Geometriae Dedicata

, Volume 172, Issue 1, pp 399–411 | Cite as

Closed geodesics on orbifolds of nonpositive or nonnegative curvature

Original Paper


In this note we prove existence of closed geodesics of positive length on compact developable orbifolds of nonpositive or nonnegative curvature. We also include a geometric proof of existence of closed geodesics whenever the orbifold fundamental group contains a hyperbolic element and therefore reduce the existence problem to developable orbifolds with \(\pi _1^{orb}\) infinite and having finite exponent and finitely many conjugacy classes.


Orbifolds Closed geodesics Group actions Torsion groups 

Mathematics Subject Classification (2000)

57R18 53C22 58E40 20K10 



I would especially like to thank my advisor Hans U. Boden for his continuous support and encouragement. This research is part of the author’s work as a graduate student at McMaster University. My warm thanks go also to the members of the mathematics department for providing such a welcoming environment. Last but not least, I thank the referee for the helpful remarks.


  1. 1.
    Adem, A., Leida, J., Ruan, Y.: Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics, vol. 171. Cambridge University Press, Cambridge (2007). doi: 10.1017/CBO9780511543081 CrossRefGoogle Scholar
  2. 2.
    Alexandrino, M.M., Javayoles, M.A.: On closed geodesics in the leaf spaces of singular Riemannian foliations. Glasgow Math. J. 53(03), 555–568 (2011)MATHCrossRefGoogle Scholar
  3. 3.
    Armstrong, M.A.: The fundamental group of the orbit space of a discontinuous group. Proc. Camb. Philos. Soc. 64, 299–301 (1968)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of Nonpositive Curvature, Progress in Mathematics, vol. 61. Birkhäuser Boston Inc., Boston (1985)CrossRefGoogle Scholar
  5. 5.
    Bieberbach, L.: Über einige Extremalprobleme im Gebiete der konformen Abbildung. Math. Ann. 77(2), 153–172 (1916). doi: 10.1007/BF01456900 MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)Google Scholar
  7. 7.
    Charlap, L.S.: Bieberbach Groups and Flat Manifolds. Universitext. Springer, New York (1986)CrossRefGoogle Scholar
  8. 8.
    Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971/72)Google Scholar
  9. 9.
    Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96, 413–443 (1972)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dragomir, G.C.: Existence of closed geodesics on orbifolds in dimensions 3, 5 and 7. (in preparation)Google Scholar
  11. 11.
    Dragomir, G.C.: Closed Geodesics on Compact Developable Orbifolds. Ph.D. Thesis, McMaster University. Open Access Dissertations and Thesis. Paper 6253. (2011).
  12. 12.
    Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Bartlett Publishers, Boston (1992)MATHGoogle Scholar
  13. 13.
    Gromoll, D., Meyer, W.: On complete open manifolds of positive curvature. Ann. Math. 90, 75–90 (1969)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Guruprasad, K., Haefliger, A.: Closed geodesics on orbifolds. Topology 45(3), 611–641 (2006). doi: 10.1016/ MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kapovich, M.: Hyperbolic Manifolds and Discrete Groups, Progress in Mathematics, vol. 183. Birkhäuser Boston Inc., Boston (2001)Google Scholar
  16. 16.
    Klingenberg, W.: Lectures on closed geodesics. Springer, Berlin. Grundlehren der Mathematischen Wissenschaften, Vol. 230 (1978)Google Scholar
  17. 17.
    Lyusternik, L.A., Fet, A.I.: Variational problems on closed manifolds. Doklady Akad. Nauk SSSR (N.S.) 81, 17–18 (1951)MATHMathSciNetGoogle Scholar
  18. 18.
    Milnor, J.: Morse theory. Based on Lecture Notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton (1963)Google Scholar
  19. 19.
    Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Myers, S.B., Steenrod, N.E.: The group of isometries of a Riemannian manifold. Ann. Math. 40(2), 400–416 (1939)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Satake, I.: On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. USA 42, 359–363 (1956)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Swenson, E.L.: A cut point theorem for CAT(o) groups. J. Differ. Geom. 53(2), 327–358 (1999)MATHMathSciNetGoogle Scholar
  23. 23.
    Thorbergsson, G.: Closed geodesics on non-compact Riemannian manifolds. Math. Z. 159(3), 249–258 (1978)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Thurston, W.P.: The geometry and topology of 3-manifolds (1978–1979). (electronic edition:
  25. 25.
    Toponogov, V.A.: The metric structure of Riemannian spaces with non-negative curvature which contain straight lines. In: Am. Math. Soc. Translations Ser. 2, Vol. 70: 31 Invited Addresses (8 in Abstract) at the Internat. Congr. Math. (Moscow, 1966), pp. 225–239. Am. Math. Soc., Providence (1968)Google Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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