Darboux chart on Projective limit of weak symplectic Banach manifold

TL;DR

This paper establishes conditions under which Darboux charts exist on projective limits of weak symplectic Banach manifolds, extending symplectic geometry concepts to infinite-dimensional settings.

Contribution

It proves the existence of Darboux charts on projective limits of weak symplectic Banach manifolds under specific conditions, advancing infinite-dimensional symplectic geometry.

Findings

Darboux charts exist when H_x are locally constant.

Compatibility conditions ensure the existence of Darboux charts.

Extension of finite-dimensional symplectic results to infinite-dimensional manifolds.

Abstract

Suppose M be the projective limit of weak symplectic Banach manifolds \{(M_i,\phi_{ij})\}_{i,j\in\mathbb N}, where M_i are modeled over reflexive Banach space and \sigma is compatible with the inverse system(defined in the article). We associate to each point x\in M, a Fr\'{e}chet space H_x(defined in section 3). We prove that if H_x are locally constant, then with certain smoothness and boundedness condition, there exists Darboux chart for the weak symplectic structure.