AN INTEGRABILITY CRITERION FOR A PROJECTIVE LIMIT OF BANACH DISTRIBUTIONS

HAL Id: hal-03207038

https://hal.archives-ouvertes.fr/hal-03207038

Preprint submitted on 23 Apr 2021

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

AN INTEGRABILITY CRITERION FOR A

PROJECTIVE LIMIT OF BANACH DISTRIBUTIONS

Fernand Pelletier

To cite this version:

Fernand Pelletier. AN INTEGRABILITY CRITERION FOR A PROJECTIVE LIMIT OF BANACH

DISTRIBUTIONS. 2021. �hal-03207038�

BANACH DISTRIBUTIONS

FERNAND PELLETIER

Abstract. We give an integrability criterion for a projective limit of Banach dis- tributions on a Fr´echet manifold which is a projective limit of Banach manifolds.

This leads to a result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds. This can be seen as a version of Frobenius Theorem in Fr´echet setting. As consequence, we obtain a version of the third Lie theorem for a Fr´echet-Lie group which is a submersive projective limit of Banach Lie groups. We also give an application to a sequence of prolongations of a Banach Lie algebroid.

2010 MSC:53C30, 53Z05, 46T05. secondary 18A30, 58B20, 58B25.

Keywords: Banach manifold, Fr´echet manifold, projective limit of Banach manifolds, pro- jective limit of Banach bundles, integrability of projective limit of Banach distributions, projective limit of Banach Lie subalgebras.

1. introduction

In classical differential geometry, adistributionon a smooth manifoldM, is an assign- ment ∆ :x7→∆x ⊂TxM onM, where ∆x is a subspace ofTxM. This distribution is integrable if, for anyx∈M, there exists an immersed submanifoldf:L→M such that x∈f(L) and for anyz ∈L, we haveT f(TzL) = ∆f(z). On the other hand, ∆ is called involutive if, for any vector fieldsX andY onM tangent to ∆, their Lie bracket [X, Y] is also tangent to ∆.

On a finite dimensional manifold, when ∆ is a subbundle ofT M, the classical Frobe- nius Theorem gives an equivalence between integrability and involutivity. In the other cases, the distribution is singular and, even under assumptions of smoothness on ∆, in general, the involutivity is not a sufficient condition for integrability (one needs some more additional local conditions). These problems were clarified and resolved essentially in [20]

and [19].

In the context of Banach manifolds, the Frobenius Theorem is again true for distribu- tions which are complemented subbundles in the tangent bundle (cf. [16]). For singular Banach distributions closed and complemented (i.e. ∆x is a complemented Banach sub- space of TxM) we also have the integrability property under some natural geometrical conditions (see [5] for instance). In a more general way, weak Banach distributions ∆ (i.e.

∆x which is a Banach subspace ofTxM not necessary complemented), the integrability property is again true under some additional geometrical assumptions (see [18] or [4] for more details).

The proof of this last results is essentially based on the existence of the flow of a local vector field. In a more general infinite dimensional context as distributions on convenient manifolds or on locally convex manifolds, in general the local flow for a vector field does not exist. Analog results exists in such previous settings: [12], [13], [8], [21] [2] for instance.

But essentially, all these integrability criteria are proved under strong assumptions which, either implies the existence of a family of vector fields which are tangent and generate locally a distribution and each one of these vector fields have a local flow, or implies the

1

existence of an implicit function theorem in such a setting.

The purpose of this paper is to give an integrability criterion for projective limit of Banach distributions on a Fr´echet manifold which is a projective limit of Banach man- ifolds. The precise assumptions on the distribution are presented in assumptions (*) in Definition 1) and this criterion is formulated in a local way in Theorem 2 and in a global way in Theorem 3. These results are obtained under conditions which permits to used a Theorem of existence and unicity of solution of ODE in a Fr´echet space proved in [17], and which can be reformulated in the context of projective limit of Banach spaces (cf.

Appendix C). Using such a Theorem, this proof needs, in the one hand, an adaptation of some arguments used in the proof of Theorem 1 in [18] for closed distributions on Banach manifolds, and on the other hand, some properties of the Banach Lie group of ”uniformly bounded” automorphisms of a Fr´echet space (cf. Appendix B). As application, this cri- terion permits to obtain a kind of projective limit of ”Banach Frobenius theorem” for submersive projective limit of involutive bundles on a submersive projective sequence of Banach manifolds (cf. Theorem 4). By the way, as consequence, a submersive projective limit of complemented Banach Lie subalgebras of a submersive projective limit of Banach Lie group algebras is the Lie algebra of a F´echet Lie group (cf. Theorem 5). This result can be considered as a version of the third Lie Theorem for a Fr´echet-Lie group which is a submersive projective limit of Banach Lie groups. We also give an application to a sequence of prolongations of a Banach Lie algebroid (cf. Theorem 7). We complete these results, by an example of integrable Fr´echet distribution which is a projective limit of non integrable distributions but which satisfies the assumptions (*).

This paper is organized as follows. The first paragraph of the next section describes the context and the assumptions of this criterion of integrability used in Theorem 2 and Theorem 3. In order to present these Theorems in a more accessible way for a quick reading, the useful definitions and results take place in Appendix A, and we formulate the assumptions with precise references to this Appendix. The Theorem 3 permits to show that, under some natural conditions, the projective limit E of a submersive projective sequence of involutive subbundlesEiof the tangent bundleT Miof a submersive projective sequence of Banach manifoldMi, is a Fr´echet involutive and integrable subbundle of the tangent bundleT M of the Fr´echet manifoldM= lim←−Mi.

A first application of these results, is the existence of a F´echet Lie group whose Lie algebra is a submersive projective limit of complemented Banach Lie subalgebra of the a submersive projective limit of Banach Lie groups Lie algebras (cf. Theorem 5 in §3.1).

In§3.2, we give an application of Theorem 2 to a sequence of prolongations of a Banach- Lie algebroid (see [3]) and we end this paragraph by the announced contre-exemple of Theorem 2 and Theorem 3.

The proof of the basic Theorem 2 is located in§4.

All properties concerning the set of uniformly bounded endomorphisms of a Fr´echet space are developed in Appendix B. Also in a series of other Appendices, we expose all the definitions and results needed in the statements of Theorems and in the proof of Theorem 2.

2. An integrability criterion for a submersive projective limit of Banach distributions

2.1. The criterion and its corollaries. The context needed in this section is detailed in Appendix A.

Let Mi, δ^j_i

j≥i be a sequence of projective Banach manifolds with projective limitM = lim←−Mi1. In order to give a criterion of integrability for projective limits of local Banach

1cf. section A.4

bundles onM under some additional assumptions, we need to introduce some notations.

Let ν (resp. µ) be a norm on a Banach space E (resp. M). We denote by k k^op the associated norm on the linear space of continuous linear mappingsL(E,M).

Then we have:

Definition 1. Let Mi, δ_i^j

j≥i be a projective sequence of Banach manifolds where the mapsδ_i^jare submersions and M = lim←−Miits projective limit.

A closed distribution∆onM will be called a submersive projective limit of local anchored bundles if the following property is satisfied:

(*) For any x = lim

←−xi ∈ M, there exists an open neighbourhood U = lim

←−Ui of x, a submersive projective sequence of anchored Banach bundles(Ei, πi, Ui, ρi) ^{2 3 4} fulfilling the following properties for anyz= lim←−zi∈U:

1. lim←−ρi((Ei)z_i) = ∆z, for anyz∈U.

2. The kernel of (ρi)z_i is complemented in (Ei)z_i and the range of (ρi)z_i is closed, for alli∈N.

3. There exists a constantC >0 and a Finsler norm|| ||^Ei (resp. || ||^Mi) on (Ei)|U_i (resp. T Mi|U_i) such that:

∀i∈N, ||(ρi)z_i||^op_i ≤C, ∀zi∈Ui.⁵ We have the following criterion of integrability:

Theorem 2. LetMbe a projective limit of a submersive projective sequence(Mi, δ^j_i)j≥iof Banach manifolds and∆be a local projective limit of local Banach bundles onM. Assume that, under the property (*), there exists a Lie bracket [., .]i on (Ei, πi, Ui, ρi) such that (Ei, πi, Ui, ρi,[., .]i)is a submersive projective sequence of Banach-Lie algebroids⁶. Then the distribution ∆is integrable and the maximal integral manifold N throughx= lim←−xi is a closed Fr´echet submanifold of M which is a submersive projective limit of the set of maximal leavesNi ofρi(Ei)throughxi inMi.

The proof of Theorem takes place in section 4.

Now, we have the following consequences of Theorem 2:

Theorem 3. Let (Ei, πi, Mi, ρi,[., .]i) be a submersive projective sequence of split Lie algebroids⁷. Then we have:

1.

E:= lim←−Ei, π:= lim←−πi, M := lim←−Mi, ρ= lim←−ρi

is Fr´echet anchored bundle and

∆ =ρ(E) is a closed distribution onM

2 If(ρi) satisfies the condition (3) in Definition 1, then∆ is integrable and each leafL of∆is a projective limit of leaves Li of∆i.

Proof The property (1) is a consequence of Proposition 29. From this property, it follows that locally ∆ satisfies assumption (1) and (2) of Definition 1 so if the assumption (3) is satisfied, the result is a direct consequence of Theorem 2.

Theorem 4. Let Mi, δ^j_i

j≥i be a submersive projective sequence of Banach manifold and(Ei, πi, Mi)an involutive subbundle of T Mi whereπi is the restriction of the natural projection pM_i :T Mi →Mi. Assume that the restrictionT δ_i^j :Ej →Ei is a surjective

2see:Definition 25 and Notations 24 for a submersive sequence of projective Banach bundles 3see:Definition 27 for an anchored Banach bundle

4 a sequence of projective anchored bundle (E_i, π_i, M_i, ρ_i) is a projective sequence of Banach bundles which satisfies assumption(PSBLA 2)in Definition 28

5 More precisely,||(ρi)_zi||^op_i = sup{||(ρi)_zi(u)||^Mi,||u||^Ei≤1}

6cf. Definition 28 7cf. Definition 28

bundle morphism for all i ∈N and j ≥i. Then(Ei, πi, Mi) is a submersive projective sequence of Banach bundles, and (E = lim←−Ei, π = lim←−πi, M = lim←−Mi) is an integrable Fr´echet subbundle ofT M whose each leafLof E in M is a projective limit of leavesLi

ofEi inMi

Proof Since δ_i^j : Mj → Mi is a surjective submersion, so isT δ^j_i :T Mj → T Mi. If T δ_i^j:Ej→Eiis a surjective morphism, this implies thatT δ^j_i is a submersion ontoEiand so (Ei, πi, Mi) is a submersive projective sequence of Banach bundles. Letιi:Ei→T Mi

the natural inclusion and [, ]ithe restriction of Lie bracket of vector fields to (local) sec- tions ofEi. Then (Ei, Mi, ιi,[, ]i) is a Banach Lie algebroid and sinceT δ^j_i◦ιj=ιi◦T δ^j_i it follows that (Ei, Mi, ιi,[ , ]i) is a is a submersive projective sequence of Banach-Lie algebroids. Fix somex= lim←−xi∈M = lim←−Mi. According to Theorem 3 we have only to show that the condition (3) of Definition 1 is satisfied byιi.

Given any norm|| ||^MionTx_iMiwe denote by|| ||^Eithe induced norm on the fiber{Ei}x_i, then, for the associated norm operator we have||{ιi}x_i||^op_i = 1. So all the assumption of Theorem 2 are satisfied which ends the proof.

3. Some applications and contre-example

3.1. Application to submersive projective sequence of Banach Lie groups. Let Gi, δ_i^j

j≥ibe a submersive a projective sequence of Banach-Lie groups whereGiis mod- elled on Gi (cf. Definition 19). We denote by L(Gi) the Lie algebra of Gi. Then L(Gi) ≡ TeGi is isomorphic to Gi. If we set ¯δ_i^j := Teδ_i^j, then each ¯δ^j_i is a surjective linear map fromL(Gj) toL(Gi) whose kernel is complemented.

Consider a sequencehi of complemented sub-Lie algebra of L(Gi) such that the restric- tion of ¯δ_i^jtohjis a continuous surjective map. Then (hi,δˆ_i^j)j≥iis a submersive projective sequence of Banach Lie algebra and soh= lim←−hi is a Fr´echet Lie algebra (cf. [4] chapter 4). Now from classic result on Banach Lie groups (cf. [16]), by left translation eachhi

gives rise to a complemented involutive subbundle ofHiofT Gi and the leafHithrough the neutralei inGi has a structure of connected Banach Lie group so that the inclusion ιi:Hi→Giis a Banach Lie morphism. Note thatii:=Te_iιiis nothing but else that the inclusion ofhi inL(Gi) and which induces the natural inclusion ˆιiofHiinT Gi. Moreover, since (hi,ˆδ^j_i)j≥iis a submersive projective sequence of Banach Lie algebra and

Gi, δ_i^j

j≥i be a submersive a projective sequence of Banach-Lie groups, it follows that (Hi, Gi,bii,[, ]i)⁸is a submersive projective sequence of split Banach Lie algebroids.

On the other hand, from Theorem 21 the Lie algebraL(G) ofG = lim←−Gi is lim←−LGi

and soh= lim←−hi is a closed complemented Lie subalgebra ofL(G). As in the context of Banach setting, by left translation,h gives rise to an involutive Fr´echet subbundleHof T Gwhich is clearly the projective limit of the submersive sequence (Hi, Gi,bii,[, ]i). So from Theorem 4 by same arguments as in Banach Lie groups, we obtain:

Theorem 5. LetG= lim←−Gi be a the projective limit of a submersive projective sequence of Banach-Lie groups Gi, δ^j_i

j≥i and for each i ∈ N, consider a closed complemented Banach Lie subalgebrahiofL(Gi). Assume that the restriction ofδ¯_i^jtohjis a continuous surjective map. Then there exists a Fr´echet Lie groupH inGsuch thatL(H)is isomor- phic tohandH is the projective limit of(Hi, δ_i^j_|H

j)j≥i.

Remark 6. The reader will also find an application of Theorem 2 in the proof of The- orem 8.23 on submersive projective limit of a projective sequence of Banach groupoids in

8here [,]idenote again the restriction to sections ofHiof the Lie bracket of vector fields onGi

[4]which is a kind of generalization of Theorem 5 to Lie groupoids setting .

3.2. Application to sequences of prolongations of a Banach-Lie algebroid over a Banach manifold. Consider an anchored Banach bundle (A, π, M, ρ) with typical fiber A. LetVA ⊂ TA be the vertical subbundle ofpA : TA → A. If Ax := π⁻¹(x) is the fiber overx∈M, according to [3], the prolongationTAof the anchored Banach bundle (A, π, M, ρ) overAis the set{(x, a, b, c),(x, b)∈ Ax, (x, a, c)∈V(x,a)A}. It is a Banach vector bundle ˆp:TA → Awith typical fiberA×Aand we have an anchor ˆρ:TA →TA given by

ˆ

ρ(x, a, b, c) = (x, a, ρx(b), c)∈T(x,a)A.

From now on, we fix a Banach Lie algebroid (A, π, M, ρ,[., .]A) such that the typical fiberAof Ais finite dimensional. By the way, we have a Banach Lie algebroid structure (TA,p,ˆ A,ρ,ˆ[., .]TA) (cf. [3] Corollary 44 )

We denote (A1, π1,A0, ρ1,[., .]1) the Banach-Lie algebroid (A, π, M, ρ,[., .]A) over a Banach manifoldA⁰=M. Thus we have the following commutative diagram:

A1

ρ₁ //

π₁

!!

TA0 p_A0

||A0

(1)

According to the notations of Theorem 43 and Corollary 44 in [3], we set A2 = TA1, ρ2 = ˆρ, [., .]2 = [., .]TA₁ and π2 = ˆp. Then we have the following commuta- tive diagram:

A2

ρ₂ //

π₂

!!

TA1 p_A1

||

T π1

##A1

ρ₁ //

π₁

""

TA0

p_A

{{ 0

A0

(2)

Fix somex∈ A0 and a norm|| ||0 (resp.|| ||1) on the fibre TxA0 ≡A0 (resp. Ax = π⁻¹₁ (x)≡A¹). Since the fiberT(x,a)A1 (resp. T(x,a)A1) is isomorphic to A¹×A¹ (resp.

A0×A1), it follows that sup{|| ||1,|| ||1}(resp. sup{|| ||0,|| ||1}gives rise to a norm on T(x,a)A1 (resp. T(x,a)A1). Then for the associated operator norm|| ||^opwe have

||(ρ2)_(x,a)||^op≤sup(||(ρ1)x||^op,1) (3) By induction, fori≥1, again according to notations of Theorem 43 and Corollary 44 in [3] , we we set

Aⁱ⁺¹=T^AiAi,ρi+1= ˆρi, [., .]i+1= [., .]_TAiA_i and πi+1= ˆpand, as before, we have the following commutative diagrams:

Ai+1

ρ_i+1 //

π_i+1

""

TAi p_A

||

T π_i

$$Ai

ρ_i //

π_i

""

TAi−1 p_Ai−1

zzAi−1

(4)

Also, by same arguments as for (3) we obtain:

||(ρi+1)(x,a₁,...,a_i+1)||^op≤sup(||(ρi)(x,a₁,...,a_i)||^op,1) (5) It follows that we have a submersive projective sequence of Banach-Lie algebroids (Ai,Ai−1, ρi,[., .]i) over a submersive projective sequence of Banach manifolds (Ai)_i∈

N

which satisfies the assumptions of Theorem 2. Thus, we obtain:

Theorem 7. Under the previous context,

(A= lim←−^i≥1Ai,M= lim←−^j≥0Aj, ρ= lim←−^i≥1ρi,[., .] = lim←−^i≥1[., .]i)

is a Fr´echet Lie algebroid on the Fr´echet manifold M and the distribution ρ(A) is in- tegrable. Each leaf L is a projective limit of a projective sequence of leaves of type(Li) defined by induction in the following way:

L0 is a leaf ofρ1(A1)and if Li is a leaf ofρi(Ai)thenLi+1= (Ai)|L_i.

3.3. A contre-example. In this subsection we give a Example of an integrable distribu- tion on a Fr´echet bundle over a finite dimensional manifold which satisfies the assumptions (*) in Definition 1 but is a projective limit of a submersive sequence of Banachnot inte- grabledistributions.

LetE=M×R^mthe trivial bundle over a manifoldM of dimensionn. The setJ^k(E) of the k-jets of section of E over M is a finite dimensional manifold which is a vector bundleπ^k :J^k(E)→M and whose typical fiber is is the space

k

Y

j=0

L^jsym(Rⁿ,R^m) where L^jsym(Rⁿ,R^m) is the space of continuousj-linear symmetric mappingsRⁿ →R^m. Then each projectionπ^l_k:J^l(E)→J^k(E) defined, forl≥k, by

πk^l

h

j^l(s) (x)i

=j^k(s) (x) is a smooth surjection.

Proposition 8. ([4]) J^k(E), π_k^l

is a submersive projective sequence of Banach vector bundles and the projective limit

J^∞(E) = lim←−J^k(E)

can be endowed with a structure of Fr´echet vector bundle whose fibre is isomorphic to the Fr´echet space

∞

Y

j=0

L^jsym(Rⁿ,R^m).

Letsbe a section ofπon a neighbourhood U of x∈M. Forξ =j^k(s) (x)∈J^k(E), the n-dimensional subspaceR(s, x) of TξJ^k(E) equals to the tangent space atξ to the submanifoldj^k(s) (U)⊂J^k(π) is called anR-plane.

The Cartan subspaceC^k(E) ofTξJ^k(E) is the linear subspace spanned by allR-planes R(s⁰, x) such that j^k(s⁰) (x) = ξ. So it is the hull of the union of j^k(s)

∗(x) (TxM) wheresis any local section ofπaround dex.

The Cartan subspaces form a smooth distribution on J^k(π) called Cartan distribu- tionand denoted C^k. Then C^k is a regular distribution which a contact distribution and so is not integrable (cf. [14]). We have a submersive projective limit of bundle (C^k, T π_k^l, j^k(E)) whose projective limit C = lim←−C^k is called the Cartan distribution on J^∞(E). In factC is integrable (cf. [14]). Note that sinceC^k is a subbundle ofT J^k(E), from the proof of Theorem 4, the condition (3) of Definition 1 is satisfied.

4. Proof of Theorem 2

Fix somex∈M. According to the property (*) and the Definition 1, we can choose a submersive projective sequence of charts

Ui, δ^j_i|U

j

j≥i and a submersive projective sequence of Banach bundles (Ei, λ^j_i)j≥i, such that:

– U = lim←−Uiis an open neighbourhood ofxinM;

– Ifx= lim←−(xi), eachUi is the contractile domain of a chart (Ui, φi) aroundxiin Mi and (U = lim←−(Ui), φ= lim←−(φi)) is a projective limit chart inM aroundx.

– the projective sequence of Banach bundles (Ei, λ^j_i)j≥isatisfies the assumption (*) onU.

Step 1: The kernel ofρx is supplemented.

There exists a trivializationτi:Ei→Ui×Eⁱwhich satisfies the compatibility condition:

(δ^j_i×λ^j_i)◦τj=τi◦λ^j_i (6) where (Ei, λ^j_i)j≥i is the projective sequence of Banach spaces on which (Ei, λ^j_i)j≥i is modeled.

Under these conditions, without loss of generality we may assume that, for eachi∈N, we have

– xi≡0∈Mⁱ;

– Ui is an open subset ofMiand soT Ui=Ui×Mi; – Ei=Ui×Eⁱ.

The projectionδ^j_i at pointxj ≡0 (resp. λ^j_i in restriction to the fibre ofEj overxj≡0) is denoted d^j_i (resp. `^j_i). The morphismρi in restriction to the fibreEi over xi ≡0 is denoted ri and ρx is denoted r, so that r = lim←−ri. Now, according to the context of Assumption 1 in Definition 1, we have the following result:

Lemma 9. There exists a decompositionE= kerr⊕F⁰ with the following property:

if (νn⁰)(resp. (µn)is the graduation onF⁰ (resp. (µn) onM) induced by the norm || ||^Ei (resp. || ||^Mi) on(Ei)x_i (resp. Tx_iMi), then the restriction ofrtoF⁰ is a closed uniformly bounded operator according to these graduations⁹.

Proof of Lemma 9

At first, in such a context we haveTxjδ_i^j≡d^j_iand the following compatibility condition:

d^j_i◦rj=ri◦`^j_i. (7)

We set Fi = ri(Ei) for alli ∈ N. From Definition 1 assumption 1, Fi is a Banach subspace ofEⁱand there exists a decompositionEⁱ= kerri⊕F⁰i. Thus the restrictionri⁰

ofritoF⁰i is an isomorphism ontoFi. Now, from (7), we have

∀(i, j)∈N² :j > i, d^j_i◦r_j⁰ =r⁰_i◦`^j_i.

But since eachr_i⁰is an isomorphism, the restriction (`<sup>j</sup><sub>i</sub>)<sup>0</sup>of`^j_i toF⁰jtakes values inF⁰ifor all (i, j)∈N² such thatj≥i. Moreover, asδ^j_i is surjective, according to (7) again, this implies thatd^j_i(F^j) =Fⁱand so`<sup>j</sup><sub>i</sub>(F<sup>0</sup>j) =F<sup>0</sup>i. Since (E<sup>i</sup>, `^j_i)j≥iis a projective sequence, this implies that (F⁰i,(`^j_i)⁰)j≥i is a surjective projective system. The vector spaceF⁰ = lim←−F⁰i

is then a Fr´echet subspace ofE.

On the other hand, let (`<sup>j</sup><sub>i</sub>)<sup>00</sup>be the restriction of`^j_i to kerrj. Always from (7), we have

∀(i, j)∈N² :j > i, (`^j_i)⁰⁰(kerrj)⊂kerri.

9cf. Appendix B

By same argument, it implies that (kerri,(`^j_i)⁰⁰)j≥i is a projective sequence and kerr= lim←−kerri. Moreover, since Ei = kerri⊕F⁰i, it follows that E= kerr⊕F⁰ and also the restrictionr⁰ ofrtoF⁰is obtained asr⁰= lim←−r⁰_iandr⁰is an injective continuous operator r⁰:F⁰→Mwhose range is the closed subspaceF= lim←−Fi. It remains to show thatr⁰ is uniformly bounded.

According to the context of assumption 2 of Definition 1, there exists a constantC >0 and, for each i ∈ N, we have a normk k^Ei onEⁱ and a norm k k^Mi on Mⁱ such that krik^op_i ≤C. AsF⁰iis a closed Banach subspace ofEi, it follows that for the induced norm onF⁰i we have

∀i∈N, ||r⁰i||^op_i ≤C. (8) Set`i= lim←−`^j_ianddi= lim←−d^j_i . By construction we have`i(F<sup>0</sup>) =F<sup>0</sup>ianddi(M) =Mi. The normk k<sup>Ei</sup> onEiinduces a normk k<sup>F0i</sup> on the Banach subspaceF<sup>0</sup>i and we get a natural graduation (νi<sup>0</sup>) on F<sup>0</sup> given by νi<sup>0</sup>(u) = ||`i(u)||^F0i (cf. Appendix B (29)). In the same way, the normk k^Mi induces a graduation (µi) on Mgiven by µi(v) =||di(v)||^Mi. Now (8) implies thatr⁰is uniformly bounded andr⁰(F⁰) =r(E) = ∆xis closed by assumption.

Therefore, the proof of Lemma 9 is complete.

Step 2: There exists a neighbourhoodV ⊂U of xsuch that the map ρ⁰=ρ|U×F⁰ takes values inIHb(F⁰,M)¹⁰and isK-Lipschitz onV for some K >0.

Since Ei =Ui×Ei, it follows thatE =U ×Eand so ρ : E → T U can be seen as a smooth map from U into H(E,M). Let ρ⁰ be the restriction of ρ to U ×F⁰ and so considerρ⁰ as a smooth map fromU toH(F⁰,M). From the definition of a Finsler norm, Assumption 2 in Definition 1 and Lemma 9, the mapx7→ρ⁰_xtakes value inHb(F⁰,M).

Lemma 10. There exists a neighbourhood V1 ⊂ V of 0 such that the map ρ⁰ : V1 → Hb(F⁰,M)is Lipschitz, that is:

there existsK >0such that ˆ

µ^op_i (ρ⁰_z−ρ⁰_x)≤Kµˆi(z−z⁰), ∀(z, z⁰)∈V₁²

11

Proof of Lemma 10Let ( ˆν⁰i)i∈N (resp. (ˆµi)_i∈

N) be the canonical increasing graduation associated to the graduation (ν_i⁰)i∈N onM(resp. (ˆµi)_i∈

N) (cf. Appendix (B 29)). Since the map x 7→ ρ⁰_x is a smooth map from U to Hb(F⁰,M) it follows that for each x the differential mapdxρ⁰ is a continuous linear map fromMto the Banach space Hb(F⁰,M) and so there existsi0∈Nand a constantAx>0 such that

kdxρ⁰(u)k∞≤Axνˆ⁰i₀(u) (9) for allu∈F⁰ and so

kdxρ⁰(u)k∞≤Axνˆ⁰i(u) (10) for allu∈F⁰ and andi≥i0 and according to Remark 32 we set

||dxρ⁰||^op_i := sup{µˆi(dxρ⁰(u)) : ˆν⁰i(u)≤1} ≤Ax, ∀i≥i0 (11) On the other hand, for 1 ≤ i < i0 we set Cⁱ_x = ||dxρ⁰||^op_i . Then dxρ⁰ belongs to Hb(M,Hb(F⁰,M)) for allx∈U since we have

||dxρ⁰||∞:= sup

i∈N

||dxρ⁰||^op_i ≤sup{Ax, Cx¹,· · ·, Cxⁱ⁰⁻¹}.

10cf. Appendix B 11cf. Remark 32

We setC =||d0ρ⁰||∞. By continuity, there exists an open neighbourhoodV1 of 0 such that

||dxρ⁰||∞≤2C (12)

By choosingK= 2C, from the definition of||dxρ⁰||∞ it implies the announced result.

Step 3: Local flow of the vector fieldXu=ρ⁰(u).

Consider a neighbourhoodV as announced in step 2. Asδi is surjective,Vi =δi(V) is an open set ofUi and so we have V = lim←−Vi. For eachu ∈F⁰, let Xu =ρ⁰(u) be the vector field onV. Ifu= lim←−uiwithui∈F⁰i, thenXu_i =ρ⁰i(ui) is a vector field onViand Xu= lim←−Xu_i. Now sinceρ⁰takes values inIHb(F⁰,M), from our assumption, there exists a constantC >0 such that||(ρ⁰i)z_i||^op_i ≤C for allzi∈Vi. Therefore, ifubelongs to F⁰ andui=λ(u), thenkXu_ik^Mi ≤Ckuk^Ei and, from Lemma 10 and the definition of ˆµ^op_i we have

∀ xi, x⁰_i

∈(δi(V))², kXu_i(xi)−Xu_i(x⁰_i)k^Mi≤Kkuik^Ei||xi−x⁰_i||^Mi. (13) Now, recall that we have providedF⁰ andMwith seminorms ( ˆν⁰n) and (ˆµn) respectively defined by

νn(u) =kλi(u)k^Ei andµn(x) =kδi(x)k^Mi.

Sinceδi(Xu)(x) =Xu_i(δi(x)) and in this way, from (13), for all n∈N, we have

∀(x, x⁰)∈V², µn(Xu(x)−Xu(x⁰))≤Kνn(u)µn(x−x⁰). (14) Therefore,Xusatisfies the assumption of Corollary 37. Let >0 such that the pseudo-ball

B_M(0,2) ={x∈M, : ˆµn_i(x)<2,1≤i≤k}

is contained inV and set C1:= max

1≤i≤k{Kνn_i(u)}=K max

1≤i≤kνn_i(u);

C2:= sup

x∈B_M(0,)

1≤i≤kmaxµn_i(X(x))

≤C max

1≤i≤kνn_i(u).

By application of Corollary 37, for anyusuch that max

1≤i≤kνn_i(u)≤1, there existsα >0 such that

αe^2αK≤ 2C,

such that the local flow Fl^u_t is defined on the pseudo-ball B_M(0, ) for allt∈[−α, α] for alluwhich satisfied the previous inequality

Note that for anys∈Rwe haveXsu=sXu. Therefore, from the classical properties of a flow of a vector field,

there existsη >0 such that the local flow Fl^u_t is defined on [−1,1], for alluin the open pseudo-ball

B_F⁰(0, η) :=

u∈F, : ν⁰_ni(u)≤η, 1≤i≤k . (cf. for instance proof of Corollary 4.2 in [5]).

We setB_Mi(0, ) = δi(B_M(0, )). Then Xu_i is a vector field on Vi =δi(V) and Fl^u_tⁱ :=

δi◦(Fl^u_t)◦δiis the local flow of Xu_i which is defined onB_Mi(0, ) for allt∈[−1,1] and Fl^u_tⁱ(xi) belongs toVi for allxi∈B_Mi(0, ) andt∈[−1,1] and, from (15), we have

Fl^ut = lim←−Fl^u_tⁱ. (15)

Step 4: Existence of an integral manifold.

Since (Ei, πi, Ui, ρi,[., .]i) is a Banach-Lie algebroid, the Lie bracket [Xu_i, X_u⁰

i] is tangent to ∆i and so by Definition 3.2 and Lemma 3.6 in [18] we have

∀t∈[−1,1], (TFl^u_tⁱ)((∆i)x_i) = (∆i)_Flui t (x_i).

Therefore, according to the notations at the end of step 3, for eachi∈N, we set:

∀ui∈B_F⁰

i(0, η) =λi(B_F⁰(0, η)), Φi(ui) = (Fl^u₁ⁱ)(0);

∀u∈B_F⁰(0, η), Φ(u) = (Fl^u₁)(0);.

Lemma 11. Φ = lim←−Φi is smooth and there exists0< η⁰≤ηsuch that the restriction of ΦtoB_F0(0, η⁰)is injective and TuΦbelongs toIHb(F⁰,M) for allu∈B_F0(0, η⁰).

Recall that, for eachi∈N,B_F⁰

i(0, η⁰)) is an open ball inF⁰iand δi(V) is open neigh- bourhood of 0∈ Mⁱ. From Lemma 11, since Φ is injective and each differential TuΦ is injective, it follows that the same is true for each Φi:B_Fi(0, η⁰))→δi(V). Thus we can apply the proof of Theorem 1 in [18] for Φi. By the way,Ui= Φi(B_F⁰

i(0, η⁰)) is an integral manifold ofρi(F⁰i) and (Ui,Φ⁻¹_i ) is a (global) chart for this integral manifold modeled on F⁰i. As Φ = lim←−Φi, B_F0(0, η⁰) = lim←−B_F0

i(0, η⁰)) Φi◦λ^j_i =δ_i^j◦Φj for allj ≥i, it follows that (Ui, δ^j_i)j≥iis a surjective projective sequence and soU= lim←−Uiis a Fr´echet manifold modeled onF⁰. This last result clearlyends the proof of Theorem 2.

Proof of Lemma 11

According to step 3, Φi is well defined onB_F0

i(0, η) and Φ = lim

←−Φi.

Now, for everyx∈V,ρ⁰_x belongs toIHb(F⁰,M) and so is injective; after shrinkingV, if necessary, there exists someM >0 such that

∀x∈V, kρ⁰xk∞≤M. (16) Now, by construction, we haveT0Φ(u) =ρ⁰0(u) and soT0Φ is injective.

Claim 12. The mapu7→TuΦis a smooth map fromB⁰(0, η)toHb(F⁰,M).

Proof of Claim 12

We will use some argument of the proof of Lemma 2.12 of [18]. We fix the indexiand, for anyy∈B_F⁰

i(0, ),v∈F⁰iwe set : Xv(y) =ρ⁰i(y, v)

: ϕ(t, v) = Fl^X_t^v(0)

: A(t) =∂1ρ⁰_i(ϕ(t, v), v) (partial derivative relative to the first variable) : B(t) =ρ⁰i(ϕ(t, v), .)

Note thatA and B are smooth fields on [0,1] of operators inL(Mi,Mi) and L(F⁰i,Mi) respectively. Therefore the differential equation

S˙=A◦S+B

has a unique solutionStwith initial condition S0= 0 given by St=

Z t 0

Gt−s◦B(s)ds (17)

whereGtis the unique solution of

G˙ =A◦G with initial conditionG0= Id_M. Given by

Gt= Id_M+ Z t

0

A◦Gsds. (18)

Under these notations, from [6], Chapter X§7, we have∂2ϕ(t, v)(.) =St.

On the one hand, from the choice of,ϕ(t, v) belongs toViand so from (12), we have kA(t)k^op_i ≤K for anyt∈[−1,1] and from (16) we havekB(t)k^op_i ≤M. Thus from (18), using Gronwal equality, we obtain

kGtk^op_i ≤e^K, And so from (17) we obtain

kStk^op_i ≤M e^K This implies that

k∂2ϕ(t, v)k^op_i ≤M e^K. We setM1=M e^K. Since Φi(ui) =ϕ(1, ui) it follows that:

kTu_iΦ(vi)k^Mi ≤M1kvik^Ei (19) from (19) we obtain:

µi(TuΦ(v))≤M1νi(v). (20)

But, since TuΦ(F⁰) = ∆Φ(u) ⊂ {Φ(u)} ×M, it follows that the map u 7→ TuΦ can be considered as a continuous linear map fromF⁰toMwhich takes values inHb(F⁰,M). Now as each Φi is a smooth map formB_Fi(0, η⁰)) toδi(V) and Φ = lim←−Φi, this imply that Φ is a smooth map onB_F⁰(0, η⁰)) = lim←−B_F⁰

i(0, η⁰)) toV = lim←−δi(V) which ends the proof of the Claim.

End of the proof of Lemma 10.

At first, from Claim 12, the mapu7→TuΦ takes values in the Banach spaceHb(F⁰,M), as in step 2 forρ, we can show that this map is Lipschitz onB_F⁰(0, η) forηsmall enough. As T0Φ =ρ⁰0, from Proposition 34, it follows that, again forηsmall enough, TΦ is injective onB_F(0, η⁰), and we have (cf. (19) )

∀u∈B_F(0, η⁰), ∀v∈F⁰µi(TuΦ(v))≤M1νi(v) (21) using the fact that the range of TuΦ is always closed for u∈ B_F⁰(0, η⁰). Moreover, for u∈B_F⁰(0, η⁰), as for the relation (33) in the proof of Proposition 34, we obtain:

1

`u

νi(v)≤µi(TuΦ(v)))≤`u.νn(v) (22) for alli∈N, where`u=kTuΦk∞≤M1.

Finally we obtain:

∀i∈N, 1 M1

νi(v)≤µi(TuΦ(v))≤M1.νi(u) (23) Suppose that, for any 0≤η⁰ ≤η, the restriction of each Φi toB_F⁰

i(0, η) is not injective.

Consider any pair (u, v)∈[B_F0(0, η)]²such thatu6=vbut Φ(u) = Φ(v), we seth=v−u.

For anyα∈M^∗, we consider the smooth curvecα: [0,1]→Rdefined by:

cα(t) =< α,(Φ(u+th)−Φ(u))> . Of course, we have ˙cα(t) =< α, Tu+thΦ(h)>.

Denote by ]u, v[ the set of points{w=u+th, t∈]0,1[}. As we havecα(0) =cα(1) = 0, from Rolle’s Theorem, there existsuα∈]u, v[ such that

< α, δl(Tu_αΦ(h))>= 0 (24) Note that, for any t∈R, this relation is also true for any th. From our assumption, it follows that, for eachk∈N\ {0}, there existsuk andvkinB_F(0,^η_k) so thatuk6=vkbut with Φ(uk) = Φ(vk). So from the previous argument, for anyα∈M^∗, we have

< α,(Tu_α,kΦ(hk))>= 0 (25)

for someuα,k∈]uk, vk[ andhk=vk−uk. From (25), for anyi∈N, anyt∈Rand any αi∈M^∗i, ifα=δ_i^∗(αi), we have:

|< α, T0Φ(thk)>|=|< αi, δi([T0Φ−Tu_αΦ](thk))>|=|< αi,[T0Φi−Tδ_i(u_α)Φi](λi(thk))|

Denote byk.k^M∗i the canonical norm onM^∗i associated tok.k^Mi. Then from (25) and since u7→TuΦ isK1-Lipschitz onB⁰(0, η) (for some constantK1) we obtain:

|< α, T0Φ(thk)>| ≤(||αi||^M∗i.K1.||δi(uα)||^Mi||.||λi(thk)||^Ei

≤(||αi||^M∗i.K1.||λi(thk)||^Eiη

k. (26)

Sinceuk6=vk, there must exist at least one integeri∈Nsuch thatλi(hk)6= 0. Thus, by takingt= uk−vk

νi((uk−vk)) in (26), we may assumet= 1 and kλi(hk)k^Ei = 1 In this way, for this choice ofhk, we have a 1-form ¯βk,ion the linear space generated byλi(hk) inF⁰i

such that<β¯k,i, λi(hk)>= 1 and with (kβ¯k,ik^Ei)^∗= 1. From the Hahn-Banach Theorem, we can extend this linear form to a formβk,i ∈M^∗i such that < βk,i, λi(hk)>= 1 and (kβk,ik^Ei)^∗ = 1. But since eachT0Φ is injective, this implies that T0Φi is injective and so the adjoint T₀^∗Φi is surjective. This implies that there exists αk,i ∈ M^∗i such that T0^∗Φi(αk,i) =βk,i. Thus from (26) we obtain

1 =|< βk,i, λi(hk)>| =|< αk,i, T0Φi(λi(hk))>|

=|< δi^∗αk,i, T0Φ(hk)>| ≤ kαk,ik^M∗iK1

η k

(27) But, on the other hand since the operation ”adjoint” is an isometry, from (23) we have

1 M1

kαk,ik^M∗i ≤ kT₀^∗Φαk,ik^M∗i = 1 (28) which gives a contradiction with (27) forklarge enough.

AppendixA. Projective limits A.1. Projective limits of topological spaces.

Definition 13. A projective sequence of topological spaces is a sequence Xi, δ_i^j

(i,j)∈N², j≥i where

(PSTS 1): For alli∈N, Xi is a topological space;

(PSTS 2): For all(i, j)∈N² such thatj≥i,δ_i^j:Xj→Xi is a continuous map;

(PSTS 3): For alli∈N,δ_iⁱ=IdX_i;

(PSTS 4): For all(i, j, k)∈N³ such thatk≥j≥i,δ_i^j◦δ^kj =δ^ki. Notation 14. For the sake of simplicity, the projective sequence Xi, δ^j_i

(i,j)∈N², j≥i

will be denoted Xi, δ_i^j

j≥i. An element (xi)_i∈

Nof the productY

i∈N

Xiis called athreadif, for allj≥i,δ_i^j(xj) =xi. Definition 15. The setX = lim←−Xi of all threads, endowed with the finest topology for which all the projections δi:X →Xi are continuous, is called the projective limit of the sequence Xi, δ^j_i

j≥i.

A basis of the topology of X is constituted by the subsets (δi)⁻¹(Ui) where Ui is an open subset ofXi(and soδiis open wheneverδi is surjective).

Definition 16. Let Xi, δ_i^j

j≥iand Yi, γ_i^j

j≥i be two projective sequences whose respec- tive projective limits areX andY.

A sequence(fi)_i∈

N of continuous mappings fi:Xi→Yi, satisfying, for all(i, j)∈N², j≥i,the coherence condition

γ_i^j◦fj=fi◦δ_i^j is called a projective sequence of mappings.

The projective limit of this sequence is the mapping

f: X → Y

(xi)_i∈

N 7→ (fi(xi))_i∈

N

The mappingf is continuous if all thefi are continuous (cf. [1]).

A.2. Projective limits of Banach spaces. Consider a projective sequence Ei, δ^j_i

j≥i

of Banach spaces.

Remark 17. Since we have a countable sequence of Banach spaces, according to the properties of bonding maps, the sequence δ^j_i

(i,j)∈N², j≥iis well defined by the sequence of bonding maps δ_iⁱ⁺¹

i∈N.

A.3. Projective limits of differential maps. The following proposition (cf. [9], Lemma 1.2 and [4], Chapter 4) is essential

Proposition 18. Let Eⁱ, δ_i^j

j≥i be a projective sequence of Banach spaces whose projec- tive limit is the Fr´echet spaceF =lim←−Eⁱ and (fi:Eⁱ→Eⁱ)_i∈

N a projective sequence of differential maps whose projective limit isf= lim←−fi. Then the following conditions hold:

(1) f is smooth in the convenient sense (cf. [15]) (2) For allx= (xi)_i∈

N,dfx= lim←−(dfi)_x

i. (3) df= lim←−dfi.

A.4. Projective limits of Banach manifolds and Banach Lie groups.

Definition 19. [9] The projective sequence Mi, δ^j_i

j≥i is called projective sequence of Banach manifolds if

(PSBM 1): Mi is a manifold modelled on the Banach spaceMⁱ; (PSBM 2):

Mi, δ^j_i

j≥i is a projective sequence of Banach spaces;

(PSBM 3): For all x= (xi) ∈ M = lim←−Mi, there exists a projective sequence of local charts(Ui, ξi)_i∈

N such thatxi∈Ui where one has the relation ξi◦δ_i^j=δ_i^j◦ϕj;

(PSBM 4): U= lim←−Uiis a non empty open set inM.

Under the assumptions(PSBM 1)and(PSBM 2)in Definition 19, the assumptions (PSBM 3)] and (PSBM 4)aroundx∈M is calledthe projective limit chart property aroundx∈M and (U = lim←−Ui, φ= lim←−φi) is called aprojective limit chart.

The projective limitM = lim←−Mi has a structure of Fr´echet manifold modelled on the Fr´echet space M= lim←−Mi and is called a PLB-manifold. The differentiable structure is definedviathe charts (U, ϕ) whereϕ= lim←−ξi:U→(ξi(Ui))_i∈

N.

ϕ is a homeomorphism (projective limit of homeomorphisms) and the charts changings ψ◦ϕ⁻¹

|ϕ(U)= lim

←−

ψi◦(ξi)⁻¹

|ξ_i(U_i)

between open sets of Fr´echet spaces are smooth in the sense of convenient spaces.