Mathematische Annalen

, Volume 364, Issue 3–4, pp 1469–1503 | Cite as

Completeness of compact Lorentzian manifolds with abelian holonomy



We address the problem of finding conditions under which a compact Lorentzian manifold is geodesically complete, a property, which always holds for compact Riemannian manifolds. It is known that a compact Lorentzian manifold is geodesically complete if it is homogeneous, or has constant curvature, or admits a time-like conformal vector field. We consider certain Lorentzian manifolds with abelian holonomy, which are locally modelled by the so called pp-waves, and which, in general, do not satisfy any of the above conditions. We show that compact pp-waves are universally covered by a vector space, determine the metric on the universal cover, and prove that they are geodesically complete. Using this, we show that every Ricci-flat compact pp-wave is a plane wave.

Mathematics Subject Classification

53C50 53C29 53C12 53C22 


  1. 1.
    Baum, H., Lärz, K., Leistner, T.: On the full holonomy group of Lorentzian manifolds. Math. Z. 277(3–4), 797–828 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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). Proceedings of Symposium in Pure Mathematics, vol. 54, pp. 27–40. American Mathematical Society, Providence (1993)Google Scholar
  3. 3.
    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)MathSciNetMATHGoogle Scholar
  4. 4.
    Berger, M.: Les espaces symétriques noncompacts. Ann. Sci. École Norm. Sup. 3(74), 85–177 (1957)MATHGoogle Scholar
  5. 5.
    Blanco, O.F., Sánchez, M., Senovilla, J.M.M.: Structure of second-order symmetric Lorentzian manifolds. J. Eur. Math. Soc. (JEMS) 15(2), 595–634 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brinkmann, H.W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 94(1), 119–145 (1925)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cahen, M., Wallach, N.: Lorentzian symmetric spaces. Bull. Am. Math. Soc. 79, 585–591 (1970)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Calabi, E., Markus, L.: Relativistic space forms. Ann. Math. 2(75), 63–76 (1962)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Candela, A.M., Flores, J.L., Sánchez, M.: On general plane fronted waves. Geodesics. Gen. Relativ. Gravit. 35(4), 631–649 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Candela, A.M., Romero, A., Sánchez, M.: Completeness of the trajectories of particles coupled to a general force field. Arch. Ration. Mech. Anal. 208(1), 255–274 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Candela, A.M., Romero, A., Sánchez, M.: Remarks on the completeness of trajectories of accelerated particles in Riemannian manifolds and plane waves. In: Folio, D. (ed.) International Meeting on Differential Geometry (Córdoba, November 15–17, 2010), pp. 27–38 (2013)Google Scholar
  12. 12.
    Carrière, Y.: Autour de la conjecture de L. Markus sur les variétés affines. Invent. Math. 95(3), 615–628 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Conlon, L.: Differentiable Manifolds, 2nd edn. Birkhäuser, Boston (2008)Google Scholar
  14. 14.
    Derdziński, A., Roter, W.: On compact manifolds admitting indefinite metrics with parallel Weyl tensor. J. Geom. Phys. 58(9), 1137–1147 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Derdziński, A., Roter, W.: The local structure of conformally symmetric manifolds. Bull. Belg. Math. Soc. Simon Stevin 16(1), 117–128 (2009)MathSciNetMATHGoogle Scholar
  16. 16.
    Derdziński, A., Roter, W.: Compact pseudo-Riemannian manifolds with parallel Weyl tensor. Ann. Global Anal. Geom. 37(1), 73–90 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Di Scala, A.J., Olmos, C.: The geometry of homogeneous submanifolds of hyperbolic space. Math. Z. 237(1), 199–209 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dumitrescu, S., Zeghib, A.: Géométries lorentziennes de dimension 3: classification et complétude. Geom. Dedic. 149, 243–273 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ehlers, J., Kundt, W.: Exact solutions of the gravitational field equations. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, pp. 49–101. Wiley, New York (1962)Google Scholar
  20. 20.
    Fischer, M.: Lattices of oscillator groups (2013). ArXiv e-printsGoogle Scholar
  21. 21.
    Flores, J.L., Sánchez, M.: On the geometry of pp-wave type spacetimes. In: Analytical and Numerical Approaches to Mathematical Relativity. Lecture Notes in Physics, vol. 692, pp. 79–98. Springer, Berlin (2006)Google Scholar
  22. 22.
    Galaev, A.S.: Metrics that realize all Lorentzian holonomy algebras. Int. J. Geom. Methods Mod. Phys. 3(5–6), 1025–1045 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Galaev, A.S., Leistner, T.: Holonomy groups of Lorentzian manifolds: classification, examples, and applications. In: Recent Developments in Pseudo-Riemannian Geometry. ESI Lectures in Mathematics and Physics, pp. 53–96. European Mathematical Society, Zürich (2008)Google Scholar
  24. 24.
    Globke, W., Leistner, T.: Locally homogeneous pp-waves (2014, preprint). arXiv:1410.3572
  25. 25.
    Hull, C.M.: Exact pp-wave solutions of 11-dimensional supergravity. Phys. Lett. B 139(1–2), 39–41 (1984)MathSciNetGoogle Scholar
  26. 26.
    Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford Univerity Press, New York (2000)MATHGoogle Scholar
  27. 27.
    Kamishima, Y.: Completeness of Lorentz manifolds of constant curvature admitting Killing vector fields. J. Differ. Geom. 37(3), 569–601 (1993)MathSciNetMATHGoogle Scholar
  28. 28.
    Kath, I., Olbrich, M.: Compact quotients of Cahen–Wallach spaces (2015, preprint). arXiv:1501.01474
  29. 29.
    Klingler, B.: Complétude des variétés lorentziennes à courbure constante. Math. Ann. 306(2), 353–370 (1996)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 2. Interscience Wiley, New York (1969)MATHGoogle Scholar
  31. 31.
    Kulkarni, R.S.: Proper actions and pseudo-Riemannian space forms. Adv. Math. 40(1), 10–51 (1981)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lärz, K.: Global aspects of holonomy in pseudo-Riemannian geometry. Ph.D. thesis, Humboldt-Universität zu Berlin (2011).
  33. 33.
    Leistner, T.: Lorentzian manifolds with special holonomy and parallel spinors. Rend. Circ. Mat. Palermo (2) Suppl. (69), 131–159 (2002)Google Scholar
  34. 34.
    Leistner, T.: Screen bundles of Lorentzian manifolds and some generalisations of pp-waves. J. Geom. Phys. 56(10), 2117–2134 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Leistner, T.: On the classification of Lorentzian holonomy groups. J. Differ. Geom. 76(3), 423–484 (2007)MathSciNetMATHGoogle Scholar
  36. 36.
    Marsden, J.: On completeness of homogeneous pseudo-Riemannian manifolds. Indiana Univ. J. 22, 1065–1066 (1972/73)Google Scholar
  37. 37.
    Medina, A., Revoy, P.: Les groupes oscillateurs et leurs réseaux. Manuscr. Math. 52(1–3), 81–95 (1985)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Milnor, J.: Morse Theory. Based on Lecture Notes by M. Spivak and R. Wells. Annals of Mathematics Studies, vol. 51. Princeton University Press, Princeton (1963)Google Scholar
  39. 39.
    O’Neill, B.: Semi-Riemannian Geometry. Academic Press, London (1983)Google Scholar
  40. 40.
    Palais, R.S.: A global formulation of the Lie theory of transformation groups. Mem. Am. Math. Soc. 22:iii + 123 (1957)Google Scholar
  41. 41.
    Romero, A., Sánchez, M.: On the completeness of geodesics obtained as a limit. J. Math. Phys. 34(8), 3768–3774 (1993)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Romero, A., Sánchez, M.: On completeness of certain families of semi-Riemannian manifolds. Geom. Dedic. 53(1), 103–117 (1994)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Romero, A., Sánchez, M.: On completeness of compact Lorentzian manifolds. In: Geometry and Topology of Submanifolds, VI (Leuven, 1993/Brussels, 1993), pp. 171–182. World Science Publications, River Edge (1994)Google Scholar
  44. 44.
    Romero, A., Sánchez, M.: Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field. Proc. Am. Math. Soc. 123(9), 2831–2833 (1995)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Sánchez, M.: Structure of Lorentzian tori with a Killing vector field. Trans. Am. Math. Soc. 349(3), 1063–1080 (1997)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Sánchez, M.: On the completeness of trajectories for some mechanical systems. In: Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden. Fields Institute Communications Series (2013)Google Scholar
  47. 47.
    Schimming, R.: Riemannsche Räume mit ebenfrontiger und mit ebener Symmetrie. Math. Nachr. 59, 128–162 (1974)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Schliebner, D.: On the full holonomy of Lorentzian manifolds with parallel Weyl tensor (2012, preprint). arXiv:1204.5907
  49. 49.
    Schliebner, D.: On Lorentzian manifolds with highest first Betti number. Annal. de l’Inst. Fourier (2015, to appear). arXiv:1311.6723
  50. 50.
    Teschl, G.: Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics, vol. 140. American Mathematical Society, Providence (2012)Google Scholar
  51. 51.
    Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds. London Mathematical Society Lecture Note Series, vol. 83. Cambridge University Press, Cambridge (1983)Google Scholar
  52. 52.
    Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill Book Co., New York (1967)MATHGoogle Scholar
  53. 53.
    Yurtsever, U.: A simple proof of geodesical completeness for compact space-times of zero curvature. J. Math. Phys. 33(4), 1295–1300 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

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

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