Skip to main content
Log in

On the number of invariant closed geodesics

  • Published:
Acta Mathematica

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Bott, R., On the iteration of closed geodesics and the Sturm intersections theory.Comm. Pure Appl. Math., 9 (1956), 176–206.

    MathSciNet  Google Scholar 

  2. Fet, A. I. &Lyusternik, L. A., Variational problems on closed manifolds.Dokl. Acad. Nauk SSSR (N.S.), 81 (1951), 17–18 (Russian).

    MathSciNet  MATH  Google Scholar 

  3. Flaschel, P. &Klingenberg, W.,Riemannsche Hilbertmannigfaltigkeiten. Periodische Geodätische. Lecture Notes in Math., Vol. 282, Springer, Berlin (1972).

    Google Scholar 

  4. Friedlander, E., Griffith, P. A. & Morgan, J.,Homotopy theory and differential forms. Lectures notes from Seminario di Geometria at the Instituto Math., Firenze 1972.

  5. Gromoll, D., Klingenberg, W. &Meyer, W.,Riemannsche Geometrie im Grossen. Lecture Notes in Math., Vol. 55, Springer, Berlin, (2. ed. 1975).

    MATH  Google Scholar 

  6. Gromoll, D. &Meyer, W., On differentiable functions with isolated critical points.topology, 8 (1969), 361–369.

    Article  MathSciNet  MATH  Google Scholar 

  7. —, Periodic geodesics on compact riemannian manifolds.J. Differential Geometry, 3 (1969), 493–510.

    MathSciNet  MATH  Google Scholar 

  8. Grove, K., Condition (C) for the energy integral on certain path-spaces and applications to the theory of geodesics.J. Differential Geometry, 8 (1973), 207–223.

    MATH  MathSciNet  Google Scholar 

  9. — Isometry-invariant geodesics.Topology, 13 (1974), 281–292.

    Article  MATH  MathSciNet  Google Scholar 

  10. —, Involution-invariant geodesics.Math. Scand., 36 (1975), 97–108.

    MATH  MathSciNet  Google Scholar 

  11. Grove, K., Halperin, S. & Vigué-Poirrier, M., The rational homotopy theory of certain path-spaces, with applications to geodesics.Acta Math., to appear.

  12. Grove, K. &Tanaka, M., On the number of invariant closed geodesics.Bull. Amer. Math. Soc., 82 (1976), 497–498.

    Article  MathSciNet  MATH  Google Scholar 

  13. Klingenberg, W., Lecture notes on closed geodesics. Bonn, 1977.

  14. Klingmann, M., Das Morse’sche Index Theorem bei Allgemeinen Randbedingungen.J. Differential Geometry, 1 (1967), 371–380.

    MATH  MathSciNet  Google Scholar 

  15. Morse, M., The calculus of variations in the large.Amer. Math. Soc. Colloq. Publ. 18 (1934).

  16. Sakai, T., On the index theorem for isometry-invariant geodesics.Japan J. Math., 1 (1975), 383–391.

    MATH  MathSciNet  Google Scholar 

  17. Serre, J. P., Homologie singulière des espace fibrés.Ann. of Math., 54 (1951), 425–505.

    Article  MATH  MathSciNet  Google Scholar 

  18. Spanier, E. H.,Algebraic Topology. McGraw-Hill, New York (1966).

    MATH  Google Scholar 

  19. Sullivan, D., Differential forms and topology of manifolds.Proceedings of the conference on Manifolds, Tokyo (1973), 37–49.

  20. Sullivan, D., Infinitesimal computations in topology.Inst. Hautes Études Sci. Publ. Math., to appear.

  21. Sullivan, D. & Vigué-Poirrier, M., The homology theory of the closed geodesic problem.J. Differential Geometry., to appear.

  22. Tanaka, M., On invariant closed geodesics under isometries.Ködai Math. Sem. Rep., 28 (1977), 262–277.

    MATH  Google Scholar 

  23. Tanaka, M., Invariant closed geodesics under isometries of prime power ordor.Ködai Math. Sem. Rep., to appear.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grove, K., Tanaka, M. On the number of invariant closed geodesics. Acta Math. 140, 33–48 (1978). https://doi.org/10.1007/BF02392302

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02392302

Keywords

Navigation