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Global Differential Geometry pp 3–37Cite as

Holonomy Groups and Algebras

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Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 17))

Abstract

An affine connection is one of the basic objects of interest in differential geometry. 4 It provides a simple and invariant way of transferring information from one 5 point of a connected manifold M to another and, not surprisingly, enjoys lots of 6 applications in many branches of mathematics, physics and mechanics.

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References

  1. Alekseevskii, D.V.: Classification of quaternionic spaces with a transitive solvable group of motions. Math. USSR Izv. 9, 297–339 (1975)

    Article  Google Scholar 

  2. Ambrose, W., Singer, I.M.: A Theorem on holonomy, Trans. Amer. Math. Soc. 75, 428–443 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baguis, F., Cahen, M.:, A construction of symplectic connections through reduction. Let. Math. Phys 57, 149–160 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baum, H.: Eichfeldtheiorie. Springer (2009)

    Google Scholar 

  5. Baum, H., Kath, I.: Parallel spinors and holonomy groups on pseudo-Riemannian spin-manifolds, Ann. Glob. Anal. Geom. 17(1), 1–17 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Beauville, A.: Variétés Kählériennes dont la première classe de Chern est nulle. Jour. Diff. Geom. 18, 755–782 (1983)

    MATH  MathSciNet  Google Scholar 

  7. Bérard-Bergery, L., Ikemakhen, A.: On the holonomy of Lorentzian manifolds. Geometry in mathematical physics and related topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., AMS, 54(2), 27–40 (1993)

    Google Scholar 

  8. Bérard-Bergery, L., Ikemakhen, A.:, Sur l’holonomie des variétés pseudo- riemaniennes de signature (n,n). Bull. Soc. Math. France 125(1), 93–114 (1997)

    Google Scholar 

  9. Berger, M.: Sur les groupes d’holonomie des variétés à connexion affine et des variétés Riemanniennes. Bull. Soc. Math. France 83, 279–330 (1955)

    MATH  MathSciNet  Google Scholar 

  10. Berger, M.: Les espaces symétriques noncompacts. Ann. Sci. Ecol. Norm. Sup. 74, 85–177 (1957)

    Google Scholar 

  11. Besse, A.L.: Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 10, Springer-Verlag, Berlin, New York (1987)

    Google Scholar 

  12. Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J., Schwachhöfer, L.J.: Symplectic Connections, International Journal of Geometric Methods in Modern Physics, 3(3), 375–420 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bochner, S.: Curvature and Betti numbers, II, Ann. Math. 50, 77–93 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  14. Borel, A., Lichnerowicz, A.: Groupes d’holonomie des variétés riemanniennes. C.R. Acad. Sci. Paris 234, 1835–1837 (1952)

    Google Scholar 

  15. Bourgeois, F., Cahen, M.: A variational principle for symplectic connections. J. Geom. Phys 30, 233–265 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bryant, R.:, Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)

    MATH  Google Scholar 

  17. Bryant, R.: Two exotic holonomies in dimension four, path geometries, and twistor theory, Proc. Symp. in Pure Math. 53, 33–88 (1991)

    Google Scholar 

  18. Bryant, R.: Classical, exceptional, and exotic holonomies: A status report. Actes de la Table Ronde de Géométrie Différentielle en l’Honneur de Marcel Berger, Collection SMF Séminaires and Congrès 1 (Soc. Math. de France) 93–166 (1996).

    Google Scholar 

  19. Bryant, R.: Recent Advances in the Theory of Holonomy, Séminaire Bourbaki, exposé no. 861, Paris: Société Mathématique de France, Astérisque. 266, 351–374 (2000)

    Google Scholar 

  20. Bryant, R.: Bochner-Kähler metrics, J. Amer. Math. Soc. 14, 623–715 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Bryant, R.: Geometry of manifolds with Special Holonomy: “100 years of Holonomy” Cont. Math. 395, 29–38 (2006)

    Google Scholar 

  22. Bryant, R., Salamon, S.: On the construction of some complete metrics with exceptional holonomy, Duke. Math. Jour. 58, 829–850 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. Cahen, M., Gutt, S., Rawnsley, J.: Symmetric symplectic spaces with Ricci-type curvature. In: Dito, G., Sternheimer, D. (eds.) Conférence Moshé Flato 1999, Vol. II, Math. Phys. Stud. 22, 81–91 (2000)

    Google Scholar 

  24. Cahen, M., Gutt, S., Horowitz, J., Rawnsley, J.: Homogeneous symplectic manifolds with Ricci-type curvature. J. Geom. Phys. 38, 140–151 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Cahen, M., Gutt, S., Schwachhöfer, L.J.: Construction of Ricci-type connections by reduction and induction, In: “The Breadth of Symplectic and Poisson Geometry”, Festschrift in Honour of Alan Weinstein, pp. 41–58. Birkhäuser (2004)

    Google Scholar 

  26. Cahen, M., Schwachhöfer, L.J.: Special Symplectic Connections. J. Diff. Geom. 83(2), 229–271 (2009)

    MATH  Google Scholar 

  27. Cahen, M., Wallach, N.: Lorentzian symmetric spaces. Bull. Amer. Math. Soc. 76, 585–591 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  28. Calabi, E.: Métriques kähleriennes et fibrés holomorphes, Ann. Ec. Norm. Sup. 12, 269–294 (1979)

    MathSciNet  Google Scholar 

  29. Cartan, É.: Les groupes de transformations continus, infinis, simples. Ann. Éc. Norm. 26, 93–161 (1909)

    MathSciNet  Google Scholar 

  30. Cartan, E.: Sur les variétés à connexion affine et la théorie de la relativité généralisée I & II, Ann. Sci. Ecol. Norm. Sup. 40, 325–412 (1923); et 41, 1–25 (1924) ou Oeuvres complètes, tome III, 659–746 et 799–824

    Google Scholar 

  31. Cartan, E.: La géométrie des espaces de Riemann, Mémorial des Sciences Mathématiques. vol. 5, Paris, Gauthier-Villars (1925)

    Google Scholar 

  32. Cartan, E.:, Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. France 54, 214–264 (1926); 55, 114–134 (1927) ou Oeuvres complètes, tome I, vol. 2, 587–659

    Google Scholar 

  33. Cartan, E.: Les groupes d’holonomie des espaces généralisés, Acta. Math. 48, 1–42 (1926); ou Oeuvres complètes, tome III, vol. 2, 997–1038

    Google Scholar 

  34. Chi, Q.-S., Merkulov, S.A., Schwachhöfer, L.J.: On the existence of infinite series of exotic holonomies. Inv. Math. 126, 391–411 (1996)

    Article  MATH  Google Scholar 

  35. Chi, Q.-S., Merkulov, S.A., Schwachhöfer, L.J.: Exotic holonomies e7 (a). Int. Jour. Math. 8, 583–594 (1997)

    Article  MATH  Google Scholar 

  36. Cortés, V.: Alekseevskian spaces. Diff. Geom. Appl. 6, 129–168 (1996)

    Article  MATH  Google Scholar 

  37. de Rham, G.: Sur la réductibilité d’un espace de Riemann. Comm. Math. Helv. 26, 328–344 (1952)

    Article  MATH  Google Scholar 

  38. Di Scala, A., Olmos, C.: The geometry of homogeneous submanifolds in hyperbolic space, Math. Zeit. 237(1), 199–209 (2001)

    Article  MATH  Google Scholar 

  39. Galaev, K.: Remark on holonomy groups of pseudo-Riemannian manifolds of signature (2, n + 2), arXive: math.DG/0406397 (2004)

    Google Scholar 

  40. Galaev, A.: Metrics that realize all Lorentzian holonomy algebras. Int. J. Geom. Methods Mod. Phys. 3(5–6), 1025–1045 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Galaev, K.: Holonomy groups and special geometric structures of pseudo-K ählerian manifolds of index 2, arXive: math.DG/0612392 (2006)

    Google Scholar 

  42. Galicki, K.: A generalization of the momentum mapping construction for quaternionic Kähler manifolds. Comm. Math. Phys. 108, 117–138 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  43. Galicki, K., Lawson, H.B.: Quaternionic reduction and quaternionic orbifolds. Math. Ann. 282, 1–21 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  44. Guillemin, V.: The integrability problem for G-structures. Trans. Amer. Math. Soc. 116, 544–560 (1965)

    MATH  MathSciNet  Google Scholar 

  45. Ikemakhen, A.: Sur l’holonomie des variétés pseudo-riemaniennes de signature (2, 2 + n). Publ. Math. 43(1), 55–84 (1999)

    MATH  MathSciNet  Google Scholar 

  46. Hano, J., Ozeki, H.: On the holonomy groups of linear connections. Nagoya Math. Jour. 10, 97–100 (1956)

    MATH  MathSciNet  Google Scholar 

  47. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics, Vol. 34, AMS, Providence, RI (2001)

    Google Scholar 

  48. Joyce, D.: Compact 8-manifolds with holonomy G 2: I & II. Jour. Diff. Geo. 43, 291–375 (1996)

    MATH  MathSciNet  Google Scholar 

  49. Joyce, D.: Compact Riemannian 7-manifolds with holonomy Spin(7). Inv. Math. 123, 507–552 (1996)

    MATH  MathSciNet  Google Scholar 

  50. Joyce, D.: Compact manifolds with Special holonomy. Oxford Mathem. Monographs, Oxford Sci. Pub. (2000)

    Google Scholar 

  51. Kamishima, Y.: Uniformization of Kähler manifolds with vanishing Bochner tensor. Acta Math. 172, 229–308 (1994)

    Article  MathSciNet  Google Scholar 

  52. Kath, I., Olbrich, M.: On the structure of pseudo-Riemannian symmetric spaces. Transform. Groups 14(4), 847–885 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  53. Kobayashi, S., Nagano, K.: On filtered Lie algebras and geometric structures II J. Math. Mech. 14, 513–521 (1965)

    MathSciNet  Google Scholar 

  54. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, I, Interscience publishers (1963)

    Google Scholar 

  55. Leistner, T.: On the classification of Lorentzian holonomy groups. Jour. Diff. Geo. 76(3), 423–484 (2007)

    MATH  MathSciNet  Google Scholar 

  56. Merkulov, S.A., Schwachhöfer, L.J.: Classification of irreducible holonomies of torsion free affine connections. Ann. Math. 150, 77–149 (1999); Addendum: Classification of irreducible holonomies of torsion-free affine connections, Ann. Math. 150, 1177–1179 (1999)

    Google Scholar 

  57. O’Brian, N.R., Rawnsley, J. Twistor spaces. Ann. Glob. Anal. Geom 3, 29–58 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  58. Ochiai, T.: Geometry associated with semi-simple flat homogeneous spaces. Trans. Amer. Math. Soc. 152, 159–193 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  59. Olmos, C.: A geometric proof of the Berger holonomy theorem. Ann. Math. 161(1), 579–588 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  60. Panák, M., Schwachhöfer, L.: Bochner-Kähler metrics and connections of Ricci type. Proceedings of the 10th International Conference on Differential Geometry and Its Applications 2007, Differential geometry and its applications, 339–352 (2008)

    Google Scholar 

  61. Salamon, S.: Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics, no. 201, Longman Scientific & Technical, Essex (1989)

    Google Scholar 

  62. Salamon, S.: Quaternionic Kähler manifolds. Inv. Math. 67, 143–171 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  63. Schwachhöfer, L.J.: On the classification of holonomy representations. Habilitationsschrift, Universität Leipzig (1998)

    Google Scholar 

  64. Schwachhöfer, L.J.: Homogeneous connections with special symplectic holonomy. Math. Zeit. 238, 655–688 (2001)

    Article  MATH  Google Scholar 

  65. Schwachhöfer, L.J.: Connections with irreducible holonomy representations. Adv. Math. 160, 1–80 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  66. Simons, J.: On transitivity of holonomy systems. Ann. Math. 76, 213–234 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  67. Vaisman, I.: Variations on the theme of twistor spaces. Balkan J. Geom. Appl. 3, 135–156 (1998)

    MATH  MathSciNet  Google Scholar 

  68. Wang, M.: Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7(1), 59–68 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  69. Wilking, B.: On compact Riemannian manifolds with noncompact holonomy groups. J. Diff. Geom. 52(2), 223–257 (1999)

    MATH  MathSciNet  Google Scholar 

  70. Wu, H.: On the de Rham decomposition theorem. Illinois. J. Math. 8, 291–311 (1964)

    MATH  MathSciNet  Google Scholar 

  71. Wu, H.: Holonomy groups of indefinite metrics. Pac. J. Math. 20(2), 351–392 (1967)

    MATH  Google Scholar 

  72. Yau, S.T.: On the Ricci curvature of a compact Kähler mainfold and the complex Monge-Ampère equation I. Com. Pure and Appl. Math 31, 339–411 (1978)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44221, Dortmund, Germany

    Lorenz J. Schwachhöfer

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  1. Lorenz J. Schwachhöfer
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Correspondence to Lorenz J. Schwachhöfer .

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Editors and Affiliations

  1. Inst. Mathematik, Universität Potsdam, Potsdam, Brandenburg, Germany

    Christian Bär

  2. , Mathematisches Institut, Universität Münster, Einsteinstraße 62, Münster, 48149, Germany

    Joachim Lohkamp

  3. , Fakultät für Mathematik und Informatik, Universität Leipzig, Johannisgasse 26, Leipzig, 04103, Sachsen, Germany

    Matthias Schwarz

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Schwachhöfer, L.J. (2012). Holonomy Groups and Algebras. In: Bär, C., Lohkamp, J., Schwarz, M. (eds) Global Differential Geometry. Springer Proceedings in Mathematics, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22842-1_1

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