A Comparison of Hofer's Metrics on Hamiltonian Diffeomorphisms and Lagrangian Submanifolds

Details A Comparison of Hofer's Metrics on Hamiltonian Diffeomorphisms and Lagrangian Submanifolds A Comparison of Hofer's Metrics on Hamiltonian Diffeomorphisms and Lagrangian Submanifolds

2002-07-08

arxiv.org/abs/math/0207070v1

Mathematics

Symplectic Geometry

Latex, 8 pages

Scientific paper

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold M. The first space is the group of Hamiltonian diffeomorphisms. The second space L consists of all Lagrangian submanifolds of $M \times M$ which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with $\pi_2(M) = 0$, the canonical embedding of Ham(M) into L, f $\mapsto$ graph(f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

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