Skip to main content
SpringerLink
Log in

Obstructions to toric integrable geodesic flows in dimension 3

  • Original Article
  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T * Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.

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. Banyaga, A., Molino, P.: Complete integrability in contact geometry. Penn State preprint PM 197 (1996)

  2. Colin de Verdiére, Y.: Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. II. Le cas intégrablé. Math. Z. 171(1), 51–73 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lerman, E.: Homotopy groups of K-contact toric manifolds. Trans. Amer. Math. Soc. 356(10), 4075–4083 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lerman, E.: Contact toric manifolds. J. Symplectic Geom. 1(4), 785–828 (2003)

    Google Scholar 

  5. Lerman, E., Shirokova, N.: Completely integrable torus actions on symplectic cones. Math. Res. Lett. 9(1), 105–115 (2002)

    Google Scholar 

  6. Madsen, I., Thomas, C.B., Wall, C.T.C.: The topological spherical space form problem II–Existence of free actions. Topology 15, 375–382 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mayer, J., Hyam Rubinstein, J.: Period three actions on the three sphere. Geom. Topol. 7, 329–397 (2003)

    Article  MathSciNet  Google Scholar 

  8. McCord, C., Meyer, K.R., Offin, D.: Are Hamiltonian flows geodesic flows? Trans. Amer. Math. Soc. 355(3), 1237–1250 (2002)

    Article  MathSciNet  Google Scholar 

  9. Milnor, J.W., Stasheff, J.D.: Characteristic Classes, Annals of Mathematical Studies, vol. 76, Princeton University Press, Princeton (1974)

    Google Scholar 

  10. Oh, H.S.: Toral actions on 5-manifolds. Trans. Amer. Math. Soc. 278(1), 233–252 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Parker, P.E.: On some theorems of geroch and stiefel. J. Math. Phys. 25(3), 597–599 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  12. Toth, J.A., Zelditch, S.: L p norms of eigenfunctions in the completely integrable case. Ann. Henri Poincaré 4(2), 343–368 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Yamazaki, T.: On a surgery of K-contact manifolds. Kodai Math J. 24, 214–225 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, U.S.A.

    Christopher R. Lee

Authors
  1. Christopher R. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher R. Lee.

Additional information

Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, C.R. Obstructions to toric integrable geodesic flows in dimension 3. Ann Glob Anal Geom 30, 397–406 (2006). https://doi.org/10.1007/s10455-006-9031-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-006-9031-y

Keywords

Search

Navigation