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Lengths of Contact Isotopies and Extensions of the Hofer Metric

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Abstract

Using the Hofer metric, we construct, under a certain condition, a bi-invariant distance on the identity component in the group of strictly contact diffeomorphisms of a compact regular contact manifold. We also show that the Hofer metric on Ham(M) has a right-invariant (but not left invariant) extension to the identity component in the groups of symplectic diffeomorphisms of certain symplectic manifolds.

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References

  1. Banyaga, A.: Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), 174–227.

    Article  MATH  MathSciNet  Google Scholar 

  2. Banyaga, A.: The Structure of Classical Diffeomorphisms Groups, Mathematics and Its Applications, Vol. 400. Dordrecht, The Netherlands: Kluwer Academic Publisher's Group 1997.

    Google Scholar 

  3. Banyaga, A.: The group of diffeomorphisms preserving a regular contact form, Topology and Geometry, Ens. Math. (1979), 47–53.

  4. Blair, D. E.: Riemannian geometry of contact and symplectic manifolds, Progress in Math. Vol. 204, Birkhauser (2002).

  5. Earle, C. and Eells, J.: A fiber bundle description of Teichmuller theory, J. Diff. Geom. 3 (1969), 19–43.

    MATH  MathSciNet  Google Scholar 

  6. Gramain, A.: Le type d'homotopie des difféomorphismes d'une surface compacte, Ann. scient. Éc. Norm. Sup. 4° série, t. 6, 1973, pp. 53–66.

  7. Hofer, H.: On the topological properties of symplectic maps, Proc. Royal Soc. Edimburgh 115A (1990), 25–38.

  8. Hofer, H. and Zehnder, E.: Symplectic Invariants and Hamiltonnian Dynamics, Birkhauser Advanced Texts, Birkhauser Verlag 1994.

  9. Lalonde, F. and McDuff, D.: The geometry of symplectic energy, Ann. Math. 141 (1995), 349–371.

    Article  MATH  MathSciNet  Google Scholar 

  10. Moser, J.: On the volume elements on manifolds, Trans. Amer. Soc. 120 (1965), 286–294.

    Article  MATH  Google Scholar 

  11. Polterovich, L.: The Geometry of the group of symplectic diffeomorphisms, Lect. in Math., ETH Zurich, Birkhauser (2001).

  12. Polterovich, L.: Symplectic displacement energy for Lagrangian submanifolds, Erg. Th and Dynamical Systems 13 (1993), 357–367.

    MATH  MathSciNet  Google Scholar 

  13. Viterbo, C.: Symplectic topology as the geometry of generating functions, Math. Annalen 292 (1992), 685–710.

    Article  MATH  MathSciNet  Google Scholar 

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Mathematics Subject classification (2000): 53C12, 53C15.

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Banyaga, A., Donato, P. Lengths of Contact Isotopies and Extensions of the Hofer Metric. Ann Glob Anal Geom 30, 299–312 (2006). https://doi.org/10.1007/s10455-005-9011-7

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  • DOI: https://doi.org/10.1007/s10455-005-9011-7

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