A comparison of Hofer's metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds

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Abstract

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold (M, ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π 2(M) = 0, the canonical embedding of Ham(M) into ℒ, f → graph(f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

Original languageEnglish
Pages (from-to)803-811
Number of pages9
JournalCommunications in Contemporary Mathematics
Volume5
Issue number5
DOIs
StatePublished - Oct 2003

Keywords

  • Hofer's metric
  • Lagrangian submanifolds
  • Symplectic manifolds

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