Skip to main content

Real Conformal Spin Structures

  • Chapter
  • 1241 Accesses

Part of the book series: Progress in Mathematical Physics ((PMP,volume 50))

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • Ahlfors L.V., Old and new in Möbius groups, Ann. Acad. Sci. Fenn., serie A.1 Math. 9, pp. 93–105, 1984.

    MathSciNet  MATH  Google Scholar 

  • 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 

  • 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.

    MathSciNet  Google Scholar 

  • Albert A., Structures of Algebras, American Mathematical Society, vol XXIV, New York, 1939.

    Google Scholar 

  • Anglès P., Les structures spinorielles conformes réelles, Thesis, Université Paul Sabatier.

    Google Scholar 

  • 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 

  • 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 

  • 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.

    MATH  Google Scholar 

  • 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 

  • Anglès P., Real conformal spin structures on manifolds, Scientiarum Mathematicarum Hungarica, vol 23, pp., Budapest, Hungary, 1988.

    Google Scholar 

  • 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 

  • Artin E., Geometric Algebra, Interscience, 1954; or in French, Algèbre géométrique, Gauthier-Villars, Paris, 1972.

    Google Scholar 

  • Atiyah M. F., R. Bott and A. Shapiro, Clifford Modules, Topology, vol 3, pp. 3–38, 1964.

    Article  MathSciNet  Google Scholar 

  • Barbance Ch., Thesis, Paris, 1969.

    Google Scholar 

  • 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 

  • Bateman H., The transformation of the Electrodynamical Equations, J. of London Mathematical Society, 8, 223, 1909.

    Article  Google Scholar 

  • Benedetti R. and C. Petronio, Lectures on hyperbolic geometry, Springer, pp. 7–22, 1992.

    Google Scholar 

  • 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 

  • Berger M., Géométrie Différentielle, Armand Colin, Paris, 1972.

    MATH  Google Scholar 

  • Berger M., Géométrie, vol. 1–5, Cedic Nathan, Paris, 1977.

    Google Scholar 

  • Bourbaki N., Algèbre—Chapitre 9: Formes sesquilineaires et quadratiques, Hermann, Paris, 1959.

    Google Scholar 

  • Bourbaki N., Elements d’histoire des Mathématiques, Hermann, Paris, p. 173, 1969.

    MATH  Google Scholar 

  • Brauer R. andWeyl H., American J. of Math., pp. 57–425, 1935.

    Google Scholar 

  • Cartan E., Annales de l’ E.N.S., 31, pp. 263–355, 1914.

    MathSciNet  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Cartan E., Les espaces à connexions conformes, Annales de la Société polonaise de Maths., 2, pp. 171–221, 1923.

    Google Scholar 

  • Cartan E., Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. de France, 41, pp. 1–53, 1931.

    Google Scholar 

  • Cartan E., Leçons sur la théorie des spineurs, Hermann, Paris, 1938.

    Google Scholar 

  • Cartan E., The theory of Spinors, Hermann, Paris, 1966.

    MATH  Google Scholar 

  • Chevalley C., Theory of Lie groups, Princeton University Press, 1946.

    MATH  Google Scholar 

  • Chevalley C., The Algebraic theory of Spinors, Columbia University Press, NewYork, 1954.

    MATH  Google Scholar 

  • 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 

  • Constantinescu-Cornea, Ideale Ränder Riemannscher Flächen, Springer-Verlag, Berlin, 1963.

    Google Scholar 

  • Crumeyrolle A., Structures spinorielles, Ann. I.H.P., Sect. A, vol. XI, no 1, pp. 19–55, 1964.

    Google Scholar 

  • Crumeyrolle A., Groupes de spinorialité, Ann. I.H.P., Sect. A, vol. XIV, no 4, pp. 309–323, 1971.

    MathSciNet  Google Scholar 

  • Crumeyrolle A., Dérivations, formes, opérateurs usuels sur les champs spinoriels, Ann. I.H.P., Sect. A, vol. XVI, no 3, pp. 171–202, 1972.

    MathSciNet  Google Scholar 

  • Crumeyrolle A., Algèbres de Clifford et spineurs, Université Toulouse III, 1974.

    Google Scholar 

  • Crumeyrolle A., Fibrations spinorielles et twisteurs généralisés, Periodica Math. Hungarica, vol. 6-2, pp. 143–171, 1975.

    Article  MathSciNet  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Cunningham E., The principle of relativity in Electrodynamics and an extension Thereof, J. of London Mathematical Society, 8, 77, 1909.

    Article  Google Scholar 

  • D’Auria R., Ferrara S., Lledó MA., Varadarajan VS., Spinor algebras, J. Geom. Phys., 40, pp. 101–128, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  • Deheuvels R., Formes Quadratiques et groupes classiques, Presses Universitaires de France, Paris, 1981.

    MATH  Google Scholar 

  • Deheuvels R., Groupes conformes et algèbres de Clifford, Rend. Sem. Mat. Univers. Politech. Torino, vol. 43, 2, pp. 205–226, 1985.

    MathSciNet  MATH  Google Scholar 

  • Dieudonné J., Les determinants sur un corps non commutatif, Bull. Soc. Math. de France, 71, pp. 27–45, 1943.

    MATH  Google Scholar 

  • Dieudonné J., On the automorphisms of the classical groups, Memoirs of Am. Math. Soc., n° 2, pp. 1–95, 1951.

    Google Scholar 

  • Dieudonné J., On the structure of Unitary groups, Trans. Am. Math. Soc., 72, 1952.

    Google Scholar 

  • Dieudonné J., La géométrie des groupes classiques, Springer-Verlag, Berlin, Heidelberg, New York, 1971.

    MATH  Google Scholar 

  • Dieudonné J., Eléments d’analyse, Tome 4, Gauthier-Villars, 1971.

    Google Scholar 

  • Dieudonné J., Sur les groupes classiques, Hermann, Paris, 1973.

    Google Scholar 

  • Dirac P. A. M., Annals Mathematics, pp. 37–429, 1936.

    Google Scholar 

  • Ehresmann C., Les connections infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Brussels, pp. 29–55, 1950.

    Google Scholar 

  • Elstrodt J., Grunewald F. and Mennicke J., Vahlen’s groups of Clifford matrices and spin groups, Math. Z., 196, pp. 369–390, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  • Ferrand J., Les géodésiques des structures conformes, CRAS Paris, t. 294, May 17 1982.

    Google Scholar 

  • FialkowA., The conformal theory of curves, Ann. Math. Soc.Trans., 51, pp. 435–501, 1942.

    Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Greub, Halperin, Vanstone, Connections, curvature and cohomology, vol. 1, Academic Press, 1972.

    Google Scholar 

  • Greub, Halperin, Vanstone, Connections, curvature and cohomology, vol. 2, Academic Press, 1972.

    Google Scholar 

  • 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 

  • 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 

  • Helgason S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London, 1962.

    MATH  Google Scholar 

  • Hepner W.A., The inhomogeneous linear group and the conformal group, Il Nuevo Cimento, vol. 26, pp. 351–368, 1962.

    MATH  MathSciNet  Google Scholar 

  • Hermann R., Gauge fields and Cartan–Ehresmann connections, Part A, Math. Sci. Press, Brookline, 1975.

    MATH  Google Scholar 

  • Hestenes D. and G. Sobczyk, Clifford algebras to geometric calculus, Reidel, Dordrecht, 1984, 1987.

    Google Scholar 

  • Husemoller D., Fiber bundles, McGraw Inc., 1966.

    Google Scholar 

  • 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 

  • Kahan T., Théorie des groupes en physique classique et quantique, Tome 1, Dunod, Paris, 1960.

    Google Scholar 

  • Karoubi M., Algèbres de Clifford et K-théorie, Annales Scientifiques de l’E.N.S., 4° série, tome 1, pp. 14-270, 1968.

    MathSciNet  Google Scholar 

  • Kobayashi S. and Nomizu K., Foundations of differential geometry, vol. 1, Interscience Publishers, New York, 1963.

    MATH  Google Scholar 

  • Kobayashi S., Transformation groups in differential geometry, Springer-Verlag, Berlin, 1972.

    MATH  Google Scholar 

  • Kosmann-Schwarzbach Y., Dérivée de Lie des spineurs, Thesis, Paris, 1969; Annali di Mat. Pura Applicata, IV, vol. 91, pp. 317–395, 1972.

    Article  MathSciNet  Google Scholar 

  • Kosmann-Schwarzbach Y., Sur la notion de covariance en relativité générale, Journées relativistes de Dijon, 1975.

    Google Scholar 

  • Kuiper N.H., On conformally flat spaces in the large, Ann. of Math., vol. 50, no 4, pp. 916–924, 1949.

    Article  MathSciNet  Google Scholar 

  • Lam T.Y., The algebraic theory of quadratic forms,W.A. Benjamin Inc., 1973.

    Google Scholar 

  • Lichnerowicz A., Eléments de calcul tensoriel, A. Colin, Paris, 1950.

    MATH  Google Scholar 

  • Lichnerowicz A., Théories relativistes de la gravitation et de l’électromagnétisme, Masson.

    Google Scholar 

  • LichnerowiczA., Champs spinoriels et propagateurs en relativité générale, Bull. Soc. Math. France, 92, pp. 11–100, 1964.

    Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Lichnerowicz A., Cours du Collège de France, ronéotypé non publié, 1963–1964.

    Google Scholar 

  • Lounesto P., Spinor valued regular functions in hypercomplex analysis, Thesis, Report HTKK-Math-A 154, Helsinki University of Technology, 1–79, 1979.

    Google Scholar 

  • Lounesto P., Latvamaa E., Conformal transformations and Clifford algebras, Proc. Amer. Math. Soc., 79, pp. 533–538, 1980.

    Google Scholar 

  • 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 

  • Lounesto P., Clifford algebras and spinors, second edition, Cambridge University Press, London Mathematical Society, Lectures Notes Series, 286, 2001.

    Google Scholar 

  • Maia M.D., Isospinors, Journal of Math. Physics, vol. 14, no 7, pp. 882–887, 1973.

    Article  MATH  Google Scholar 

  • Maia M.D., Conformal spinors in general relativity, Journal of Math. Physics, vol. 15, no. 4, pp. 420–425, 1974.

    Article  MathSciNet  Google Scholar 

  • Maks J. G., Modulo (1,1) periodicity of Clifford algebras and the generalized (anti-)Möbius transformations, Thesis, Technische Universiteit, Delft, 1989.

    Google Scholar 

  • Maass H., Automorphe Funktionen von mehreren Veränderlichen und Dirichletsche Reihen, Abh. Math. Sem. Univ. Hamburg, 16, pp. 72–100, 1949.

    MathSciNet  MATH  Google Scholar 

  • Milhorat J. L., Sur les connections conformes, Thesis, Université Paul Sabatier, Toulouse, 1985.

    Google Scholar 

  • Milnor J., Spin structure on manifolds, Enseignement mathématique, Genève, 2 série 9, pp. 198–203, 1963.

    MATH  MathSciNet  Google Scholar 

  • Murai Y., On the group of transformations of six dim. spaces, Prog. of Th. Physics, vol. 9, pp. 147–168, 1953.

    Article  MATH  MathSciNet  Google Scholar 

  • Murai Y., Conformal groups in Physics, Prog. of Th. Physics, vol. 11, no 45, pp. 441–448, 1954.

    Article  MATH  MathSciNet  Google Scholar 

  • Ogiue K., Theory of conformal connections, Kodai Math. Sem. Rep., 19, pp. 193–224, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  • O’Meara O.T., Introduction to quadratic forms, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1973.

    MATH  Google Scholar 

  • Penrose R., Twistor algebra, J. of Math. Physics, t. 8, pp. 345–366, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  • Penrose R., Twistor quantization and curved space-time, Int. J. of Th. Physics, (I), 1968.

    Google Scholar 

  • Pham Mau Quan, Introduction à la géométrie des variétés différentiables, Dunod, Paris, 1969.

    MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Porteous I.R., Topological geometry, 2ν δ gdition, Cambridge University Press, 1981.

    Google Scholar 

  • Postnikov M., Leçons de géométries: Groupes et algèbres de Lie, Trad. Française, Ed. Mir, Moscou, 1985.

    Google Scholar 

  • Riescz M., Clifford numbers and spinors, Lectures series no 38, University of Maryland, 1958.

    Google Scholar 

  • Ryan J., Conformal Clifford manifolds arising in Clifford analysis, Proc. R. Irish Acad., Section 1.85, pp. 1–23, 1985.

    Google Scholar 

  • Ryan J., Clifford matrices, Cauchy–Kowalewski extension and analytic functionals, Proc. Centre Math. Annal Aust. Natl. Univ., 16, pp. 286–299, 1988.

    Google Scholar 

  • Satake I., Algebraic structures of symmetric domains, Iwanami Shoten publishers and Princeton University Press, 1981.

    Google Scholar 

  • Schouten J. A. and D. J. Struik, Einführung in die nueren Methoden der Differential- Geometrie, Groningen, Noordhoff, vol. 2, p. 209, 1938.

    Google Scholar 

  • 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 

  • 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 

  • Steenrod N., The topology of fiber bundles, Princeton University Press, New Jersey, 1951.

    Google Scholar 

  • Sternberg S., Lectures on differential geometry, P. Hall, New-York, 1965.

    Google Scholar 

  • Sudbery A., Division algebras, pseudo-orthogonal groups and spinors, J. Phys. A. Math. Gen. 17, pp. 939–955, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • Tanaka N., Conformal connections and conformal transformations, Trans. A.M.S., 92, pp. 168–190, 1959.

    Article  MATH  Google Scholar 

  • Toure A., Divers aspects des connections conformes, Thesis, Université Paris VI, 1981.

    Google Scholar 

  • Vahlen K.-Th., Über Bewegungen und complexen Zahlen, Math. Ann., 55, pp. 585– 593, 1902.

    Article  MathSciNet  MATH  Google Scholar 

  • Van der Waerden B.L., Nachr. Ges.Wiss., Göttingen, 100, I, 1929.

    Google Scholar 

  • Wall C.T.C., Graded algebras anti-involutions, simple groups and symmetric spaces, Bull. Am. Math. Soc., 74, pp. 198–202, 1968.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Wolf J. A., Spaces of constant curvature, Publish or Perish. Inc. Boston, 1974.

    MATH  Google Scholar 

  • Wybourne B.G., Classical groups for Physicists, JohnWiley and sons, Inc. NewYork, 1974.

    MATH  Google Scholar 

  • Yano K., Sur les circonferences généralisées dans les espaces à connexion conforme, Proc. Imp. Acad. Tokyo, 14, pp. 329–332, 1938.

    Article  MATH  Google Scholar 

  • 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 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Birkhäuser Boston

About this chapter

Cite this chapter

Anglè, P. (2008). Real Conformal Spin Structures. In: Anglès, P. (eds) Conformal Groups in Geometry and Spin Structures. Progress in Mathematical Physics, vol 50. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4643-1_2

Download citation

Publish with us

Policies and ethics