Advertisement

Real Conformal Spin Structures

  • Pierre Anglè
Part of the Progress in Mathematical Physics book series (PMP, volume 50)

Keywords

Vector Bundle Clifford Algebra Principal Bundle Conformal Group Horizontal Lift 
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.

Bibliography

  1. Ahlfors L.V., Old and new in Möbius groups, Ann. Acad. Sci. Fenn., serie A.1 Math. 9, pp. 93–105, 1984.MathSciNetMATHGoogle Scholar
  2. Ahlfors L.V., Möbius transformations and Clifford numbers, pp. 65–73 in I. Chavel, M.M. Farkas (eds.), Differential geometry and complex analysis, Springer, Berlin, 1985.Google Scholar
  3. Ahlfors L.V., Möbius transformations in R n expressed through 2 × 2 matrices of Clifford numbers, Complex Variables Theory Appl., 5, pp. 215–224, 1986.MathSciNetGoogle Scholar
  4. Albert A., Structures of Algebras, American Mathematical Society, vol XXIV, New York, 1939.Google Scholar
  5. Anglès P., Les structures spinorielles conformes réelles, Thesis, Université Paul Sabatier.Google Scholar
  6. Anglès P., Construction de revêtements du groupe conforme d’un espace vectoriel muni d’une métrique de type (p,q), Annales de l’I.H.P., section A, vol XXXIII no 1, pp. 33–51, 1980.Google Scholar
  7. Anglès P., Géométrie spinorielle conforme orthogonale triviale et groupes de spinorialité conformes, Report HTKK Mat A 195, pp. 1–36, Helsinki University of Technology, 1982.Google Scholar
  8. Anglès P., Construction de revêtements du groupe symplectique réel C Sp(2r R) Géométrie conforme symplectique réelle. Définition des structures spinorielles conformes symplectiques réelles, Simon Stevin (Gand-Belgique), vol 60 no 1, pp. 57–82, Mars 1986.MATHGoogle Scholar
  9. Anglès P., Algèbres de Clifford C r,s des espaces quadratiques pseudo-euclidiens standards E r,s et structures correspondantes sur les espaces de spineurs associés. Plongements naturels de quadriques projectives Q(E r,s) associés aux espaces E r,s. Nato ASI Séries vol 183, 79–91, Clifford algebras édité par JSR Chisholm et A.K. Common D. Reidel Publishing Company, 1986.Google Scholar
  10. Anglès P., Real conformal spin structures on manifolds, Scientiarum Mathematicarum Hungarica, vol 23, pp., Budapest, Hungary, 1988.Google Scholar
  11. Anglès P. and R. L. Clerc, Operateurs de creation et d’annihilation et algèbres de Clifford, Ann. Fondation Louis de Broglie, vol. 28, no 1, pp. 1–26, 2003.Google Scholar
  12. Artin E., Geometric Algebra, Interscience, 1954; or in French, Algèbre géométrique, Gauthier-Villars, Paris, 1972.Google Scholar
  13. Atiyah M. F., R. Bott and A. Shapiro, Clifford Modules, Topology, vol 3, pp. 3–38, 1964.CrossRefMathSciNetGoogle Scholar
  14. Barbance Ch., Thesis, Paris, 1969.Google Scholar
  15. Bateman H., The conformal transformations of a space of four dimensions and their applications to geometrical optics, J. of London Mathematical Society, 8, 70, 1908.Google Scholar
  16. Bateman H., The transformation of the Electrodynamical Equations, J. of London Mathematical Society, 8, 223, 1909.CrossRefGoogle Scholar
  17. Benedetti R. and C. Petronio, Lectures on hyperbolic geometry, Springer, pp. 7–22, 1992.Google Scholar
  18. Berg M., DeWitt-Morette C., Gwo S. and Kramer E., The Pin groups in physics: C, P and T, Rev. Math. Phys., 13, 2001.Google Scholar
  19. Berger M., Géométrie Différentielle, Armand Colin, Paris, 1972.MATHGoogle Scholar
  20. Berger M., Géométrie, vol. 1–5, Cedic Nathan, Paris, 1977.Google Scholar
  21. Bourbaki N., Algèbre—Chapitre 9: Formes sesquilineaires et quadratiques, Hermann, Paris, 1959.Google Scholar
  22. Bourbaki N., Elements d’histoire des Mathématiques, Hermann, Paris, p. 173, 1969.MATHGoogle Scholar
  23. Brauer R. andWeyl H., American J. of Math., pp. 57–425, 1935.Google Scholar
  24. Cartan E., Annales de l’ E.N.S., 31, pp. 263–355, 1914.MathSciNetGoogle Scholar
  25. Cartan E., La déformation des hypersurfaces dans l’espace conforme réel à ν gε g dimensions, Bull. Soc. Math. France, 45, pp. 57–121, 1917.MathSciNetMATHGoogle Scholar
  26. Cartan E., Les espaces à connexions conformes, Annales de la Société polonaise de Maths., 2, pp. 171–221, 1923.Google Scholar
  27. Cartan E., Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. de France, 41, pp. 1–53, 1931.Google Scholar
  28. Cartan E., Leçons sur la théorie des spineurs, Hermann, Paris, 1938.Google Scholar
  29. Cartan E., The theory of Spinors, Hermann, Paris, 1966.MATHGoogle Scholar
  30. Chevalley C., Theory of Lie groups, Princeton University Press, 1946.MATHGoogle Scholar
  31. Chevalley C., The Algebraic theory of Spinors, Columbia University Press, NewYork, 1954.MATHGoogle Scholar
  32. Cnops J., Vahlen matrices for non-definite matrices, pp. 155–164 in R. Ablamowicz, P. Lounesto, J. M. Parra (eds.), Clifford algebras with numery and symbolic computations, Birkhäuser, Boston, MA, 1996.Google Scholar
  33. Constantinescu-Cornea, Ideale Ränder Riemannscher Flächen, Springer-Verlag, Berlin, 1963.Google Scholar
  34. Crumeyrolle A., Structures spinorielles, Ann. I.H.P., Sect. A, vol. XI, no 1, pp. 19–55, 1964.Google Scholar
  35. Crumeyrolle A., Groupes de spinorialité, Ann. I.H.P., Sect. A, vol. XIV, no 4, pp. 309–323, 1971.MathSciNetGoogle Scholar
  36. Crumeyrolle A., Dérivations, formes, opérateurs usuels sur les champs spinoriels, Ann. I.H.P., Sect. A, vol. XVI, no 3, pp. 171–202, 1972.MathSciNetGoogle Scholar
  37. Crumeyrolle A., Algèbres de Clifford et spineurs, Université Toulouse III, 1974.Google Scholar
  38. Crumeyrolle A., Fibrations spinorielles et twisteurs généralisés, Periodica Math. Hungarica, vol. 6-2, pp. 143–171, 1975.CrossRefMathSciNetGoogle Scholar
  39. Crumeyrolle A., Algèbres de Clifford dégénérées et revêtements des groupes conformes affines orthogonaux et symplectiques, Ann. I.H.P., Sect. A, vol. XXIII, no 3, pp. 235–249, 1980.MathSciNetGoogle Scholar
  40. Crumeyrolle A., Bilinéarité et géométrie affine attachées aux espaces de spineurs complexes Minkowskiens ou autres, Ann. I.H.P., Sect. A, vol. XXXIV, no 3, pp. 351–371, 1981.MathSciNetGoogle Scholar
  41. Cunningham E., The principle of relativity in Electrodynamics and an extension Thereof, J. of London Mathematical Society, 8, 77, 1909.CrossRefGoogle Scholar
  42. D’Auria R., Ferrara S., Lledó MA., Varadarajan VS., Spinor algebras, J. Geom. Phys., 40, pp. 101–128, 2001.MATHCrossRefMathSciNetGoogle Scholar
  43. Deheuvels R., Formes Quadratiques et groupes classiques, Presses Universitaires de France, Paris, 1981.MATHGoogle Scholar
  44. Deheuvels R., Groupes conformes et algèbres de Clifford, Rend. Sem. Mat. Univers. Politech. Torino, vol. 43, 2, pp. 205–226, 1985.MathSciNetMATHGoogle Scholar
  45. Dieudonné J., Les determinants sur un corps non commutatif, Bull. Soc. Math. de France, 71, pp. 27–45, 1943.MATHGoogle Scholar
  46. Dieudonné J., On the automorphisms of the classical groups, Memoirs of Am. Math. Soc., n° 2, pp. 1–95, 1951.Google Scholar
  47. Dieudonné J., On the structure of Unitary groups, Trans. Am. Math. Soc., 72, 1952.Google Scholar
  48. Dieudonné J., La géométrie des groupes classiques, Springer-Verlag, Berlin, Heidelberg, New York, 1971.MATHGoogle Scholar
  49. Dieudonné J., Eléments d’analyse, Tome 4, Gauthier-Villars, 1971.Google Scholar
  50. Dieudonné J., Sur les groupes classiques, Hermann, Paris, 1973.Google Scholar
  51. Dirac P. A. M., Annals Mathematics, pp. 37–429, 1936.Google Scholar
  52. Ehresmann C., Les connections infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Brussels, pp. 29–55, 1950.Google Scholar
  53. Elstrodt J., Grunewald F. and Mennicke J., Vahlen’s groups of Clifford matrices and spin groups, Math. Z., 196, pp. 369–390, 1987.MATHCrossRefMathSciNetGoogle Scholar
  54. Ferrand J., Les géodésiques des structures conformes, CRAS Paris, t. 294, May 17 1982.Google Scholar
  55. FialkowA., The conformal theory of curves, Ann. Math. Soc.Trans., 51, pp. 435–501, 1942.Google Scholar
  56. Fillmore J.P. and A. Springer, Möbius groups over general fields using Clifford algebras associated with spheres, Int. J. Theor. Phys., 29, pp. 225–246, 1990.MATHCrossRefMathSciNetGoogle Scholar
  57. Gilbert J. and M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, Cambridge University Press, pp. 34–38 and pp. 278–296, 1991.Google Scholar
  58. Greub, Halperin, Vanstone, Connections, curvature and cohomology, vol. 1, Academic Press, 1972.Google Scholar
  59. Greub, Halperin, Vanstone, Connections, curvature and cohomology, vol. 2, Academic Press, 1972.Google Scholar
  60. Greub W. and Petry R., On the lifting of structure groups, Lecture notes in mathematics, no 676. Differential geometrical methods in mathematical physics, Proceedings, Bonn, pp. 217–246, 1977.Google Scholar
  61. Haantjes J., Conformal representations of an n-dimensional Euclidean space with a non-definitive fundamental form on itself, Nedel. Akad. Wetensch. Proc. 40, pp. 700–705, 1937.Google Scholar
  62. Helgason S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London, 1962.MATHGoogle Scholar
  63. Hepner W.A., The inhomogeneous linear group and the conformal group, Il Nuevo Cimento, vol. 26, pp. 351–368, 1962.MATHMathSciNetGoogle Scholar
  64. Hermann R., Gauge fields and Cartan–Ehresmann connections, Part A, Math. Sci. Press, Brookline, 1975.MATHGoogle Scholar
  65. Hestenes D. and G. Sobczyk, Clifford algebras to geometric calculus, Reidel, Dordrecht, 1984, 1987.Google Scholar
  66. Husemoller D., Fiber bundles, McGraw Inc., 1966.Google Scholar
  67. Jadczyk A. Z., Some comments on conformal connections, Preprint no 443, I.F.T. UniwersytetuWroclawskiego, Wroclaw, November 1978. Proceedings Differential Geometrical Methods in Maths. Physics, Springer, LNM 836, 202–210, 1979.Google Scholar
  68. Kahan T., Théorie des groupes en physique classique et quantique, Tome 1, Dunod, Paris, 1960.Google Scholar
  69. Karoubi M., Algèbres de Clifford et K-théorie, Annales Scientifiques de l’E.N.S., 4° série, tome 1, pp. 14-270, 1968.MathSciNetGoogle Scholar
  70. Kobayashi S. and Nomizu K., Foundations of differential geometry, vol. 1, Interscience Publishers, New York, 1963.MATHGoogle Scholar
  71. Kobayashi S., Transformation groups in differential geometry, Springer-Verlag, Berlin, 1972.MATHGoogle Scholar
  72. Kosmann-Schwarzbach Y., Dérivée de Lie des spineurs, Thesis, Paris, 1969; Annali di Mat. Pura Applicata, IV, vol. 91, pp. 317–395, 1972.CrossRefMathSciNetGoogle Scholar
  73. Kosmann-Schwarzbach Y., Sur la notion de covariance en relativité générale, Journées relativistes de Dijon, 1975.Google Scholar
  74. Kuiper N.H., On conformally flat spaces in the large, Ann. of Math., vol. 50, no 4, pp. 916–924, 1949.CrossRefMathSciNetGoogle Scholar
  75. Lam T.Y., The algebraic theory of quadratic forms,W.A. Benjamin Inc., 1973.Google Scholar
  76. Lichnerowicz A., Eléments de calcul tensoriel, A. Colin, Paris, 1950.MATHGoogle Scholar
  77. Lichnerowicz A., Théories relativistes de la gravitation et de l’électromagnétisme, Masson.Google Scholar
  78. LichnerowiczA., Champs spinoriels et propagateurs en relativité générale, Bull. Soc. Math. France, 92, pp. 11–100, 1964.Google Scholar
  79. Lichnerowicz A., Champ de Dirac, champ du neutrino et transformations C. P. T. surun espace-temps courbe, Ann. Inst. H. Poincaré 6 Sect. A.N.S., 1, pp. 233–290, 1964.MathSciNetGoogle Scholar
  80. Lichnerowicz A., Cours du Collège de France, ronéotypé non publié, 1963–1964.Google Scholar
  81. Lounesto P., Spinor valued regular functions in hypercomplex analysis, Thesis, Report HTKK-Math-A 154, Helsinki University of Technology, 1–79, 1979.Google Scholar
  82. Lounesto P., Latvamaa E., Conformal transformations and Clifford algebras, Proc. Amer. Math. Soc., 79, pp. 533–538, 1980.Google Scholar
  83. Lounesto P. and A. Springer, Möbius transformations and Clifford algebras of Euclidean and anti-Euclidean spaces, in Deformations of Mathematical Structures, J. Lawrynowicz, ed., Kluwer Academic, Dordrecht, pp. 79–90, 1989.Google Scholar
  84. Lounesto P., Clifford algebras and spinors, second edition, Cambridge University Press, London Mathematical Society, Lectures Notes Series, 286, 2001.Google Scholar
  85. Maia M.D., Isospinors, Journal of Math. Physics, vol. 14, no 7, pp. 882–887, 1973.MATHCrossRefGoogle Scholar
  86. Maia M.D., Conformal spinors in general relativity, Journal of Math. Physics, vol. 15, no. 4, pp. 420–425, 1974.CrossRefMathSciNetGoogle Scholar
  87. Maks J. G., Modulo (1,1) periodicity of Clifford algebras and the generalized (anti-)Möbius transformations, Thesis, Technische Universiteit, Delft, 1989.Google Scholar
  88. Maass H., Automorphe Funktionen von mehreren Veränderlichen und Dirichletsche Reihen, Abh. Math. Sem. Univ. Hamburg, 16, pp. 72–100, 1949.MathSciNetMATHGoogle Scholar
  89. Milhorat J. L., Sur les connections conformes, Thesis, Université Paul Sabatier, Toulouse, 1985.Google Scholar
  90. Milnor J., Spin structure on manifolds, Enseignement mathématique, Genève, 2 série 9, pp. 198–203, 1963.MATHMathSciNetGoogle Scholar
  91. Murai Y., On the group of transformations of six dim. spaces, Prog. of Th. Physics, vol. 9, pp. 147–168, 1953.MATHCrossRefMathSciNetGoogle Scholar
  92. Murai Y., Conformal groups in Physics, Prog. of Th. Physics, vol. 11, no 45, pp. 441–448, 1954.MATHCrossRefMathSciNetGoogle Scholar
  93. Ogiue K., Theory of conformal connections, Kodai Math. Sem. Rep., 19, pp. 193–224, 1967.MATHCrossRefMathSciNetGoogle Scholar
  94. O’Meara O.T., Introduction to quadratic forms, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1973.MATHGoogle Scholar
  95. Penrose R., Twistor algebra, J. of Math. Physics, t. 8, pp. 345–366, 1967.MATHCrossRefMathSciNetGoogle Scholar
  96. Penrose R., Twistor quantization and curved space-time, Int. J. of Th. Physics, (I), 1968.Google Scholar
  97. Pham Mau Quan, Introduction à la géométrie des variétés différentiables, Dunod, Paris, 1969.MATHGoogle Scholar
  98. Popovici I. and D. C. Radulescu, Characterizing the conformality in a Minkowski space, Annales de l’I.H.P., section A, vol. XXXV, no 1, pp. 131–148, 1981.MathSciNetGoogle Scholar
  99. Porteous I.R., Topological geometry, 2ν δ gdition, Cambridge University Press, 1981.Google Scholar
  100. Postnikov M., Leçons de géométries: Groupes et algèbres de Lie, Trad. Française, Ed. Mir, Moscou, 1985.Google Scholar
  101. Riescz M., Clifford numbers and spinors, Lectures series no 38, University of Maryland, 1958.Google Scholar
  102. Ryan J., Conformal Clifford manifolds arising in Clifford analysis, Proc. R. Irish Acad., Section 1.85, pp. 1–23, 1985.Google Scholar
  103. Ryan J., Clifford matrices, Cauchy–Kowalewski extension and analytic functionals, Proc. Centre Math. Annal Aust. Natl. Univ., 16, pp. 286–299, 1988.Google Scholar
  104. Satake I., Algebraic structures of symmetric domains, Iwanami Shoten publishers and Princeton University Press, 1981.Google Scholar
  105. Schouten J. A. and D. J. Struik, Einführung in die nueren Methoden der Differential- Geometrie, Groningen, Noordhoff, vol. 2, p. 209, 1938.Google Scholar
  106. Segal I., Positive energy particle models with mass splitting, Proced. of the Nat. Ac. Sc. of U.S.A., Vol. 57, pp. 194–197, 1967.Google Scholar
  107. Serre J.P., Applications algébriques de la cohomologie des groupes, II. Théorie des algèbres simples, Séminaire H. Cartan, E.N.S., 2 exposés 6.01, 6.09, 7.01, 7.11, 1950–1951.Google Scholar
  108. Steenrod N., The topology of fiber bundles, Princeton University Press, New Jersey, 1951.Google Scholar
  109. Sternberg S., Lectures on differential geometry, P. Hall, New-York, 1965.Google Scholar
  110. Sudbery A., Division algebras, pseudo-orthogonal groups and spinors, J. Phys. A. Math. Gen. 17, pp. 939–955, 1984.CrossRefMathSciNetMATHGoogle Scholar
  111. Tanaka N., Conformal connections and conformal transformations, Trans. A.M.S., 92, pp. 168–190, 1959.MATHCrossRefGoogle Scholar
  112. Toure A., Divers aspects des connections conformes, Thesis, Université Paris VI, 1981.Google Scholar
  113. Vahlen K.-Th., Über Bewegungen und complexen Zahlen, Math. Ann., 55, pp. 585– 593, 1902.CrossRefMathSciNetMATHGoogle Scholar
  114. Van der Waerden B.L., Nachr. Ges.Wiss., Göttingen, 100, I, 1929.Google Scholar
  115. Wall C.T.C., Graded algebras anti-involutions, simple groups and symmetric spaces, Bull. Am. Math. Soc., 74, pp. 198–202, 1968.MATHMathSciNetCrossRefGoogle Scholar
  116. Weil A., Algebras with involutions and the classical groups, Collected papers, vol. II, pp. 413–447, 1951–1964; reprinted by permission of the editors of Journal of Ind. Math. Soc., Springer-Verlag, New York, 1980.Google Scholar
  117. Wolf J. A., Spaces of constant curvature, Publish or Perish. Inc. Boston, 1974.MATHGoogle Scholar
  118. Wybourne B.G., Classical groups for Physicists, JohnWiley and sons, Inc. NewYork, 1974.MATHGoogle Scholar
  119. Yano K., Sur les circonferences généralisées dans les espaces à connexion conforme, Proc. Imp. Acad. Tokyo, 14, pp. 329–332, 1938.MATHCrossRefGoogle Scholar
  120. Yano K., Sur la théorie des espaces à connexion conformes, Journal of Faculty of Sciences, Imperial University of Tokyo, vol. 4, pp. 40–57, 1939.Google Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  • Pierre Anglè
    • 1
  1. 1.Laboratoire Emile Picard Institut de Mathématiques de ToulouseUniversité Paul SabatierFrance

Personalised recommendations