A Comparison between Hofer's metric and C^1-topology

TL;DR

This paper compares Hofer's metric with the C^1-topology on Hamiltonian diffeomorphism groups, showing that Hofer's metric induces a weaker topology on closed symplectic manifolds.

Contribution

It establishes a relationship between Hofer's metric topology and the C^1-topology, revealing that Hofer's metric is weaker on closed symplectic manifolds.

Findings

Hofer's metric induces a weaker topology than C^1-topology on closed symplectic manifolds.

The result clarifies the topological relationship between Hofer's metric and classical differentiable topology.

The paper provides insights into the structure of Hamiltonian diffeomorphism groups under different topologies.

Abstract

Hofer's metric is a bi-invariant metric on Hamiltonian diffeomorphism groups. Our main result shows that the topology induced from Hofer's metric is weaker than C^1-topology if the symplectic manifold is closed.