Annals of Global Analysis and Geometry

, Volume 37, Issue 2, pp 173–184 | Cite as

Three-dimensional manifolds all of whose geodesics are closed

Original Paper

Abstract

We present some results concerning the Morse Theory of the energy function on the free loop space of the three sphere for metrics all of whose geodesics are closed. We also explain how these results relate to the Berger conjecture in dimension three.

Keywords

Berger conjecture Morse theory Manifolds all of whose geodesics are closed Three sphere 

References

  1. 1.
    Besse A.L.: Manifolds All of Whose Geodesics are Closed, volume 93 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer, Berlin (1978)Google Scholar
  2. 2.
    Bökstedt M., Ottosen I.: The suspended free loop space of a symmetric space. Preprint Aarhus University 1(18), 1–31 (2004)Google Scholar
  3. 3.
    Bredon G.E.: Introduction to Compact Transformation Groups. Academic Press, New York (1972)MATHGoogle Scholar
  4. 4.
    Ballmann W., Thorbergsson G., Ziller W.: Closed geodesics on positively curved manifolds. Ann. Math. 116(2), 213–247 (1982)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Ballmann W., Thorbergsson G., Ziller W.: Existence of closed geodesics on positively curved manifolds. J. Differential Geom. 18(2), 221–252 (1983)MATHMathSciNetGoogle Scholar
  6. 6.
    Gromoll, D., Grove, K.: On metrics on S 2 all of whose geodesics are closed. Invent. Math 65(1), 175–177 (1981/1982)Google Scholar
  7. 7.
    Hingston N.: Equivariant Morse theory and closed geodesics. J. Differential Geom. 19(1), 85–116 (1984)MATHMathSciNetGoogle Scholar
  8. 8.
    Klingenberg W.: Lectures on Closed Geodesics. Springer, Berlin (1978)MATHGoogle Scholar
  9. 9.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry, Vol. II. Wiley, New York (1969)MATHGoogle Scholar
  10. 10.
    Tsukamoto Y.: Closed geodesics on certain Riemannian manifolds of positive curvature. Tôhoku Math. J. 18(2), 138–143 (1966)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Wilking B.: Index parity of closed geodesics and rigidity of Hopf fibrations. Invent. Math. 144(2), 281–295 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ziller W.: The free loop space of globally symmetric spaces. Invent. Math. 41(1), 1–22 (1977)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RochesterRochesterUSA

Personalised recommendations