Projective limit of a sequence of compatible weak symplectic forms on a sequence of Banach bundles and Darboux Theorem

TL;DR

This paper investigates conditions under which weak symplectic forms on a sequence of Banach bundles induce a symplectic form on their projective limit and explores the applicability of Darboux's theorem in this context.

Contribution

It establishes necessary and sufficient conditions for the preservation of weak symplectic structures and Darboux's theorem in the projective limit of Banach manifolds and discusses limitations of Moser's method.

Findings

Weak symplectic forms can induce a form on the projective limit under certain conditions.

Darboux's theorem may not hold on the projective limit without strong assumptions.

An example shows Darboux's theorem can fail on the projective limit despite holding on each manifold.

Abstract

Given a projective sequence of Banach bundles, each one provided with a of weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the projective limit bundle. Then we apply this results to the tangent bundle of a projective limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the projective limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general, the Moser's method can not work on projective limit of Banach weak symplectic Banach manifolds without very strong conditions like Kumar 's results ([17]). In particular we give an example of a projective sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true on each manifold, but is not true on the projective limit of these manifolds.