On the generalization of the Darboux theorem

We provide sufficient conditions for the existence of Darboux charts on weakly symplectic bounded Fr\'{e}chet manifolds by using the Moser's trick.


Introduction
The Darboux theorem has been extended to weakly symplectic Banach manifolds by using Moser's method, see [1]. The essence of this method is to obtain an appropriate isotopoy generated by a time dependent vector field that provides the chart transforming of symplectic forms to constant ones. In order to apply this method to more general context of Fréchet manifolds we need to establish the existence of the flow of a vector filed which in general does not exist. One successful approach to the differential geometry in Fréchet context is in terms of projective limits of Banach manifolds (see [3]). In this framework, a version of the Darboux theorem is proved in [4].
Another approach to Fréchet geometry is to use the stronger notion of differentiability (see [2]). This differentiability leads to a new category of generalized manifolds, the so call bounded (or MC k ) Fréchet manifolds. In this paper we prove that in that context the flow of a vector field exists (Theorem 2.2) and we will apply the Moser's method to obtain the Darboux theorem (Theorem 3.1).
The obtained theorem might be useful to study the topology of the space of Riemannian metrics M as it has the structure of a bounded Fréchet manifold. Theorem [ §48.9, [7]] asserts that if pM, σq is a smooth weakly symplectic convenient manifold which admits smooth partitions of unity in C 8 σ pM, Rq, and which admits 'Darboux chart', then the symplectic cohomology equals to the De Rham cohomology: H k σ pMq " H k DR pMq. The manifold M admits smooth partition of unity in C 8 σ pM, Rq (it follows from Theorem [7,Theorem 16.10] and Definition [7, Definition 16.1]) so it is interesting to ask if it has a Darboux chart. This, in turn, rises other question: how to construct on Fréchet manifolds weak symplectic forms. It is known that (see [5]) expect Hilbert manifolds an infinite dimensional manifold may not admit a Lagrangian splitting so in general the Weinstein's construction ( [10]) is not applicable. Moreover, the Marsden's idea to construct a symplectic form on a manifold by using the canonical form on its cotangent bundle also is not applicable as there is no natural smooth vector bundle structure on the cotangent bundle [8,Remark I.3.9]. It seems that a symplectic form on M might arise from a weak Riemannian metric and complex structure, however, that would require different version of the Darboux theorem. This is an interesting topic for further studies.

Bounded differentiability
In this section we prove the existence of the local flow of a MC k -vector field, we refer to [2] for more details on bounded Fréchet geometry. We denote by pF, ρq a Fréchet space whose topology is defined by a complete translational-invariant metric ρ. We consider only metrics with absolutely convex balls. Note that every Fréchet space admits such a metric, cf [2]. One reason to choose this particular metric is that a metric with this property can give us a collection of seminorms that defines the same topology. More precisely: Theorem 3.4). Assume that pF, ρq is a Fréchet space and ρ is a metric with absolutely convex balls. Let B ρ These Minkowski functionals are continuous seminorms on F . A collection t v i u iPN of these seminorms gives the topology of F .
In the sequel we assume that a Fréchet space F is graded with the collection of seminorms v n F " ř k"n k"1 v k that defines its topology. Let pE, gq be another Fréchet space. Let L g,ρ pE, F q be the set of all linear maps L : E Ñ F such that The transversal-invariant metric D g,ρ : L g,ρ pE, F qˆL g,ρ pE, F q ÝÑ r0, 8q, pL, Hq Þ Ñ LippL´Hq g,ρ , (2.1) on L ρ,g pE, F q turns it into an Abelian topological group. Let U an open subset of E, and P : U Ñ F a continuous map. If P is Keller-differentiable, d P ppq P L ρ,g pE, F q for all p P U, and the induced map d P ppq : U Ñ L ρ,g pE, F q is continuous, then P is called bounded differentiable. We say P is MC 0 and write P 0 " P if it is continuous. We say P is an MC 1 and write P p1q " P 1 if it is bounded differentiable. We define for pk ą 1q maps of class MC k recursively, see [2]. If ϕptq is a continuous path in a Fréchet space we denote its derivative by d dt ϕptq.
Within this framework we define MC k (bounded) Fréchet manifolds, MC k -maps of manifolds and tangent bundles and their MC k -vector fields. A MC k -vector field X on a MC k - A vector field on an infinite dimensional Fréchet manifold may have no, one ore multiple integral curves. However, a MC k -vector field always has a unique integral curve. neighborhood U 0 of p 0 and a positive real number α such that for every q P U 0 there exists a unique integral curve ℓptq " F t pqq satisfying ℓp0q " q and ℓ 1 ptq " Xpℓptqq for all t P p´α, αq.
There exists a real number α ą 0 such that for each x P U there exists a unique integral curve ℓ x ptq satisfying ℓ x p0q " x for all t P I " p´α, αq. Furthermore, the mapping F : IˆU Ñ F given by F t pxq " Fpt, xq " ℓ x ptq is of class MC k .
Proof. The first part of the proof follows from Corollary 3.4. We now proof the second part. Let x, y P U be arbitrary and define the maps ϕ n ptq " Fpt, xq´Fpt, yq n F , @n P N. Since X is MC k , so it is globally Lipschitz. Let β ą 0 be its Lipschitz constant we then have Since X is MC k , for given ε ą 0 there is a δ ą 0 such that h n F ă δp@n P Nq yields that the second term is less than ż t 0 ε Fps, x`hq´Fps, xq n F , @n P N, The right-hand sides are MC K´1 , so are the solutions by induction. Thus Fpt, xq is MC k .

Darboux charts
In general for a Fréchet manifold differential forms cannot be defined as the sections of its cotangent bundle since in general we can not define a manifold structure on the cotangent bundle, see [8,Remark I.3.9]. To define differential forms we follow the approach of Neeb [8]. Let σ P Ω p pM, Rq, Y P VpMq and F t the local flow of Y . We define the usual Lie derivative for by which of course coincides by pL Y ωqpX 1 , . . . , X p q " Y.ωpX 1 , . . . , X p q´p ÿ j"1 ωpX 1 , . . . , rY, X j s, . . . , X p q for X i P VpUq, U Ă M open. For each X P VpMq and p ě 1 we define a linear map where pi v ω x qpv 1 , . . . , v p´1 q :" ω x pv, v 1 , . . . , v p´1 q. For ω P Ω 0 pM, Rq " C 8 pM, Rq, we put for all x P M and v The Darboux theorem is a local result so we consider the case where M is an open set U of F . For the simplicity we suppose that 0 P U.
Let F 1 b be the strong dual of F and define the map ω # x : where x¨,¨y is a duality pairing. Condition 3.4 implies that ω # is injective.
Let x P U be fixed and define H x -tω x py, .q | y P F u, this is a subset of F 1 b and its topology is induced from it. Henceforth we assume that all Fréchet spaces are reflexive.
Proof. Obviously ω # x is injective. The space F is reflexive so it is distinguished and therefore the strong dual is barrelled therefore by open mapping theorem and the restriction of the domain to H x the inverse is continuous. Since F is reflexive the natural embedding ı : E Ñ E 2 , x Ñx wherexpℓq " ℓpxq for ℓ P F 1 is onto. Assume that ω # x is not surjective and R is its range, i.e. R ‰ H x . Form the continuity of the inverse it follows that R is closed. By the Hahn-Banach theorem there is a 0 ‰ φ P F 2 such that φpRq " t0u. Let φ " ıpvq. Then for any w P F , ωpv, wq " xv, ω # x pwqy " φpω # x pwqq " 0. Therefore v " 0 and so φ " 0 which is the contradiction.
We will need the following result. Let ω P Ω K`1 pU, V q be a V -valued closed pk`1q-form. Then ω is exact. Moreover, ω " d dR α for some α P Ω K pU, V q with αp0q " 0 given by αpxqpv 1 ,¨¨¨v k q " ż 1 0 t k ωptkqpx, v 1 ,¨¨¨v k qdt.
Theorem 3.1. Let pM, ωq be a weakly symplectic smooth bounded Fréchet manifold modeled on F . Let ω t " ω 0`t pω´ω 0 q for t P r0, 1s. Suppose that following hold (1) There exits an open star-shaped neighborhood U of zero such that for all x P U the map ω t# x : F Ñ H x is isomorphism for each t, (2) for x P U the map pω t# x q´1 : H x Ñ E is smooth for each t. Then there exists a coordinate chart pV, ϕq around zero such that ϕ˚ω " ω 0 .
Proof. By Lemma 2.2 there exist 1-form α such that d dR α " ω´ω 0 and αp0q " 0. Consider a time-dependent vector field X t : U Ñ F such that i Xt ω t "´α.
By Condition 1 for x P U and all t we have ω t# x is isomorphism hence X t pxq " pω t# x q´1pα x q is well defined and it is smooth by Condition 2. Thus, by Theorem 2.2 there exists the smooth isotopy F t on F generated by X t and for t P r0, 1s it satisfies Ft ω t " ω 0 (3.5) To solve 3.5, we need to solve d dt Ft ω t " 0. (3.6) We have by product rule of derivative and the Cartan formula d dt Ft ω t " Ft pL Xt ω t q`Ft d dt ω t (3.7) " Ft p d dt ω t´d dR pi Xt ω t qq (3.8) " Ft p´dα`ω 0´ω q " 0. (3.9) Thus, F1ω 1 " F0ω 0 so F1ω " ω 0 .