On Darboux Theorem for symplectic forms on direct limits of symplectic Banach manifolds

TL;DR

This paper investigates whether the Darboux theorem extends to direct limits of ascending sequences of weak symplectic Banach manifolds, showing that it generally does not, but providing specific positive examples in certain Sobolev manifold contexts.

Contribution

It demonstrates that the Darboux theorem does not generally hold on direct limits of weak symplectic Banach manifolds, and provides explicit examples where it does.

Findings

Darboux theorem fails in general on direct limits

Counterexample of Darboux theorem validity in specific Sobolev loop manifolds

Positive example of Darboux theorem on direct limit in a particular setting

Abstract

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show in general, without very strong conditions, the answer is negative. In particular we give an example of an ascending weak symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In a second part, we illustrate this discussion in the context of an ascending sequences of Sobolev manifolds of loops in symplectic finite dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux theorem is true around any point.