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

TL;DR

This paper compares Hofer's metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds, revealing that the natural embedding between these spaces does not preserve distances, despite preserving path lengths.

Contribution

It demonstrates that the canonical embedding of Hamiltonian diffeomorphisms into Lagrangian submanifolds is not isometric under Hofer's metric in certain symplectic manifolds.

Findings

The embedding is not isometric in the case of closed symplectic manifolds with trivial second homotopy.

The embedding preserves Hofer's length of smooth paths.

The comparison provides insights into the geometric structure of these symplectic spaces.

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.