Skip to main content
SpringerLink
Log in

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

  • Published:
Boletim da Sociedade Brasileira de Matemática Aims and scope Submit manuscript

Abstract

Given a rational homology classh in a two dimensional torusT 2, we show that the set of Riemannian metrics inT 2 with no geodesic foliations having rotation numberh isC k dense for everyk ∈ N. We also show that, generically in theC 2 topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V. I.: Mathematical Methods of Classical Mechanics. Second Edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, Berlin, Heidelberg.

  2. Ballmann, W., Brin, M., Eberlein, P.: Structure of manifolds with nonpositive curvature I. Annals of Math. 122 (1985) 171–203.

    Article  MathSciNet  MATH  Google Scholar 

  3. Ballmann, W., Brin, M., Spatzier, R.: Structure of manifolds with nonpositive curvature II. Annals of Math. 122 (1985) 205–235.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bangert, V.: Mather sets for twist maps and geodesics on tori. Dynamics reported, Vol. 1. U. Kirchgraber, H. O. Walter Editors, 1988 John Wiley and Sons and B. G. Teubner.

  5. Bangert, V., Schröeder, V.: Existence of flat tori in analytic manifolds of nonpositive curvature. Ann. Sci. Ec. Norm. Sup. 24 (1991) 605–634.

    MathSciNet  MATH  Google Scholar 

  6. Birkhoff, G. D.: Surface transformations and their dynamical applications. Acta Math., 43 (1922), 1–119.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonk, M.: Quasi-geodesic segments and Gromov hyperbolic spaces. Geom. Dedicata 62 (1996) 281–298.

    Article  MathSciNet  MATH  Google Scholar 

  8. Eberlein, P.: Geodesic flows in certain manifolds without conjugate points. Trans. Amer. Math. Soc. 167 (1972) 151–170.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gromov, M.: Hyperbolic groups. Essays in group theory 75–263. Gersten Ed. Springer-Verlag, New York.

  10. Hedlund, G.: Geodesics on a two dimensional Riemannian manifold with periodic coefficients. Annals of Math. 33 (1932) 719–739.

    Article  MathSciNet  MATH  Google Scholar 

  11. Mather, J.: Destruction of invariant circles. Ergod. Th. and Dynam. Sys. (1988), 8, 199–214. Charles Conley Memorial Volume.

    Article  MathSciNet  MATH  Google Scholar 

  12. Mather, J.: Variational construction of orbits of twist diffeomorphisms. J. Amer. Math. Soc. 4, 2 (1991) 207–263.

    Article  MathSciNet  MATH  Google Scholar 

  13. Morse, M.: A fundamental class of geodesics on any closed surface of genus greather than one. Trans. Amer. Math. Soc. 26 (1924) 25–60.

    Article  MathSciNet  MATH  Google Scholar 

  14. Moser, J. K., Siegel, C. L.: Lectures on Celestial Mechanics. Springer-Verlag, New York, Heidelberg, Berlin, 1971.

    MATH  Google Scholar 

  15. Ruggiero, R.: On the creation of conjugate points. Math. Zeitschrift (1991), 208, 41–55.

    Article  MathSciNet  MATH  Google Scholar 

  16. Ruggiero, R.: Topological stability and Gromov hyperbolicity. Erg. Th. Dyn. Sys. (1999), 19, 143–154.

    Article  MathSciNet  MATH  Google Scholar 

  17. Ruggiero, R.: On manifolds admitting continuous foliations by geodesics. Geometriae Dedicata 78: 161–170, 1999.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, PUC-Rio, Rua Marques de São Vicente 225 Gávea, Rio de Janeiro, Brasil

    Rafael Ruggiero

Authors
  1. Rafael Ruggiero
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by CNPq-GMD, FAPERJ, and the University of Freiburg.

About this article

Cite this article

Ruggiero, R. On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces. Bol. Soc. Bras. Mat 31, 93–111 (2000). https://doi.org/10.1007/BF01377597

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01377597

Keywords

Search

Navigation