle-like metrics,” Ann. of Math. 69, 119–132 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  81. G. Reeb, Sur Certaines Proprietes Topologiques des Varieté Feuilletees, Herman, Paris, 1952.

    MATH  Google Scholar 

  82. G. de Rham, Varietés Differentiables, Herman, Paris, 1960.

    MATH  Google Scholar 

  83. W. Slebodzinski, Exterior Forms and their Applications, Polish Scientific Publishers, Warsaw, 1970.

    MATH  Google Scholar 

  84. M. Spivak, Calculus on Manifolds, W.A. Benjamin, New York, 1965.

    MATH  Google Scholar 

  85. M. Spivak, A Comprehensive Introduction to Differential Geometry, Brandeis Univ., Waltham, MA, 1970.

    MATH  Google Scholar 

  86. N. Steenrod, The Topology of Fiber Bundles, Princeton Univ. Press, 1951.

    Google Scholar 

  87. S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1964.

    MATH  Google Scholar 

  88. A. Trautman, “Fiber Bundles, Gauge Fields, and Gravitation,” General Relativity and Gravitation, Vol. I, A Held, (Ed.), Plenum Pub. Co., 1980.

    Google Scholar 

  89. N. Wallach, Symplectic Geometry and Fourier Analysis, Math Sci Press, 1977.

    Google Scholar 

Download references

Authors
  1. Robert Hermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rodolfo Martini Eduardus M. de Jager

Rights and permissions

Reprints and permissions

Copyright information

© 1982 Springer-Verlag

About this paper

Cite this paper

Hermann, R. (1982). Fiber spaces, connections and Yang-Mills fields. In: Martini, R., de Jager, E.M. (eds) Geometric Techniques in Gauge Theories. Lecture Notes in Mathematics, vol 926. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092656

Download citation

  • DOI: https://doi.org/10.1007/BFb0092656

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11497-0

  • Online ISBN: 978-3-540-39192-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation