Inventiones mathematicae

, Volume 210, Issue 3, pp 911–962 | Cite as

On the Berger conjecture for manifolds all of whose geodesics are closed

  • Marco Radeschi
  • Burkhard Wilking


We define a Besse manifold as a Riemannian manifold (Mg) all of whose geodesics are closed. A conjecture of Berger states that all prime geodesics have the same length for any simply connected Besse manifold. We firstly show that the energy function on the free loop space of a simply connected Besse manifold is a perfect Morse–Bott function with respect to a suitable cohomology. Secondly we explain when the negative bundles along the critical manifolds are orientable. These two general results, then lead to a solution of Berger’s conjecture when the underlying manifold is a sphere of dimension at least four.

Mathematics Subject Classification

53C22 58E10 


  1. 1.
    Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups. Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge University Press, Cambridge (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ballmann, W., Thorbergsson, G., Ziller, W.: Closed geodesics on positively curved manifolds. Ann. Math. 116(2), 213–247 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Besse, A.L.: Manifolds all of Whose Geodesics are Closed. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 93 (1978)Google Scholar
  4. 4.
    Borel, A.: Linear Algebraic Groups. Second Enlarged Edition. Springer, New York (1991)CrossRefGoogle Scholar
  5. 5.
    Bredon, G.E.: Introduction to Compact Transformation Groups, vol. 46 (Pure and Applied Mathematics). Elsevier, New York (1972)Google Scholar
  6. 6.
    Cornea, O., Lupton, G., Oprea, J., Tanré, D.: Lusternik-Schnirelmann Category. Mathematical Surveys and Monographs, vol. 103 (2003)Google Scholar
  7. 7.
    Gromoll, D., Grove, K.: On metrics on \(S^2\) all of whose geodesics are closed. Invent. Math. 65, 175–177 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Hingston, N.: Equivariant Morse theory and closed geodesics. J. Differ. Geom. 19, 85–116 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hilton, P.J., Mislin, G., Roitberg, J.: Localization of Nilpotent Groups and Spaces, Notas de Matematica. North-Holland, Amsterdam (1975)zbMATHGoogle Scholar
  10. 10.
    McCleary, J.: On the mod-p Betti numbers of loop spaces. Invent. Math. 87(3), 643–654 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Olsen, J.: Three-dimensional manifolds all of whose geodesics are closed. Ann. Glob. Anal. Geom. 37(2), 173–184 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Schwarz, A.S.: The homologies of spaces of closed curves. Trudy Moskov. Mat. Obšč. 9, 3–44 (1960)MathSciNetGoogle Scholar
  13. 13.
    Taimanov, I.: The type numbers of closed geodesics. Regul. Chaot. Dyn. 15(1), 84–100 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Wadsley, A.W.: Geodesic foliations by circles. J. Differ. Geom. 10(4), 541–549 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Weinstein, A.: On the volume of manifolds all of whose geodesics are closed. J. Differ. Geom. 9(4), 513–517 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Weinstein, A.: Symplectic V-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds. Commun. Pure Appl. Math. 30, 265–271 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Wilking, B.: Index parity of closed geodesics and rigidity of Hopf fibrations. Invent. Math. 144(2), 281–295 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ziller, W.: The free loop space of globally symmetric spaces. Invent. Math. 41, 1–22 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Ziller, W.: Geometry of the Katok examples. Ergod. Theory Dyn. Syst. 3(1), 135–157 (1983)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.University of MünsterMünsterGermany

Personalised recommendations