On the generalization of the Darboux theorem

Abstract

Darboux theorem to more general context of Frechet manifolds we face an obstacle:  in general vector fields do not have local flows. Recently, Fr\'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic  manifold $(M^{2n},\omega)$ is locally symplectomorphic to $(R^{2n}, \omega_0)$, where $\omega_0$  is the standard symplectic form on  $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ \ff_t $  generated by a time-dependent vector field $ X_t  $ on $M$ such that $ \ff_1^{*} \omega = \omega_0$. Nevertheless,  it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we  associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the  classifying space does not depend on the point of the manifold and a suitable smoothness condition holds.  If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar.  In this paper we consider the category of so-called bounded Fr\'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr\'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields  we prove the Darboux theorem.