The Bott–Samelson theorem for positive Legendrian isotopies

  • Lucas Dahinden


The classical Bott–Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space. This result on geodesic flows has been generalized to Reeb flows and partially to positive Legendrian isotopies by Frauenfelder–Labrousse–Schlenk. We prove the full theorem for positive Legendrian isotopies.


Bott–Samelson theorem Positive Legendrian loop Slow entropy 

Mathematics Subject Classification

Primary 53D35 Secondary 53D40 57R17 



I wish to thank Felix Schlenk and the anonymous referee for their valuable suggestions. This work is supported by SNF Grant 200021-163419/1.


  1. 1.
    Albers, P., Frauenfelder, U.: A variational approach to Givental’s nonlinear Maslov index. Geom. Funct. Anal. 22(5), 1033–1050 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Besse, A.: Manifolds all of whose Geodesics are Closed. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 93. Springer, Berlin (1978)CrossRefGoogle Scholar
  3. 3.
    Bott, R.: On manifolds all of whose geodesics are closed. Ann. Math. 60(3), 375–382 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cieliebak, K., Frauenfelder, U.: A Floer homology for exact contact embeddings. Pac. J. Math. 293(2), 251–316 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cieliebak, K., Frauenfelder, U., Oancea, A.: Rabinowitz–Floer homology and symplectic homology. Ann. sc. de l’ENS 43(4), 957–1015 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Frauenfelder, U., Labrousse, C., Schlenk, F.: Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems. J. Topol. Anal. 07(3), 407–451 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Frauenfelder, U., Schlenk, F.: Fiberwise volume growth via Lagrangian intersections. J. Symp. Geom. 4, 117–148 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frauenfelder, U.: The Arnold–Givental conjecture and moment Floer homology. Int. Math. Res. Not. 42, 2179–2269 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Geiges, H.: An Introduction to Contact Topology. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Merry, W.: Lagrangian Rabinowitz Floer homology and twisted cotangent bundles. Geom. Dedic. 171, 345–386 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Robbin, J., Salamon, D.: The Maslov index for paths. Topology 32, 827–844 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Samelson, H.: On manifolds with many closed geodesics. Port. Math. 22(4), 193–196 (1963)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Serre, J.-P.: Homologie singulière des espaces fibrés. Ann. Math. 54, 425–505 (1951)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité de NeuchâtelNeuchâtelSwitzerland

Personalised recommendations