Pseudo-distances on symplectomorphism groups and applications to flux theory

TL;DR

This paper introduces pseudo-distances on symplectomorphism groups derived from norms on exact 1-forms, explores their properties, and applies these concepts to flux groups and Hamiltonian diffeomorphisms.

Contribution

It generalizes Banyaga's construction to produce new pseudo-distances and defines genuine distances to analyze flux groups in symplectic geometry.

Findings

Pseudo-distances are equivalent to initial norms on exact 1-forms when restricted to Hamiltonian diffeomorphisms.

New genuine distances to Hamiltonian diffeomorphisms are established.

Applications include insights into flux groups and symplectic topology.

Abstract

Starting from a given norm on the vector space of exact 1-forms of a compact symplectic manifold, we produce pseudo-distances on its symplectomorphism group by generalizing an idea due to Banyaga. We prove that in some cases (which include Banyaga's construction), their restriction to the Hamiltonian diffeomorphism group is equivalent to the distance induced by the initial norm on exact 1-forms. We also define genuine "distances to the Hamiltonian diffeomorphism group" which we use to derive several consequences, mainly in terms of flux groups.