A Remark On Hofer-like Geometry

TL;DR

This paper proves that Banyaga's Hofer-like norm coincides with the classical Hofer norm on Hamiltonian diffeomorphisms, bridging the gap between Hofer and Hofer-like geometries on compact symplectic manifolds.

Contribution

It confirms a conjecture that the Hofer-like norm reduces to the classical Hofer norm on Hamiltonian diffeomorphisms, enabling extension of results between these geometries.

Findings

Hofer-like norm equals Hofer norm on Hamiltonian diffeomorphisms

Results from Hofer geometry extend to Hofer-like setting

Fills the theoretical gap between Hofer and Hofer-like geometries

Abstract

We show that Banyaga's Hofer-like norm, a generalization of the Hofer norm coincides with the classical Hofer norm when restricted to Hamiltonian diffeomorphisms on compact symplectic manifolds. This result proves a conjecture of Banyaga and fills the gap between Hofer and Hofer-like geometries: the refined Hofer-like structure degenerates to standard Hofer geometry within the Hamiltonian subgroup. This equality allows the straightforward extension of essential results from Hofer geometry to the Hofer-like setting.