Mathematische Zeitschrift

, Volume 277, Issue 3–4, pp 797–828 | Cite as

On the full holonomy group of Lorentzian manifolds

Article

Abstract

The classification of restricted holonomy groups of \(n\)-dimensional Lorentzian manifolds was obtained about ten years ago. However, up to now, not much is known about the structure of the full holonomy group. In this paper we study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. Based on the classification of the restricted holonomy groups of such manifolds, we prove several structure results about the full holonomy. We establish a construction method for manifolds with disconnected holonomy starting from a Riemannian manifold and a properly discontinuous group of isometries. This leads to a variety of examples, most of them being quotients of pp-waves with disconnected holonomy, including a non-flat Lorentzian manifold with infinitely generated holonomy group. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor.

Keywords

Lorentzian manifolds Holonomy groups Isometry groups  Parallel spinor fields Globally hyperbolic manifolds pp-waves 

Mathematics Subject Classification (2010)

Primary 53C29 53C50 Secondary 53C27 

References

  1. 1.
    Ambrose, W., Singer, I.M.: A theorem on holonomy. Trans. Am. Math. Soc. 75, 428–443 (1953)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bär, C., Gauduchon, P., Moroianu, A.: Generalized cylinders in semi-Riemannian and Spin geometry. Math. Z. 249(3), 545–580 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baum, H.: Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, volume 41 of Teubner-Texte zur Mathematik. Teubner-Verlagsgesellschaft (1981)Google Scholar
  4. 4.
    Baum, H.: Gauge theory. An introduction into differential geometry on fibre bundles. (Eichfeldtheorie. Eine Einführung in die Differentialgeometrie auf Faserbündeln.). Berlin: Springer. xiv, 358 p. (2009)Google Scholar
  5. 5.
    Baum, H., Müller, O.: Codazzi spinors and globally hyperbolic manifolds with special holonomy. Math. Z. 258(1), 185–211 (2008)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bérard-Bergery, L., Ikemakhen, A.: On the holonomy of Lorentzian manifolds. In Differential Geometry: Geometry in Mathematical Physics and Related Topics (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pp. 27–40. Am. Math. Soc., Providence, RI (1993)Google Scholar
  7. 7.
    Berger, M.: Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83, 279–330 (1955)MATHMathSciNetGoogle Scholar
  8. 8.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin-Heidelberg-New York (1987)CrossRefMATHGoogle Scholar
  9. 9.
    Boubel, C.: On the holonomy of Lorentzian metrics. Ann. Fac. Sci. Toulouse Math. (6) 16(3), 427–475 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Cahen, M., Kerbrat, Y.: Champs de vecteurs conformes et transformations conformes des spaces Lorentzian symmetriques. J. Math. Pures Appl. 57, 99–132 (1978)MATHMathSciNetGoogle Scholar
  11. 11.
    Cahen, M., Parker, M.: Pseudo-Riemannian symmetric spaces. Mem. AMS 24(229), 1–108 (1980)MathSciNetGoogle Scholar
  12. 12.
    Cahen, M., Wallach, N.: Lorentzian symmetric spaces. Bull. AMS 76(3), 585–591 (1970)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Candela, A.M., Flores, J.L., Sanchez, M.: On general plane fronted waves. Gen. Relat. Gravit. 35(4), 631–649 (2003)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Carrière, Y., Dal’Bo, F.: Généralisations du premier théorème de Bieberbach sur les groupes cristallographiques. Enseign. Math (2) 35(3–4), 245–262 (1989)MATHMathSciNetGoogle Scholar
  15. 15.
    Duncan, D., Ihrig, E.: Homogeneous spacetimes of zero curvature. Proc. AMS 107(3), 785–795 (1989)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    de Rham, G.: Sur la réducibilité d’un espace de Riemann. Math. Helv. 26, 328–344 (1952)CrossRefMATHGoogle Scholar
  17. 17.
    Di Scala, A.J., Olmos, C.: The geometry of homogeneous submanifolds of hyperbolic space. Math. Z. 237(1), 199–209 (2001)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Flores, J.L., Sanchez, M.: Causality and conjugated points in general plane waves. Class. Quantum Gravity 20, 2275–2291 (2003)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Fried, D., Goldman, W.M.: Three-dimensional affine crystallographic groups. Adv. Math. 47, 1–49 (1983)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Galaev, A.S.: Metrics that realize all Lorentzian holonomy algebras. Int. J. Geom. Methods Mod. Phys. 3(5–6), 1025–1045 (2006)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Hitchin, N.: Compact four-dimensional Einstein manifolds. J. Differ. Geom. 9, 435–441 (1974)MATHMathSciNetGoogle Scholar
  22. 22.
    Joyce, D.D.: Riemannian holonomy groups and calibrated geometry, volume 12 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2007)Google Scholar
  23. 23.
    Kobayashi, S., Nomizu, K.: Fondations of differential geometry, vol. I. Whiley Classics Library, London (1996)Google Scholar
  24. 24.
    Leistner, T.: Lorentzian manifolds with special holonomy and parallel spinors. Rend. Circ. Mat. Palermo 2(Suppl. 69), 131–159 (2002)MathSciNetGoogle Scholar
  25. 25.
    Leistner, T.: Screen bundles of Lorentzian manifolds and some generalisations of pp-waves. J. Geom. Phys. 56(10), 2117–2134 (2006)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Leistner, T.: On the classification of Lorentzian holonomy groups. J. Differ. Geom. 76(3), 423–484 (2007)MATHMathSciNetGoogle Scholar
  27. 27.
    Margulis, G.A.: Complete affine locally flat manifolds with a free fundamental group. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 134, 190–205, Automorphic functions and number theory, II (1984)Google Scholar
  28. 28.
    McInnes, B.: Methods of holonomy theory for Ricci-flat Riemannian manifolds. J. Math. Phys. 32(4), 888–896 (1991)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    McInnes, B.: Examples of Einstein manifolds with all possible holonomy groups in dimensions less than seven. J. Math. Phys. 34(9), 4287–4304 (1993)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Merkulov, S., Schwachhöfer, L.: Classification of irreducible holonomies of torsion-free affine connections. Ann. Math. (2) 150(1), 77–149 (1999)CrossRefMATHGoogle Scholar
  31. 31.
    Moroianu, A., Semmelmann, U.: Parallel spinors and holonomy groups. J. Math. Phys. 41(4), 2395–2402 (2000)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Neukirchner, T.: Solvable Pseudo-Riemannian Symmetric Spaces. arXiv:math/0301326 (2003)
  33. 33.
    O’Neill, B.: Semi-Riemannian Geometry. Academic Press, NY (1983)MATHGoogle Scholar
  34. 34.
    Wang, M.Y.: Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7(1), 59–68 (1989)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Wang, M.Y.: On non-simply connected manifolds with non-trivial parallel spinors. Ann. Global Anal. Geom. 13(1), 31–42 (1995)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Wilking, B.: On compact Riemannian manifolds with noncompact holonomy groups. J. Differ. Geom. 52(2), 223–257 (1999)MATHMathSciNetGoogle Scholar
  37. 37.
    Wolf, J.A.: Spaces of Constant Curvature ed. 6. Ams Chelsea Publishing, Providence, RI (2011)Google Scholar
  38. 38.
    Wu, H.: On the de Rham decomposition theorem. Ill. J. Math. 8, 291–311 (1964)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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