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

Abstract

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.

Latex, 8 pages