Higher dimensional Möbius bands and their boundaries

  • Chady El Mir
  • Jacques Lafontaine


We give a characterisation of Bieberbach manifolds which are geodesic boundaries of a compact flat manifold, and discuss the low dimensional cases, up to dimension 4.

Mathematics Subject Classification

53C20 53C22 53C23 


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The authors would like to thank Andrzej Szczepański for giving the description of 4-dimensional Bieberbach manifolds with zero first Betti number by the use of the useful CARAT package. They also thank the anonymous referee for his comments and remarks.


  1. 1.
    Brown, H., Bulow, R., Neubuser, J., Wondratschek, H., Zassenhaus, H.: Crystallographic Groups of Four-Dimensional space, 1st edn. Wiley, New York (1978)zbMATHGoogle Scholar
  2. 2.
    Charlap, L.S.: Bieberbach Groups and Flat Manifolds. Springer Universitext, Berlin (1986)CrossRefzbMATHGoogle Scholar
  3. 3.
    Elmir, C., Lafontaine, J.: Sur la Géométrie Systolique des Variétés de Bieberbach. Geom. Dedic. 136, 95–110 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Elmir, C., Lafontaine, J.: The systolic constant of orientable Bieberbach \(3\)-manifolds. Ann. Math. Toulouse Sr. 6 22(3), 623–648 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Frigerio, R., Petronio, C.: Construction and recognition of hyperbolic \(3\)-manifolds with geodesic boundary. TAMS 13, 171–184 (2001)zbMATHGoogle Scholar
  6. 6.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 3rd edn. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gordon, M.: The unoriented cobordism classes of compact flat Riemannian manifolds. J. Differ. Geom. 15(1), 81–90 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hamrick, G., Royster, D.: Flat Riemannian manifolds are boundaries. Invent. Math. 66, 405–413 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Szczepański, A.: Geometry of Cristallographic Groups, ADM. World Scientific, Singapore (2012)CrossRefGoogle Scholar
  10. 10.
    Thurston, W.P.: Three-Dimensional Geometry and Topology. In: Levy, S. (ed.) Princeton University Press, Princeton (1997)Google Scholar
  11. 11.
    Wolf, J.A.: Spaces of Constant Curvature. Publish or Perish, Boston (1974)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Applications (LaMA-Liban)Université LibanaiseTripoliLebanon
  2. 2.Institut Montpelliérain Alexander Grothendieck (UMR 5149)Université de MontpellierMontpellier Cedex 5France

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