On symplectic linearization of singular Lagrangian foliations
Eva Miranda Galcerán

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On symplectic linearization of singular Lagrangian foliations
Eva Miranda Galcer´n a

` Departament d’Algebra i Geometria Universitat de Barcelona Juny 2003

Mem`ria presentada per aspirar al grau de o Doctor en Matem`tiques per la Universitat a de Barcelona.

Certifico que la present mem`ria ha estat o realitzada per Eva Miranda Galcer´n i dia rigida per mi. Barcelona, 10 de Juny de 2003

Carlos Curr´s Bosch a

A mi madre...

“Pirata de Mar y cielo si no lo fui ya lo ser´, e si no rob´ la aurora de los mares, e si no la rob´, ya la robar´”. e e Rafael Alberti

Preface
In this thesis we study the problem of classification of symplectic structures in a neighbourhood of a singular compact orbit of a completely integrable system on a symplectic manifold (M 2n , Ω) for which the foliation determined by the moment map is generically Lagrangian. The foliation is determined by the orbits of the distribution generated by the symplectic gradients of the components of a proper moment map F : M 2n −→ Rn . We also assume that the singularity is non-degenerate in the Morse-Bott sense. Under these assumptions, we prove that any two symplectic structures for which this foliation is generically Lagrangian are equivalent in the following sense: there exists a diffeomorphism defined in a neighbourhood of a compact orbit preserving the foliation, fixing the singular orbit and sending one symplectic form to the other. In the case there exists a symplectic action of a compact Lie group preserving the moment map we prove that the diffeomorphism can be chosen to be G-equivariant. We also give an application of this result to contact geometry. We consider a contact manifold (M 2n+1 , α) for which the Reeb vector field admits n first integrals generically independent and commuting with respect to the Jacobi bracket. The horizontal parts of the contact vector fields associated to these n functions determine a foliation F. We consider the enlarged foliation F generated by this foliation and the Reeb vector field. The Reeb vector field is assumed to be the infinitesimal generator of an S 1 -action. We study the problem of classification of contact forms α in a neighbourhood of a singular orbit having the same Reeb i

ii vector field and for which F is Legendrian. Then under the assumption that the singular orbit is compact and non-degenerate in the Morse-Bott sense we prove that any two contact forms are equivalent. In other words, we show that there exists a diffeomorphism preserving the foliation F, fixing the singular orbit and taking one contact form to the other. In the case there exists a contact action of a compact Lie group preserving the functions and preserving the Reeb vector field this diffeomorphism can be chosen to be G-equivariant.

Acknowledgements
First of all, I would like to take this opportunity to thank my advisor Carlos Curr´s a Bosch for his help, endless patience and advice. In fact, the problem of studying singular Lagrangian foliations is a joint endeavour with him. I owe him lots of hours devoted to make me understand my mistakes and also a lot of “lunches” discussing some of the problems tackled in this thesis. I learned a lot from him. In the end, I hope the result fulfills his expectations on the topic. I would also like to thank Nguyen Tien Zung for his help and for being so inspiring. In fact, part of the results obtained in this thesis have been obtained jointly with him. It has been a great pleasure to collaborate with him. I also wish to thank Jaume Amor´s and Viktor Ginzburg for stimulating diso cussions about some contents in this thesis and for giving me another perspective on the topic. This is the time to thank my parents and Jordi. A very special thank must go to my mother who didn’t loose the faith on me even when I had already lost it. In particular, my mother has been all her life cheering me on my projects and I wish to thank her for that. I suppose I wouldn’t be here if I didn’t have such inspiring maths teachers ´ as I have had from the cradle. I am specially grateful to Angel P´rez, my maths e teacher at the High School for making me love mathematics so much and all the great teachers in the University. ` Although my graduate studies have been held at the Departament d’Algebra i iii

iv Geometria, I have also been working in the following maths Departments: Departament de Matem`tiques de la Universitat de Lleida, Departament de Matem`tica a a Aplicada i Telem`tica de la Universitat Polit`cnica de Catalunya, Departament de a e Matem`tica Aplicada I de la Universitat Polit`cnica de Catalunya. In those plaa e ces, I met not only some wonderful professionals but also some wonderful people to learn from. I also wish to thank my friends Marc Fransoy, Marco Castrill´n, Ferran Espuny o and Carlos Calvo for all their support. I remember having shared a lot of coffee breaks with Josep Maria Ribo, Susana Clara Lopez, Juan Carlos Naranjo and Frederic Gabern. Thanks for your advice. Thanks also to Jos´ Gil for helping me e with the typesetting problems associated to the thesis (thanks also for the fancyhdr package although I didn’t use it in the end).

Contents
Preface Acknowledgements Introduction Catalan abridged version /Resum en catal` a 0.1 0.2 0.3 i iii vii xix

Introducci´ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix o Resultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii 1 1 4 5 6 7 8 9

1 Differentiable linearization 1.1 1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 Hamiltonian vector fields and the Poisson bracket . . . . . . Completely integrable systems and regular Lagrangian foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orbit versus leaf . . . . . . . . . . . . . . . . . . . . . . . . Transversal linearization at a singular point . . . . . . . . . The linear model . . . . . . . . . . . . . . . . . . . . . . . .

The parametrized Morse lemma . . . . . . . . . . . . . . . . 13 Our notion of equivalent symplectic germs . . . . . . . . . . 15 v

vi 1.3 Differentiable equivalence in a finite normal covering

CONTENTS . . . . . . . . 16 21

2 Analytic tools and symplectic linearization in dimension 2 2.1 2.2 2.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Two special decompositions for functions . . . . . . . . . . . . . . . 22 Symplectic linearization in dimension 2 . . . . . . . . . . . . . . . . 30 37

3 Rank 1 singularities in dimension 4 3.1 3.2 3.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Recovering a Hamiltonian S 1 -action . . . . . . . . . . . . . . . . . . 40 Two proofs for theorem 3.1.1 3.3.1 3.3.2 . . . . . . . . . . . . . . . . . . . . . 46 First proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Second proof . . . . . . . . . . . . . . . . . . . . . . . . . . 49 53

4 Rank 0 singularities in dimension 4 4.1 4.2 4.3 4.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Strategy of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Three common lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 57 A common proposition . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.1 4.4.2 4.4.3 Proof of proposition 4.4.1 in the non-completely hyperbolic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Proof of proposition 4.4.1 in the completely hyperbolic cases 65 A normalization result . . . . . . . . . . . . . . . . . . . . . 67 First proof of proposition 4.5.1 . . . . . . . . . . . . . . . . . 70 The Bott-Weinstein connection and a geometrical proof of proposition 4.5.1 . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5

A special Hamiltonian for the non-completely hyperbolic cases . . . 69 4.5.1 4.5.2

4.6 4.7

A special Hamiltonian for the completely hyperbolic cases . . . . . 76 Two symplectic orthogonal distributions and symplectic linearization 77

CONTENTS 5 Higher dimensions 5.1 5.2

vii 81

Rank 0 foliations in any dimension . . . . . . . . . . . . . . . . . . 82 Rank k foliations in any dimension . . . . . . . . . . . . . . . . . . 92 97

6 Equivariant linearization and symplectic equivalence 6.1 6.2 6.3 6.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 The linear action on the linear model . . . . . . . . . . . . . . . . . 101 G-linearization for rank 0 foliations . . . . . . . . . . . . . . . . . . 103 Linearization in the neighbourhood of an orbit . . . . . . . . . . . . 115 6.4.1 6.4.2 The classical slice theorem . . . . . . . . . . . . . . . . . . . 118 The slice statement of the linearization . . . . . . . . . . . . 120

6.5

Equivariant symplectic equivalence . . . . . . . . . . . . . . . . . . 121 123

7 Contact linearization of singular Legendrian foliations 7.1 7.2 7.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Basics in contact geometry . . . . . . . . . . . . . . . . . . . . . . . 126 The foliation and its differentiable linearization . . . . . . . . . . . 128 7.3.1 7.3.2 Posing the problem . . . . . . . . . . . . . . . . . . . . . . . 128 Differentiable linearization . . . . . . . . . . . . . . . . . . . 130 . . . . . . . . . . . . . . . . . . . 139 144

7.4 7.5

Contact linearization in the model manifold . . . . . . . . . . . . . 136 Equivariant contact linearization

Bibliography

viii

CONTENTS

Introduction
“Around the new position, a circle, somewhat larger than in the former instance was now described, and we again set to work with the spades. I was dreadfully, weary, but scarcely understanding what had occasioned the change in my thoughts..... I dug eagerly, and now and then caught myself actually looking with something that very much resembled expectation for the fancied treasure.”

“The Gold Bug” by Edgar Allan Poe

This thesis is mainly concerned with the geometry underlying a completely integrable Hamiltonian system. A Hamiltonian system on a symplectic manifold (M 2n , ω) is the system defined by the symplectic gradient of a function H which is called the Hamiltonian function of the system. The study of the integrability of such systems is relevant in many areas of mathematics and has its own story. In June 29th of 1853 Joseph Liouville presented a communication entitled “Sur l’int´gration des equations diff´rentielles de la Dynamique” at the “Bureau des e e longitudes”. In the resulting note [40] he relates the notion of integrability of the system to the existence of n integrals in involution with respect to the Poisson bracket attached to the symplectic form. These systems come to the scene with the classical denomination of “completely integrable systems”. In another language, a particular choice of n-first integrals in involution determines the n components of a moment map F : M 2n −→ Rn . A lot of work has been done in the subject after ix

x

Introduction

Liouville. Let us outline some of the remarkable achievements from a geometrical and topological point of view. Consider a completely integrable Hamiltonian system. The symplectic gradients of the components of the moment map define an involutive distribution. Assume that the moment map is proper. Let L be a regular orbit of this distribution then this orbit is a Lagrangian submanifold. Moreover, it is a torus and the neighbouring orbits are also tori. Those tori are called Liouville tori. This is the topological contribution of a theorem which has been known in the literature as Arnold-Liouville theorem. The geometrical contribution of the above-mentioned theorem ensures the existence of symplectic normal forms in the neighbourhood of a compact regular orbit. To the author’s knowledge, the works of Henri Mineur [44, 45, 46] already gave the a complete description of the Hamiltonian system in a neighbourhood of a compact regular orbit. That is why we will refer to the classical Arnold-Liouville theorem as Liouville-Mineur-Arnold theorem. Let us state the theorem below, Theorem 0.0.1 (Liouville-Mineur-Arnold Theorem) Let (M 2n , ω) be a symplectic manifold and let F : M 2n −→ Rn be a proper moment map. Assume that the components fi of F are pairwise in involution with respect to the Poisson bracket associated to ω and that df1 ∧ · · · ∧ dfn = 0 on a dense set. Let N = F −1 (c), c ∈ Rn be a connected levelset. Then there exists a neighbourhood U (N ) of N and a diffeomorphism φ : U (N ) −→ Dn × Tn such that, 1. φ(N ) = {0} × Tn . 2. A set of coordinates µi in Dn and a set of coordinates βi in Tn for which, φ∗ (
n i=1

dµi ∧ dβi ) = ω.

3. F depends only on φ∗ (µi ) = pi and it does not depend on φ∗ (βi ) = θi . The new coordinates pi obtained are called action coordinates. The coordinates θi are called angle coordinates. Mineur also showed that the action functions pi

Introduction

xi

can be defined via the period integrals. Let x be a point in a small neighbourhood of N , the period integrals are defined by the following formula: pi (x) =
Γi (x)

α

(0.0.1)

where α fulfills the condition dα = ω, and Γi (x) is a closed curve which depends smoothly on x and which lies on the Liouville torus containing x. The homology classes of Γ1 (x), ..., Γn (x) form a basis of the first homology group of the Liouville torus. The existence of action-angle coordinates in a neighbourhood of a compact orbit provides a symplectic model for the Lagrangian foliation F determined by the symplectic gradients of the n component functions fi of the moment map F . In fact, Liouville-Mineur-Arnold theorem entails a “uniqueness” result for the symplectic structures making F into a Lagrangian foliation. In other words, if ω1 and ω2 are two symplectic structures defined in a neighbourhood of N for which F is Lagrangian then there exists a symplectomorphism preserving the foliation, fixing N and carrying ω1 to ω2 . This is due to the following observation: Let Xfi be the symplectic gradients of the functions fi for any 1 ≤ i ≤ n, then the Lagrangian condition implies that in fact F =< Xf1 , . . . , Xfn >, further {fj , fk }i = 0 where {., .}i stands for the Poisson bracket attached to ωi , i = 1, 2 . Then by virtue of Liouville-Mineur-Arnold theorem there exists a foliation-preserving symplectomorphism φi taking ωi to ω0 = φ−1 2
n i=1

dpi ∧ dθi . In all, the diffeomorphism

◦ φ1 does the job. It takes ω1 to ω2 , it fixes N and it is foliation preserving.

So if the orbit is regular the existence of action-angle coordinates enables to classify the symplectic germs, up to foliation-preserving symplectomorphism, for which F is Lagrangian in a neighbourhood of a compact orbit. There is just one class of symplectic germs for which the foliation is Lagrangian. One could look at the problem from a global perspective. There are topological obstructions to the existence of global action-angle coordinates as it was shown by Duistermaat in [22].

xii

Introduction The problem of classification of symplectic germs for regular Lagrangian folia-

tions can be taken further to consider the case of foliations not necessarily determined by a completely integrable system. Curras-Bosch and Molino have considered the following concomitant problem: They consider the problem of classification for germs of Lagrangian foliation defined in a neighbourhood of an torus equipped with an affine structure. The motivation for considering an affine structure on the torus is the Bott-Weinstein connection attached to the regular leaves of a Lagrangian foliation [59]. In the case the germ of Lagrangian foliation is determined by a completely integrable system this affine structure is trivial. In the above mentioned papers it is proved that there is no uniqueness result for the symplectic germ if the affine structure on the torus is non-trivial. After this review for regular Lagrangian foliations, the following question arises: What can be said about the corresponding classification problem for symplectic germs if the completely integrable systems has singularities? This question is quite natural because singularities are present in many wellknown examples of integrable systems. In fact, if the completely integrable system is defined on a compact manifold then the singularities cannot be avoided. One of the main goals of this thesis is to prove that the uniqueness result for symplectic germs for which the foliation determined by a completely integrable system is generically Lagrangian holds when L is a singular orbit. In the singular case, the problem can be posed at three different levels: 1. At the orbit level: In the neighbourhood of a compact singular orbit. 2. At a semi-local level: In the neighbourhood of a compact singular leaf. 3. At a global level. Throughout this thesis we will only deal with the first situation. We will always

Introduction

xiii

assume that the singularity is non-degenerate. In any case, let us say a few words about the semi-local and global problem first. The problem of topological classification of integrable Hamiltonian systems began with Fomenko [25] in some particular cases. Nguyen Tien Zung [61] studied the general case for the semi-local problem for non-degenerate singularities. It turns out that from a topological point of view we have a product-like description of the singularities in terms of the Williamson type. Nguyen Tien Zung also proved in [61] the existence of partial action-angle coordinates. The symplectic classification in the semi-local case for non-elliptic singularities has been studied in the hyperbolic case by Dufour, Molino and Toulet in [20]. The focus-focus case has been studied recently by San Vu Ngoc in [58]. In the hyperbolic and focus-focus case there are more invariants attached to the singularity. The symplectic germ in the hyperbolic case is determined by the jet of a function depending on a variable and in the focus-focus case is determined by the jet of a function in two variables. The singular global case has been studied by Nguyen Tien Zung in the paper [63] where the notion of Duistermaat-Chern class and monodromy (introduced by Duistermaat for regular foliations) is extended in order to include the singularities into the picture. The condition of non-degeneracy is always present in the works cited above. There are also some contributions for degenerate singularities in the world of integrable systems. A recent contribution in that direction is contained in the paper [7] by Colin de Verdi`re. In that paper, among other things, the problem of clase sification of germs of singular Lagrangian manifolds is posed for more general singularities with a special emphasis on quasi-homogeneous singularities. For instance in this paper an explicit classification is obtained in the case of the cusp. The singular achievements formerly specified often have a semiclassical version. Their semiclassical counterpart has been obtained by Colin de Verdi`re and San e Vu Ngoc in [8, 56, 57, 7].

xiv

Introduction After this digression we will focus on the orbit-like case. The main goal of this

thesis is to study problems of classification in the neighbourhood of an orbit. The singularity of the orbit can be described in terms of the singularity of the functions fi . Let us start with the case L is reduced to a point. Observe that the Poisson bracket induces a Lie algebra structure in the set of functions. Since the functions fi are in involution with respect to the Poisson bracket, the quadratic parts of the functions fi commute defining in this way an abelian subalgebra of Q(2n, R) (the set of quadratic forms on 2n-variables). In the case the singularity of the functions fi is of Morse type this subalgebra is indeed a Cartan subalgebra. We call these singularities of non-degenerate type. The problem of classification of singularities for the quadratic parts of the functions fi can be therefore converted into the problem of classification of Cartan subalgebras of Q(2n, R). The singularities for the quadratic parts are well-understood thanks to a result of Williamson [60] where Cartan subalgebras of Q(2n, R) are classified. Let us recall its precise statement, Theorem 0.0.2 (Williamson) For any Cartan subalgebra C of Q(2n, R) there is a symplectic system of coordinates (x1 , . . . , xn , y1 , . . . , yn ) in R2n and a basis f1 , . . . , fn of C such that each fi is one of the following:
2 fi = x2 + yi i

for 1 ≤ i ≤ ke ,

(elliptic) (hyperbolic)

f i = xi y i for ke + 1 ≤ i ≤ ke + kh ,   fi = xi yi+1 − xi+1 yi , (focus-focus pair)  fi+1 = xi yi + xi+1 yi+1 for

i = ke + kh + 2j − 1, 1 ≤ j ≤ kf (0.0.2)

The linear system given by the quadratic parts of the fi is called the linear model for a singularity. Williamson’s Theorem can be seen as a normal form theorem

Introduction for the linear model.

xv

We may attach a triple of natural numbers (ke , kh , kf ) to a non-degenerate singularity p of F , where ke stand for the number of elliptic components in the linear model, kh and kf the number of hyperbolic and focus-focus components in the linear model respectively. By virtue of Williamson theorem this triple is an invariant of the linear system. That is why this triple is often called the Williamson type of the singularity. Now that the classification in the linear model has been carried out a natural question arises: Can we linearize the completely integrable system symplectically in a neighbourhood of a point p? We can reformulate the question as follows, Problem 1 Consider a foliation F defined by a completely integrable system defined in a neighbourhood of a non-degenerate singular 0-dimensional orbit of F. Assume that we are given two symplectic forms ω1 and ω2 for which the foliation F is Lagrangian. Does there exist a local diffeomorphism fixing p and taking ω1 to ω2 ? This problem of symplectic linearization is closely related to another problem in the spirit of Morse lemma which was solved succesfully by Vey for analytic systems and by Vey and Colin de Verdi`re for smooth systems. e Problem 2 Given a function f : Rn −→ R with a non-degenerate singularity at the origin and let ω be a volume form on Rn and let Q be its quadratic part at the origin. Does there exist a diffeomorphism φ : (Rn , 0) −→ (Rn , 0) such that φ∗ (f ) = Q and such that ω is taken to the volume form ω0 = dx1 ∧ · · · ∧ dxn ?

xvi

Introduction In [6] Colin de Verdi`re and Vey prove that there exists a smooth function χ e

such that φ∗ (ω) = χ(Q) · ω0 . In that paper it is also proved that the function χ is characteristic of the pair (f, ω) if Q is definite, otherwise only the jet is characteristic for the pair. As a corollary of this result we obtain normal forms for foliations defined by the levelsets of f because we can find a foliation-preserving diffeomorphism sending the volume form χ(Q)·ω0 to the volume form ω0 as was observed in the paper cited above. Notice as well that this result provides an affirmative answer to Problem 1 in the case n = 2 because a volume form on a 2-dimensional manifold is a symplectic form and the Lagrangian condition for a curve is automatic in that dimension. The affirmative answer to Problem 1 in any dimension was provided by Eliasson in [23] and [24]. As a matter of fact the proof provided by Eliasson seems complete just in the case the singularity is completely elliptic ( of Williamson type (ke , 0, 0)). In this thesis we will give another proof of Eliasson theorem with all the details for singularities whose Williamson type is (ke , kh , 0). We will also sketch a proof for the focus-focus components. Observe that Eliasson’s theorem can be seen as a symplectic linearization result which ensures that the initial completely integrable system can be taken to the linear system and that the symplectic form can be taken to the standard one. As a byproduct we obtain a multiple differentiable linearization result for n commuting vector fields with singularities of non-degenerate type. The symplectic linearization in a neighbourhood of an orbit L with dim L > 0 is due to Ito in the analytic case [32]. In this thesis we present the result in the smooth case. Partial results in that direction (with dim L = 1 in a manifold of dimension 4) where obtained by Curr´s-Bosch and the author of this thesis in a [13] and independently by Colin de Verdi`re and San Vu Ngoc in [8]. The final e result in any dimension was obtained by Nguyen Tien Zung and the author of this thesis in [48]. In [48] it is also included a G-equivariant version of the symplectic

Introduction linearization.

xvii

Symmetries are present in many physical problems and therefore they show up in integrable systems theory as well. Those symmetries are encoded in actions of Lie groups. A special emphasis has been given to Hamiltonian actions of tori in symplectic geometry. Along the way many results of symplectic uniqueness are obtained. A good example of this is Delzant’s theorem [19] which enables to recover information of a compact 2n-dimensional manifold by looking at the image of the moment map of a Hamiltonian torus action which is, surprisingly, a convex polytope in Rn . A lot of contributions in the area of Hamiltonian actions of Lie groups have been done ever since. Let us mention some of the references of the large list of results in that direction: the works of Lerman and Tolman to extend those result to symplectic orbifolds ([37]) and the works of Karshon and Tolman for complexity one Hamiltonian group actions ([33], [34]). In this thesis actions of compact Lie group are also considered. We assume also that the group acts symplectically and preserves the moment map which is underlying in the foliation. We end up proving the equivariant version of the symplectic uniqueness result in a neighbourhood of a singular compact orbit. A nice consequence is the abelianity of the group of symplectomorphisms preserving the moment map. In particular, in the case the action of the group is effective then this group is Abelian, in all, since it is also compact it is a product of a torus with a finite group. In the end, in the case the group is connected we recover actions by tori in the spirit of the theorem of Delzant. Loosely speaking, the odd-dimensional counterpart of the theorems obtained would be considered in the contact case. That is we can consider foliations on a contact manifold as close as possible to the ones described by completely integrable systems on symplectic manifolds. The regular case started with Lutz ([41]) who

xviii

Introduction

studies the problem of classification for contact structures in a compact contact manifold under the constraint that they are invariant under the action of a torus. This problem is naturally linked with the analogous problem for symplectic manifolds exposed above. Recent contributions to that problem in the setting of contact orbifolds are due to Lerman [36] where a convexity result is also established. This problem has been considered by Molino and Banyaga in [3] and [4] also for singular foliations. The common property of the foliations considered by Lutz, Lerman, Molino and Banyaga is that their orbits are given by a torus action. In this thesis we prove a similar result in the neighbourhood of a compact orbit but for foliations whose orbits are not necessarily given by a torus action but fulfill hypothesis of non-degeneracy. The foliation is determined as the enlarged foliation of a Legendrian foliation described by the horizontal parts of contact vector fields together with a Reeb vector field. We also assume that the Reeb vector field is the infinitesimal generator of an S 1 -action. We study the problem of classification for Legendrian foliations under the assumption that the contact form has the same Reeb vector field. This assumption is a bit constraining. The natural generalization of this result would be to study the problem of classification under the less-constraining assumption that the Reeb vector field belongs to the enlarged foliation instead. This result has been left in the pipeline and it is not included in this thesis. It uses an adaptation of Gray’s path method in contact geometry adapted to foliations. Organization of this thesis: In Chapter 1 we make a review of the differentiable linearization result ( theorem 1.3.1) for the foliations considered. We provide our own proof for the corank 1 case. This differentiable linearization allows to work in a linear model in the covering. In Chapter 2 the analytic tools necessary to face the problem are developed. We also present our own proof for the symplectic linearization in dimension 2.

Introduction

xix

In Chapter 3 we study the corank 1 case in dimension 4. We present two proofs for the symplectic uniqueness. One of the proofs is based on the construction of a symplectic orthogonal decomposition to reduce the problem to a 2 dimensional case. The tecniques of decomposition of functions introduced in chapter 2 are used to construct the symplectic orthogonal decomposition. In Chapter 4 we study the rank 0 case in dimension 4. We prove the symplectic uniqueness again using the geometrical tecniques of symplectic orthogonal decomposition. In the construction of the symplectically orthogonal distributions we use Moser tecniques and geometrical tricks relying on the Bott-Weinstein connection. In Chapter 5 we use induction, Liouville-Mineur-Arnold Theorem and the results obtained in the previous chapters to prove the general case in any rank and in any dimension. In Chapter 6 we present the equivariant version of the symplectic uniqueness attained in Chapter 5. This equivariant version allows to conclude the symplectic linearization in a neighbourhood of the initial compact orbit considered. We also present a slice statement of the equivariant symplectic linearization result in the neighbourhood of an orbit. Finally, in Chapter 7 we consider the contact case and prove the contact linearization result in the covering. We also present the G-equivariant contact version of the theorem which yields in particular the contact linearization in the initial neighbourhood considered. Part of the results contained in this thesis are contained in the publications and preprints that we cite below, • Publications: 1. C. Curr´s-Bosch and E. Miranda, Symplectic linearization of singular a Lagrangian foliations in M 4 , Differential Geom. Appl. 18 (2003), no. 2, 195-205.

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Introduction 2. E. Miranda, On the symplectic classification of singular Lagrangian foliations. Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltr´, 2000), 239–244, Publ. R. Soc. Mat. Esp., 3, R. u Soc. Mat. Esp., Madrid, 2001. • Preprints: 1. E. Miranda and Nguyen Tien Zung, Equivariant normal forms for nondegenerate singular orbits of integrable Hamiltonian systems, preprint 2003, http://xxx.arxiv.org/abs/math.SG/0302287. 2. C. Curr´s-Bosch and E. Miranda, Symplectic germs of singular Lagrana gian Foliations, preprint 265 de la Facultat de Matem`tiques. Univera sitat de Barcelona, 1999.

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0.1 Introducci´ o

Objectius de la tesi
L’objectiu d’aquesta tesi ´s estudiar dos problemes de classificaci´ de foliacions. El e o primer problema es planteja per a foliacions definides per sistemes completament integrables a varietats simpl`ctiques. El segon problema es planteja en l’`mbit de e a les varietats de contacte per a foliacions tamb´ de tipus completament integrable e la definici´ de les quals est` fortament inspirada en el cas simpl`ctic. o a e Tot seguit anem a establir els objectius amb precisi´ que seran tractats en o aquesta tesi. Primer estudiarem el problema de classificaci´ d’estructures simpl`ctiques deo e finides a un entorn d’una `rbita compacta d’un sistema completament integrable o per les quals la foliaci´ definida per l’aplicaci´ moment ´s gen`ricament Lagrangio o e e ana. Quan diem que una foliaci´ amb singularitats ´s gen`ricament Lagrangiana o e e volem dir que les fulles regulars s´n subvarietats Lagrangianes. Per continu¨ o ıtat, les fulles singulars (de dimensi´ inferior a la meitat de la dimensi´ de la varietat) o o s´n subvarietats is`tropes. Al llarg de tota la tesi treballarem a nivell de germes. o o ´ Es a dir tots els objectes es consideren definits a un entorn tubular de l’`rbita o compacta. Suposem que l’entorn considerat ´s una varietat simpl`ctica. Sigui Ω e e una forma simpl`ctica fixada inicialment a l’entorn. La foliaci´ est` definida de la e o a xxi

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seg¨ent manera: ´s la foliaci´ determinada per les `rbites de la distribuci´ generau e o o o da pels gradients simpl`ctics respecte de la forma simpl`ctica Ω de les components e e de l’aplicaci´ moment. La forma simpl`ctica inicial Ω nom´s ´s necess`ria per a o e e e a definir la foliaci´. El tipus de singularitats que considerarem s´n no degenerades o o en el sentit de Morse-Bott. Sota aquestes hip`tesis demostrem que qualssevol dues o estructures simpl`ctiques per a les quals la foliaci´ ´s gen`ricament Lagrangiana e oe e s´n equivalents. La noci´ d’equival`ncia per al problema de classificaci´ plantejat o o e o ´s el seg¨ent: Dues formes simpl`ctiques definides a un entorn de la fulla s´n equie u e o valents si existeix un simplectomorfisme preservant la foliaci´ que envia una forma o simpl`ctica a l’altra i que fixa l’`rbita singular. En el cas que existeixi una ace o ci´ d’un grup de Lie compacte preservant l’aplicaci´ moment provem que aquesta o o equival`ncia ´s G-equivariant. e e El segon problema que ens plantegem ´s un problema de classificaci´ per a e o formes de contacte. Considerem una varietat de contacte (M 2n+1 , α) que compleix les seg¨ents hip`tesis: u o 1. El camp de Reeb ´s el generador infinitesimal d’una acci´ del grup de Lie e o S 1. 2. El camp de Reeb admet n integrals primeres fi funcionalment independents en un conjunt dens. 3. Les integrals primeres commuten respecte del par`ntesi de Jacobi. e En aquesta varietat de contacte hi considerem dues foliacions: la foliaci´ F o definida per les parts horitzontals dels camps de contacte associats a les n integrals primeres fi considerades i la foliaci´ F definida com la foliaci´ generada per F o o conjuntament amb el camp de Reeb. Un cop definida la foliaci´ anem a plantejar el problema de classificaci´: Volem o o classificar les formes de contacte α definides a un entorn d’una `rbita compacta o

0.1. Introducci´ o

xxiii

singular de la foliaci´ F que tenen el mateix camp de Reeb i per a les quals la o foliaci´ F ´s Legendriana. Sota la hip`tesis de que la singularitat sigui no degeo e o nerada en el sentit de Morse-Bott provem que qualssevol dues formes de contacte ´ verificant les condicions anteriorment esmentades s´n equivalents. Es a dir, proo vem que existeix un difeomorfisme definit a un entorn de l’`rbita que envia una o forma de contacte a l’altra, preservant la foliaci´ F i fixant l’`rbita singular . En o o el cas en qu` existeixi una acci´ d’un grup de Lie compacte preservant la forma de e o contacte α provem que es pot trobar un difeomorfisme G-equivariant, aix´ doncs, ı la equival`ncia ´s G-equivariant. e e

Ubicaci´ del problema o
Aquesta tesi es centra en l’estudi de la geometria que est` encoberta als sistea mes Hamiltonians totalment integrables. Un sistema Hamiltoni` en una varietat a simpl`ctica (M 2n , ω) ´s el sistema definit pel gradient simpl`ctic d’una funci´ H e e e o anomenada funci´ Hamiltoniana del sistema. L’estudi de la integrabilitat d’aquests o sistemes ´s rellevant en moltes `rees de les matem`tiques i t´ la seva pr`pia hist`ria. e a a e o o El 29 de Juny de 1853 Joseph Liouville va presentar una comunicaci´ titulada o “Sur l’int´gration des equations diff´rentielles de la Dynamique”al “Bureau des e e longitudes”. A la nota resultant [40] es relaciona la noci´ d’integrabilitat del siso tema amb l’exist`ncia de n integrals primeres en involuci´ respecte el par`ntesi e o e de Poisson associat a la forma simpl`ctica. Aquests sistemes apareixen amb la e denominaci´ cl`ssica de “sistemes completament integrables”. En un altre lleno a guatge l’elecci´ de n integrals determina les components de l’aplicaci´ moment o o F : M 2n −→ Rn . Els treballs de Joseph Liouville constitueixen el punt de partida de tot un seguit de treballs posteriors. Anem a destacar algunes de les fites aconseguides en aquest terreny des d’un punt de vista geom`tric i topol`gic. e o Considerem, d’entrada, un sistema Hamiltoni` completament integrable en una a varietat simpl`ctica. Els gradients simpl`ctics de les components de l’aplicaci´ moe e o

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ment defineixen una distribuci´ involutiva. Suposem que aquesta aplicaci´ moment o o ´s pr`pia. Sigui L una `rbita regular d’aquesta distribuci´, la condici´ de complee o o o o ta integrabilitat implica que aquesta `rbita ´s una subvarietat Lagrangiana. A o e m´s a m´s es tracta d’un torus i les `rbites a un entorn d’aquesta s´n tamb´ torus. e e o o e Aquests torus es diuen torus de Liouville. Aquesta ´s la contribuci´ topol`gica d’un e o o teorema conegut com teorema d’Arnold-Liouville. L’aportaci´ geom`trica d’aquest o e teorema ´s l’exist`ncia de formes normals simpl`ctiques a un entorn d’una `rbita e e e o regular compacta. De fet sembla ser que els treballs de Henri Mineur [44, 45, 46] donaven una descripci´ del sistema Hamiltoni` en un entorn d’una `rbita como a o pacta regular. Per aquest motiu ens referirem al cl`ssic teorema d’Arnold-Lioville a com a teorema de Liouville-Mineur-Arnold. Recordem-ne l’enunciat, Teorema de Liouville-Mineur-Arnold Sigui (M 2n , ω) una varietat simpl`ctica i sigui F : M 2n −→ Rn una aplicaci´ e o moment. Suposem que les components fi de F estan en involuci´ dos a dos respecte o el par`ntesi de Poisson associat a ω i que df1 ∧ · · · ∧ dfn = 0 en un conjunt dens. e Sigui N = F −1 (c), c ∈ Rn un nivell connex de l’aplicaci´ moment. Llavors existeix o un entorn U (N ) de N i un difeomorfisme φ : U (N ) −→ Dn × Tn tal que, 1. φ(N ) = {0} × Tn . 2. Existeixen coordenades µi a un disc Dn i coordenades βi definides en un torus Tn tals que, φ∗ (
n i=1

dµi ∧ dβi ) = ω.

3. F nom´s dep`n de φ∗ (µi ) = pi i no dep`n de φ∗ (βi ) = θi . e e e

Les noves coordenades pi s’anomenen coordenades acci´. Les coordenades θi o s’anomenen coordenades angle. Mineur va donar la f`rmula de les integrals de o per´ ıode que s’utilitzen per a definir les coordenades d’acci´. Sigui x un punt a un o

0.1. Introducci´ o entorn de N , definim les integrals de per´ ıode mitjan¸ant la f`rmula: c o pi (x) =
Γi (x)

xxv

α

(0.1.1)

on α ´s una forma de Liouville per a la forma simpl`ctica, ´s a dir, ve donada per e e e la condici´ dα = ω i Γi (x) ´s una corba tancada que dep`n diferenciablement de x o e e i est` continguda en un torus de Liouville. Les classes d’homologia Γ1 (x), ..., Γn (x) a formen una base del primer grup d’homologia del torus de Liouville. L’exist`ncia de coordenades acci´-angle a un entorn d’una `rbita compacta doe o o nen un model simpl`ctic per a la foliaci´ Lagrangiana determinada per les `rbites e o o del gradient simpl`ctic de les n funcions components de l’aplicaci´ moment F . e o De fet, el teorema de Liouville-Mineur-Arnold porta impl´ un resultat d’uniıcit citat simpl`ctica, llevat de simplectomorfisme preservant la foliaci´, de formes e o simpl`ctiques que fan que F sigui Lagrangiana. e Dit d’una altra manera, si ω1 i ω2 s´n dues formes simpl`ctiques definides a o e un entorn de N per a les quals la foliaci´ F ´s Lagrangiana, llavors existeix un o e simplectomorfisme preservant la foliaci´, fixant N i enviant ω1 a ω2 . Aix` es degut o o
ω a la seg¨ent observaci´: Siguin Xfik els gradients simpl`ctics de les funcions fi resu o e

pecte ωk per a qualsevol 1 ≤ i ≤ n i k = 1, 2, llavors la condici´ de Lagrangianitat o
ωk implica que de fet es t´ la seg¨ent igualtat F =< Xf1 ωk , . . . , Xfn >, k = 1, 2, a e u

m´s a m´s {fi , fj }k = 0 on {., .}k ´s el par`ntesi de Poisson associat a ωk , k = 1, 2. e e e e Llavors, pel teorema de Liouville-Mineur-Arnold existeix un simplectomorfisme φk , preservant la foliaci´ i enviant la forma simpl`ctica ωk a ω0 = o e
n i=1

dpi ∧ dθi .

Finalment el difeomorfisme φ−1 ◦ φ1 envia ω1 a ω2 , fixa N i preserva la foliaci´. o 2 Per tant si l’`rbita ´s regular l’exist`ncia de coordenades acci´ angle permet o e e o classificar els germes simpl`ctics per als quals la foliaci´ F ´s Lagrangiana llevat e o e de simplectomorfisme preservant la foliaci´. Com acabem de comprovar nom´s o e existeix una classe de germes simpl`ctics pels quals la foliaci´ ´s Lagrangiana. e oe Podem mirar aquest problema des d’un punt de vista global. Existeixen obstruccions topol`giques a l’exist`ncia de coordenades acci´-angle global tal i com o e o

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El problema de classificaci´ de germes simpl`ctics de foliacions Lagrangianes es o e pot portar m´s enll` i considerar foliacions Lagrangianes regulars que no provenen, e a necess`riament, d’un sistema completament integrable. Curr´s-Bosch i Molino ([9, a a 10, 14, 15, 16]) han considerat el problema de classificaci´ per a germes de foliacions o Lagrangianes definides a un entorn d’un torus amb una estructura af´ fixada. La ı motivaci´ per a considerar una estructura af´ al torus ´s la connexi´ de Botto ı e o Weinstein associada a les fulles regulars d’una foliaci´ Lagrangiana [59]. o En el cas que el germe de foliaci´ Lagrangiana quedi determinat per un sistema o completament integrable aquesta estructura af´ ´s trivial. Als treballs esmentats ıe anteriorment es prova que no hi ha resultat d’unicitat per als germes simpl`ctics e en el cas que la estructura af´ no sigui trivial. ı Despr´s d’aquest rep`s de resultats per a foliacions Lagrangianes regulars ens e a plantegem la seg¨ent pregunta. u Qu` podem dir sobre el problema de classificaci´ de germes simpl`ctics si el e o e sistema completament integrable t´ singularitats? e Aquesta pregunta ´s bastant natural perqu` les singularitats s´n presents en e e o molts sistemes integrables coneguts. De fet si el sistema completament integrable est` definit en una varietat compacta les singularitats s´n inevitables. a o Un dels principals objectius d’aquesta tesi ´s provar que es t´ tamb´ un resule e e tat d’unicitat per a germes simpl`ctics per als quals la foliaci´ determinada pel e o sistema completament integrable ´s Lagrangiana a un entorn d’una `rbita singular e o compacta. En el cas singular, ens podem plantejar el problema a tres nivells diferents: o o 1. A nivell d’`rbita. En un entorn d’una `rbita singular compacta. 2. A nivell semilocal. En un entorn d’una fulla singular compacta.

0.1. Introducci´ o 3. A nivell global.

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En aquesta tesi nom´s ens preocuparem de la primera situaci´. e o ´ Es a dir, farem un estudi a un entorn d’una `rbita compacta singular que o suposarem singular no-degenerada en el sentit de Morse-Bott. Abans d’endinsarnos en l’estudi a un entorn de l’`rbita, anem a fer esment r`pidament d’alguns o a resultats destacables a nivell semilocal i global. El problema de classificaci´ topol`gica dels sistemes Hamiltonians completao o ment integrables va comen¸ar amb Fomenko [25] en alguns casos particulars. L’esc tudi de la topologia d’aquests sistemes a l’entorn d’una fulla no-degenerada en el cas general es deu a Nguyen Tien Zung [61]. La descripci´ de les singularitats o des d’un punt de vista topol`gic ´s de tipus producte de singularitats el·l´ o e ıptiques, hiperb`liques i focus-focus. El tipus de Williamson ´s per tant l’unic invariant too e pol`gic semi-local. En aquest treball, Nguyen Tien Zung tamb´ prova l’exist`ncia o e e de coordenades acci´-angle parcials. La classificaci´ simpl`ctica en el cas semi-local o o e per a singularitats de tipus no el·l´ ıptic ha estat estudiat per Dufour, Molino i Toulet [20], [53] en el cas hiperb`lic i per San Vu Ngoc en el cas focus-focus [58]. La o conclusi´ d’aquests treballs ´s que hi ha m´s invariants associats a la singularitat o e e caracteritzats per jets de funcions d’una variable (en el cas hiperb`lic) i pels jets o de funcitons de dues variables (en el cas focus-focus). El cas global singular va ´sser estudiat per Nguyen Tien Zung a [63]. En aquest treball s’est´n el concepte e e de classe de Duistermaat-Chern i el concepte de monodromia en el cas singular. La condici´ de no-degeneraci´ est` sempre present als treballs anteriorment citats. o o a Per` tamb´ cal destacar algunes contribucions en el camp de sistemes integrables o e amb singularitats de tipus degenerat. En aquesta direcci´ apunta el treball de Colin o de Verdi`re [7]. En aquest article, entre altres moltes coses es planteja el problema e de classificaci´ de germes de varietats Lagrangianes singulars per a singularitats de o tipus m´s general que les no degenerades, destacant especialment les singularitats e quasi-homog`nies. Per exemple, s’estudia el problema de classificaci´ associat al e o

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La majoria de resultats esmentats tenen la seva versi´ semicl`ssica. Els resultats o a en l’`mbit semicl`ssic han estat desenvolupats per Colin de Verdi`re i San Vu Ngoc a a e a [8, 56, 57, 7].

0.2

Resultats

L’objectiu principal d’aquesta tesi ´s estudiar problemes de classificaci´ en un e o entorn de l’`rbita. o Les singularitats queden descrites en termes de les singularitats de les components de l’aplicaci´ moment fi . o Comencem pel cas que L sigui redu¨ a un punt. Observem que el par`ntesi ıda e de Poisson indueix una estructura de `lgebra de Lie en el conjunt de funcions a diferenciables. Com que les funcions fi estan en involuci´ respecte el par`ntesi o e de Poisson, les parts quadr`tiques de les funcions fi commuten definint, d’aquesta a manera, una estructura d’`lgebra abeliana al conjunt de formes quadr`tiques en 2n a a variables que denotarem per Q(2n, R). En el cas que la singularitat de les funcions sigui de tipus Morse aquesta sub`lgebra ´s una sub`lgebra de Cartan. Aquestes a e a singularitats es diuen singularitats de tipus no degenerat. Per tant el problema de classificaci´ de singularitats per les parts quadr`tiques o a de les components de l’aplicaci´ moment s’ha convertit en un problema merament o algebraic: la classificaci´ de les sub`lgebres de Cartan de Q(2n, R). La classificaci´ o a o d’aquestes singularitats es deu al seg¨ent resultat de Williamson ([60]) u Teorema (Williamson) Donada una sub`lgebra de Cartan C de Q(2n, R) existeix un sistema simpl`ctic a e de coordenades (x1 , . . . , xn , y1 , . . . , yn ) a R2n i una base f1 , . . . , fn de C tal que cada fi ´s del seg¨ent tipus: e u

0.2. Resultats

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2 fi = x2 + yi i

si 1 ≤ i ≤ ke ,

(el·l´ ıptic) (hiperb`lic) o (0.2.1)

f i = xi y i si ke + 1 ≤ i ≤ ke + kh ,   fi = xi yi+1 − xi+1 yi , (parell focus-focus )  fi+1 = xi yi + xi+1 yi+1

si i = ke + kh + 2j − 1, 1 ≤ j ≤ kf

El sistema lineal donat per les parts quadr`tiques de les fi es diu model lineal a per a les singularitats. El teorema de Williamson es pot veure com un teorema de formes normals per al model lineal. Podem adjuntar un triplet de nombres naturals (ke , kh , kf ) a una singularitat no degenerada de F on ke ´s el nombre de e components el.l´ ıptiques al model lineal i kh i kf s´n el nombre de components o hiperb`liques i focus-focus respectivament. o Com a conseq¨`ncia del teorema de Williamson aquest triplet ´s un invariant ue e del sistema lineal. Per aquest motiu s’anomena tipus de Williamson de la singularitat. Ara que ja tenim la classificaci´ al model lineal la pregunta natural ´s: o e Podem linealitzar simpl`cticament un sistema completament integrable en un e entorn d’un punt singular p? Podem reformular la pregunta de la manera seg¨ent, u Problema 1 Considerem una foliaci´ F definida per un sistema completament integrable o definit en un entorn d’una `rbita singular no degenerada de dimensi´ 0 de F. o o Suposem que tenim donades dues formes simpl`ctiques ω1 i ω2 per a les quals la e foliaci´ F ´s Lagrangiana. Existeix un difeomorfisme local fixant p i portant ω1 a o e ω2 ? El problema de linealitzaci´ simpl`ctica est` ´ o e a ıntimament relacionat amb un

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altre problema en l’ordre d’idees del lema de Morse. L’altre problema ´s el seg¨ent: e u Problema 2 Donada una funci´ f : Rn −→ R amb una singularitat no degenerada a l’origen o i sigui ω una forma de volum a Rn . Notem per Q la seva part quadr`tica a l’origen. a Existeix un difeomorfisme φ : (Rn , 0) −→ (Rn , 0) tal que φ∗ (f ) = Q i enviant la forma de volum ω a la forma ω0 = dx1 ∧ · · · ∧ dxn ? A l’article [6] Colin de Verdi`re i Vey demostren que existeix una funci´ difee o renciable χ tal que φ∗ (ω) = χ(Q) · ω0 . A l’article esmentat anteriorment es prova que la funci´ χ caracteritza el parell o (f, ω) en el cas que la forma quadr`tica Q sigui definida, en cas contrari nom´s el a e jet de la funci´ caracteritza el parell. o Com a corol·lari d’aquest resultat obtenim formes normals per a foliacions definides pels nivells de f perqu` podem trobar un difeomorfisme preservant la e foliaci´ i enviant la forma de volum χ(Q) · ω0 a la forma de volum ω0 com s’observa o a la publicaci´ anteriorment citada. o Observem que aquest resultat d´na una resposta afirmativa al Problema 1 en el o cas n = 2 perqu` una forma de volum en una varietat de dimensi´ 2 ´s una forma e o e simpl`ctica i la condici´ de Lagrangianitat en el cas d’una corba ´s autom`tica en e o e a aquesta dimensi´. o La resposta afirmativa al Problema 1 en qualsevol dimensi´ ´s conseq¨`ncia del oe ue teorema d’Eliasson [23] i [24]. De fet, cal remarcar que la demostraci´ donada per o Eliasson ´s completa nom´s en cas qu` la singularitat sigui completament el·l´ e e e ıptica ( tipus de Williamson (ke , 0, 0)). En aquesta tesi donem una altra demostraci´ amb tots els detalls per a sino gularitats que tenen tipus de Williamson (ke , kh , 0). Tamb´ donem un esbo¸ de la e c demostraci´ en el cas d’existir components focus-focus. o

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Observem que el Teorema d’Eliasson es pot mirar com un resultat de linealitzaci´ simpl`ctica que assegura que el sistema completament integrable inicial es o e pot portar a un sistema lineal amb la forma simpl`ctica est`ndard. Com a resultat e a obtenim un resultat de linealitzaci´ m´ltiple per a n camps vectorials que commuo u ten i que tenen singularitats de tipus no degenerat. La linealitzaci´ simpl`ctica a o e un entorn de l’`rbita L en el cas dim L > 0 es degut a Ito en el cas anal´ [32]. o ıtic En aquesta tesi presentem el resultat en el cas diferenciable. Resultats parcials en aquesta direcci´ (en el cas en que la dimensi´ de l’`rbita singular sigui 1 en una o o o varietat de dimensi´ 4) v`ren ser obtinguts per Curr´s-Bosch conjuntament amb o a a l’autora d’aquesta tesi conjuntament ([13]) i independentment per Colin de Verdi`re i San Vu Ngoc [8]. El resultat final en qualsevol dimensi´ ha estat obtingut e o per Nguyen Tien Zung conjuntament amb l’autora d’aquesta tesi a [48]. En aquest paper tamb´ est` continguda la versi´ G-equivariant de la linealitzaci´ simpl`ctica. e a o o e L’estudi de simetries t´ rellev`ncia en molts problemes f´ e a ısics i, en consequ`ncia, e tamb´ apareix a la teoria de sistemes completament integrables. Aquestes simetries e queden codificades en forma d’accions diferenciables de grups de Lie. El cas d’accions Hamiltonianes de torus mereix especial atenci´. En aquest teoria apareixen o molts resultats d’unicitat simpl`ctica. El Teorema de Delzant n’´s un exemple clar. e e El Teorema de Delzant permet recuperar informaci´ en una varietat compacta de o dimensi´ 2n a partir de la imatge de l’aplicaci´ moment, que curiosament, ´s un o o e politop convex. S’han produit moltes contribucions en aquest camp darrerament. En destaquem dues: Els treballs de Lerman i Tolman per extendre aques resultat al cas d’orbifolds simpl`ctiques [37] i els treballs de Karshon i Tolman per a e generalitzar aquests resultats en el cas d’accions Hamiltonianes de complexitat 1. En aquesta tesi tamb´ considerem accions de grups de Lie compactes. Supoe sarem que el grup act´a simpl`cticament i preserva l’aplicaci´ moment que est` u e o a oculta a la foliaci´. En aquesta tesi demostrem la versi´ equivariant dels resulo o tats d’unicitat simpl`ctica en un entorn de l’`rbita compacta singular. Una consee o

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qu`ncia curiosa ´s que el grup de simplectomorfismes preservant l’aplicaci´ moment e e o ´s un grup abeli`. En particular, si l’acci´ del grup G ´s efectiva d’aquest resultat e a o e en podem extreure l’abelianitat de G. Donat que el grup G ´s compacte ´s un proe e ducte d’un torus per un grup finit. En el cas que el grup G sigui connex recuperem l’acci´ d’un torus en la l´ o ınia del Teorema de Delzant. La contrapartida dels anteriors resultats en dimensi´ imparella venen donats o pel cas de contacte. Aquest ´s el segon problema de classificaci´ plantejat a la e o tesi. Considerem foliacions en una varietat de contacte semblants a les donades per sistemes completament integrables en varietats simpl`ctiques. e La motivaci´ en el cas regular va ser donada per Lutz ([41]) que va estudiar el o problema de classificaci´ per a estructures de contacte en una varietat compacta o sota la hip`tesi que siguin invariants per l’acci´ d’un torus. o o Aquest problema est` lligat amb el problema an`leg per a varietats simpl`ctiques a a e que hem exposat abans. Cal destacar les seg¨ents contribucions recents en aquest u camp: en el cas d’orbifolds de contacte destaquem els treballs de Lerman [36] on es d´na un resultat de convexitat. Aquest problema ha estat considerat per Molino i o Banyaga a [3] i [4] en el cas de foliacions singulars. El denominador com´ de les fou liacions considerades per Lutz, Lerman, Molino i Banyaga ´s que les `rbites v´nen e o e donades per l’acci´ d’un torus. En aquesta tesi demostrem un resultat similar en o l’entorn d’una `rbita compacta en el cas que la foliaci´ no estigui necess`riament o o a donada per l’acci´ d’un torus. Les singularitats les suposem no degenerades i la o foliaci´ queda deteminada com la foliaci´ donada per les parts horitzontals del o o camps de contacte de n integrals primeres del camp de Reeb conjuntament amb el camp de Reeb. En aquesta tesi suposem que el camp de Reeb ve donat com a generador infinitesimal d’una acci´ del grup S 1 . Estudiem el problema de classifio caci´ de la foliaci´ Legendriana descrita per les parts horitzontals dels camps de o o contacte sota la hip`tesi que la forma de contacte tingui el mateix camp de Reeb. o Aquesta condici´ ´s una mica restrictiva. Una generalitzaci´ natural seria estudiar oe o

0.2. Resultats

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el problema m´s general en qu` el camp de Reeb pertanyi a la foliaci´. Aquest e e o resultat que requereix les t`cniques de Gray per deformaci´ de formes de contacte e o no est` incl`s a la tesi. a o

Organitzaci´ de la tesi o
Al cap´ ıtol 1 estudiem el problema de linealitzaci´ diferenciable en un recobrio ment de l’entorn inicialment considerat. Donem tamb´ una demostraci´ diferent e o a la donada per l’Eliasson en el cas de foliacions amb corang 1. Aquest resultat de linealitzaci´ diferenciable permet treballar en un model lineal al recobridor. En o aquest cap´ donem una demostraci´ del seg¨ent teorema en el cas de singulariıtol o u tats de corang 1. Teorema 1.3.1 A U (L) la foliaci´ Lagrangiana definida pels gradients simpl`ctics de l’aplicaci´ o e o moment ´s difeomorfa a la foliaci´ linealitzada. e o On notem per U (L) un recobriment finit d’un entorn de l’`rbita singular como pacta. L’objectiu del cap´ ıtol 2 ´s doble; per una banda s’introdueixen les eines e anal´ ıtiques necess`ries per resoldre el problema tamb´ es d´na una demostraci´ de a e o o la linealitzaci´ simpl`ctica en dimensi´ 2. o e o El primer objectiu d’aquest cap´ ´s provar que donada una funci´ diferenciıtol e o able g es pot trobar una descomposici´ del tipus, o g = g1 + X(g2 ) , X(g1 ) = 0 (0.2.2)

per a determinats tipus de camps singulars, els que corresponen als camps lineals de la foliaci´ (components el.l´ o ıtiques i hiperb`liques). Aquest tipus de descomposicio ons de funcions seran utils m´s endavant per a trobar deformacions de l’estructura ´ e

xxxiv simpl`ctica “` la Moser”preservant la foliaci´. e a o

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El cas el.liptic ja havia estat considerat per Eliasson. El cas hiperb`lic en dimeno si´ 2 tamb´. Donem demostracions pels casos el.liptics i hiperb`lics en qualsevol o e o dimensi´. Els resultats principals provats s´n els seg¨ents: o o u En el cas que X sigui un camp corresponent a una singularitat el.l´ ıptica, Proposici´ 2.2.1 o Sigui M una varietat diferenciable i sigui g un germe de funci´ diferenciable o en un entorn del punt p. Donat un camp X que en coordenades locals s’expressa
∂ ∂ X = x1 ∂x2 − x2 ∂x1 llavors existeixen funcions diferenciables g1 i g2 tals que:

g = g1 (x2 + x2 , x3 . . . , xn ) + X(g2 ) 1 2

En el cas que X sigui un camp corresponent a una singularitat hiperb`lica, o Proposici´ 2.2.2 o Sigui M una varietat diferenciable i sigui g un germe de funci´ diferenciable en o un entorn del punt p. Considerem un camp vectorial X que en coordenades locals
∂ ∂ s’expressa Y = −x1 ∂x1 + x2 ∂x2 llavors existeixen funcions diferenciables g1 i g2

tals que, g = g1 (x1 x2 , x3 . . . , xn ) + Y (g2 )

En el cas hiperb`lic aquest resultat est´n el resultat en el cas en dimensi´ 2 o e o provat per Colin de Verdi`re i Vey a [6]. El cas hiperb`lic ´s m´s complicat, cal e o e e redu¨ ırse primer al cas de funcions planes al llarg d’un subespai i despr´s usar e

0.2. Resultats

xxxv

t`cniques d’integraci´ semblants a les usades al Teorema de linealitzaci´ de Sterne o o berg. Usant aquests resultats anal´ ıtics en el cas 2 dimensional i t`cniques de deformae ci´ d’estructures simpl`ctiques usant camins “` la Moser”demostrem, per ultim, o e a ´ el resultat de linealitzaci´ simpl`ctica en dimensi´ 2, o e o Teorema 2.3.1 Sigui (M 2 , ω1 ) una varietat simpl`ctica 2-dimensional amb coordenades (x, y) e i sigui F una foliaci´ Lagrangiana amb singularitats de tipus el.l´ o ıptic o hiperb`lic o a l’origen (0, 0), llavors existeix un difeomorfisme local φ preservant la foliaci´ F o tal que φ∗ (dx ∧ dy) = ω1 . Aquest resultat de linealitzaci´ simpl`ctica en dimensi´ 2 es despr´n de [6] pero o e o e en donem la nostra pr`pia demostraci´. o o Al cap´ ıtol 3 estudiem el cas de corang 1 en dimensi´ 4. Donem dues demoso tracions de la unicitat simpl`ctica. e El punt clau per a provar la unicitat simpl`ctica rau en recuperar una acci´ e o Hamiltoniana de S 1 tangent a la foliaci´. Per tal d’aconseguir aquesta acci´ usem o o el Lema de Poincar´ i provem un lema tipus Moser de deformaci´ d’estructures e o simpl`ctiques preservant la foliaci´ que citem a continuaci´: e o o Lema 3.2.3 Sigui α una 1-forma, que s’anul.la a L, i que es F0 -b`sica i sigui ω1 una forma a
4 e simpl`ctica a M0 tal que F0 ´s Lagrangiana. Llavors: e

1. La 2-forma ωo = ω1 − dα es una estructura simpl`ctica en un entorn de L e per la qual la foliaci´ ´s Lagrangiana. oe
4 2. Existeix un difeomorfisme η entre dos entorns de L a M0 preservant F0 i tal

xxxvi que η ∗ (ω1 ) = ω0 .

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El resultat principal de la segona secci´ d’aquest cap´ ´s el seg¨ent, o ıtol e u Proposici´: o Existeix una acci´ Hamiltoniana de S 1 tangent a la foliaci´. De fet, existeixen o o coordenades (θ, p, x, y) en un entorn de L tal que ω = d(pdθ + C(p, x, y)dx + D(p, x, y)dy) i l’acci´ Hamiltoniana es produeix per translacions en la coordenada o θ. Un cop provada l’exist`ncia d’aquesta acci´ donem dues demostracions del e o seg¨ent teorema, u Teorema 3.1.1
4 Sigui M0 = S 1 × D3 , amb coordenades (θ, p, x, y). Sigui F0 la foliaci´ donada o

per: Y1 = Y2 = y ∈ {−1, 1} ( = 1 cas el.l´ptic, ı ∂ ∂θ

∂ ∂ − x ∂x ∂y

= −1 cas hiperb`lic). o

Sigui L = S 1 × (0, 0, 0). Llavors qualssevol dues formes simpl`ctiques ω1 i ω2 a e M 4 per a les quals F0 ´s Lagrangiana s´n equivalents. e o La primera demostraci´ implica un treball de deformaci´ de la forma simpl`ctica o o e pel m`tode del cam´ La segona demostraci´ usa l’exist`ncia d’una acci´ Hamiltonie ı. o e o ana per a contruir una descomposici´ ortogonal simpl`ctica (lema 3.3.2) utilitzada o e per a reduir el problema a dos problemes de classificaci´ 2-dimensional. o

0.2. Resultats

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Al Cap´ ıtol 4 estudiem el cas de rang 0 en dimensi´ 4. Donem un resultat o d’unicitat simpl`ctica usant les t`cniques geom`triques de descomposici´ ortogonal. e e e o Els resultats m´s importants d’aquest cap´ s´n els seg¨ents, e ıtol o u Teorema 4.2.1 (Descomposici´ simpl`ctica ortogonal) o e Sigui ω una forma simpl`ctica per la qual F ´s gen`ricament Lagrangiana. e e e Llavors existeix un germe simpl`ctic ω equivalent a ω i existeixen dues distribucions e simpl`ctiques D1 i D2 tals que, e o e 1. D1 i D2 s´n involutives i simpl`cticament ortogonals respecte ω. 2. X1, 1 ∈ D1 i X2, 2 ∈ D2 . Teorema 4.2.2 (Unicitat simpl`ctica) e Sigui ω una forma simpl`ctica en un entorn de p per a la qual F ´s gen`ricament e e e lagrangiana llavors ω ´s equivalent a ω0 = dx1 ∧ dy1 + dx2 ∧ dy2 . e Per a demostrar l’exist`ncia de la descomposici´ ortogonal recuperem accions e o Hamiltonianes tangents a les fulles usant el m`tode del cam´ Un dels resultats e ı. b`sics en la prova d’aquest teorema es la seg¨ent Proposici´: a u o Proposici´ 4.4.1 o Existeix un germe simpl`ctic ω 1 equivalent a ω tal que, e iX1, 1 ω 1 = H1 df1 + H2 df2 . per a funcions F-b`siques H1 i H2 . a Un cop demostrada aquesta proposici´ utilitzem t`cniques de normalitzaci´ per o e o a trobar camps Hamiltonians convenients, tangents a la foliaci´. La demostraci´ o o

xxxviii

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´s diferent en el cas que la foliaci´ contingui components el.l´ e o ıptiques o en el cas que la foliaci´ sigui completament hiperb`lica. Usem al llarg de la demostraci´ els o o o resultats de el cap´ 2 i el m`tode del cam´ per a deformar formes simpl`ctiques. ıtol e ı e El seg¨ent pas per a demostrar l’exist`ncia de la descomposici´ ortogonal ´s prou e o e var que un dels camps lineal ´s paral.lel respecte de la connexi´ de Bott-Weinstein e o definida a les fulles Lagrangianes regulars properes a la fulla singular. El fet que els camps siguin paral.lels ens permet donar una demostraci´ geom`trica del teorema o e de descomposici´ ortogonal simpl`ctic. De fet en el cas que la foliaci´ tingui alguna o e o component el.l´ ıptica donem dues demostracions d’aquest fet, una basada en raonaments geom`trics usant la connexi´ de Bott-Weinstein i un altra usant la forma e o expl´ ıcita de l’estructura simpl`ctica en un entorn de la fulla i el Lema de Poincar´. e e En el cas completament hiperb`lic cal aplicar el m`tode del cam´ diverses vegades o e ı per a trobar una estructura simpl`ctica tal que el Hamiltoni` corresponent a f1 e a sigui el camp lineal X1 . Aquest proc´s constitueix el contingut en les proposicions e 4.6.1, 4.5.1 i permet demostrar el teorema 4.2.1. Per a demostrar 4.2.1 contru¨ ım dues distribucions ortogonals simpl`ctiques que contenen cadascun dels camps lie neals de la distribuci´. A partir d’aix` per a demostrar el teorema 4.2.2 usem els o o resultats d’unicitat simpl`ctica en dimensi´ 2 demostrats al cap´ 2. e o ıtol Al Cap´ 5 usem inducci´, el teorema de Liouville-Mineur-Arnold, el m`tode ıtol o e de la descomposici´ simpl`ctica ortogonal i els resultats de cap´ o e ıtols anteriors per a demostrar el cas general en qualsevol rang i qualsevol dimensi´. o En aquest cap´ demostrem els seg¨ents teoremes, ıtol u Per al cas de foliacions de rang 0, Teorema 5.1.1 Sigui ω un forma simpl`ctica definida en un entorn de l’origen i tal que la folie aci´ lineal F ´s Lagrangiana, llavors existeix un difeomorfisme local φ : (U, p) −→ o e (φ(U ), p) preservant la foliaci´ i tal que φ∗ ( o
i

dxi ∧ dyi ) = ω, essent xi , yi coorde-

0.2. Resultats nades locals a (φ(U ), p).

xxxix

Per a foliacions de rang diferent de zero provem el teorema d’unicitat simpl`ctica e seg¨ent, u Teorema 5.2.1 Siguin ω i ω0 dues formes simpl`ctiques que en un entorn d’una `rbita singue o lar compacta per a les quals la foliaci´ lineal ´s Lagrangiana llavors ω i ω0 s´n o e o equivalents. Al Cap´ ıtol 6 donem la versi´ equivariant de la unicitat simpl`ctica obtino e guda al Cap´ 5. Aquesta versi´ equivariant permet concloure la linealitzaci´ ıtol o o simpl`ctica en un entorn de l’`rbita inicial compacta. Tamb´ donem un enunciat e o e tipus “slice”de la linealitzaci´ simpl`ctica equivariant. o e Al llarg d’aquest cap´ suposem que G ´s un grup de Lie compacte que act´a ıtol e u simpl`cticament en la varietat i deixa invariant l’aplicaci´ moment. Els resultats e o principals obtinguts en aquest cap´ s´n els seg¨ents. ıtol o u En la primera secci´ es s’estudia el problema de linealitzaci´ de l’acci´ a l’entorn o o o d’un punt fix. Per a fer aix` primer estudiem els simplectomorfismes locals que deixen invao riant l’aplicaci´ moment. Provem el seg¨ent teorema, o u Teorema 6.3.2 Sigui ψ : (R2n , 0) → (R2n , 0) un simplectomorfisme local de R2n que preserva l’aplicaci´ moment en el model h. Llavors la part lineal ψ (1) ´s un simplectomoro e fisme que preserva l’aplicaci´ moment i existeix una unica funci´ diferenciable o ´ o definida en un entorn de l’origen Ψ : (R2n , 0) → R que s’anul.la a l’origen, que es una integral primera del sistema lineal definit per h i tal que ψ (1) ◦ ψ −1 es el flux a

xl

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temps 1 del camp Hamiltoni` XΨ de Ψ. Si ψ ´s real anal´ a e ıtica llavors Ψ ´s tamb´ e e real anal´ ıtica. Si ψ dep`n diferenciablement del par`metres (resp anal´ e a ıticament) Ψ tamb´. e Com a corol.lari obtenim la versi´ amb par`metres, o a Corol.lari 6.3.3 Sigui Dp un disc centrat a l’origen 0 en els par`metres p1 , . . . , pk . Notem per a p = (p1 , . . . , pk ). Suposem que ψp : (R2n , 0) → (R2n , 0) ´s un simplectomorfisme e local de R2n que preserva l’aplicaci´ quadr`tica h i que dep`n diferenciablement dels o a e par`metres p. Llavors existeix una unica funci´ local diferenciable Ψp : (R2n , 0) → a ´ o R que s’anul.la a 0 i que dep`n diferenciablement en els par`mentres p i que ´s e a e
−1 una integral primera del sistema lineal definit per h i tal que ψ0 ◦ ψp es el flux (1)

a temps 1 del camp Hamiltoni` XΨp de Ψp . En cas que ψp sigui real anal´ a ıtica i depengui anal´ ıticament en els par`metres, la funci´ Ψp tamb´. a o e Aquests resultats permetem provar el seg¨ent teorema, u Teorema 6.3.4 Existeix un canvi de coordenades a R2n que preserva el sistema (R2n , dyi , h) i que linealitza l’acci´ de G. o Tamb´ demostrem com a corol.lari la versi´ amb par`metres que queda recollida e o a al corol.lari 6.3.5. Corol.lari 6.3.5 Si l’acci´ ρp dep`n de par`metres diferenciablement (resp. anal´ o e a ıticament) existeix una transformaci´ local simpl`ctica a R2n , Φp que preserva el sistema i que o e verifica,
n i=1

dxi ∧

0.2. Resultats

xli

Φp ◦ ρp (h) = ρ0 (h)(1) ◦ Φp

Si notem G com el grup de simplectomorfismes preservant l’aplicaci´ moment o finalment provem el teorema, Teorema 6.3.6 El grup G ´s abeli`. e a Com a corol.lari en dedu¨ que si l’acci´ es efectiva el grup G ´s abeli`. ım o e a Un cop obtingut el resultat local de linealitzaci´ l’`rbita provem el teorema de o o linealitzaci´ en un entorn de l’`rbita. o o Teorema 6.4.1 Sigui G un grup compacte preservant el sistema (Dk × Tk × D2(n−k) , dθi +
n−k i=1 n−k i=1 k i=1

dpi ∧

dxi ∧ dyi , F) llavors existeix ΦG un difeomorfisme en un entorn de
k i=1

l’`rbita L = Tk que preserva el sistema (Dk × Tk × D2(n−k) , o dxi ∧ dyi , F) i que linealitza l’acci´ G. o

dpi ∧ dθi +

Tamb´ es dona com a corol.lari un resultat tipus “slice”en un entorn de l’`rbita e o (Corolari 6.4.2). Si prenem com a grup G el grup de transformacions recobridores del recobriment U (L) considerat al cap´ 1 es prova el teorema de linealitzaci´ en un entorn ıtol o de l’`rbita inicialment considerada que enunciem de manera abreujada com, o Teorema 6.5.1 La foliaci´ F ´s simpl`cticament linealitzable en un entorn d’un `rbita como e e o

xlii pacta singular.

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Finalment al cap´ ıtol 7 considerem el cas de contacte i provem el resultat de linealitzaci´ de contacte al recobridor. Tamb´ donem la versi´ G-equivariant de o e o contacte del teorema que ens d´na en particular el cas de linealitzaci´ de contacte o o a l’entorn inicialment considerat. Els resultats m´s importants d’aquest cap´ s´n els seg¨ents: e ıtol o u Primer provem un resultat per a linealitzaci´ diferenciable de foliacions legeno drianes verificant les condicions especificades a la secci´ 7.3.1. o Teorema 7.3.1 Existeixen coordenades (θ0 , . . . , θk , p1 , . . . , pk , x1 , y1 , . . . , xn−k , yn−k ) en un recobriment finit d’un entorn tubular de O tal que • El camp de Reeb ´s Z = e
∂ . ∂θ0

• Existeix un triplet de nombres naturals (ke , kh , kf ) amb ke + kh + 2kf = n − k i tal que les integrals primeres fi s´n fi = pi , o
2 fi+k = x2 + yi si 1 ≤ i ≤ ke , i

1≤i≤k i

fi+k = xi yi si ke + 1 ≤ i ≤ ke + kh , fi+k = xi yi+1 − xi+1 yi i fi+k+1 = xi yi + xi+1 yi+1 si i = ke + kh + 2j − 1, 1 ≤ j ≤ kf o o o • La foliaci´ F ve descrita per les `rbites de la distribuci´ D =< Y1 , . . . Yn > on Yi = Xi − fi Z essent Xi el camp de contacte fi respecte la forma de contacte est`ndard α = dθ0 + a
n−k 1 i=1 2 (xi dyi

− yi dxi ) +

k i=1

pi dθi .

Un cop demostrada la linealitzaci´ diferenciable procedim a provar la linealito zaci´ simpl`ctica o e Teorema 7.4.1

0.2. Resultats

xliii

2n+1 Sigui α una forma de contacte a la varietat model M0 per la qual la foliaci´ o

F ´s legendriana i tal que el camp de Reeb ´s e e enviant α a α0 .

∂ . ∂θ0

Llavors existeix un difeomorfisme

φ definit a un entorn de l’`rbita singular O = (θ0 , . . . , θk , 0, . . . , 0) preservant F i o

En el cas que existeixi una acci´ d’un grup compacte preservant la foliaci´ i el o o camp de Reeb tenim una versi´ G-equivariant del teorema anterior que enunciem o resumidament, Teorema 7.5.1 Existeix un contactomorfisme que linealitza la foliaci´ F i l’acci´ del grup. o o Si apliquem aquest resultat al cas que el grup sigui el grup de transformacions recobridores obtenim el Teorema 7.5.2 que assegura que la linelitzaci´ de contaco te ´s pot dur a terme a l’entorn inicial de l’`rbita. Enunciem aquest teorema a e o continuaci´, o Teorema 7.5.2 Sigui F una foliaci´ verificant totes les condicions especificades a la secci´ o o 7.3.1, sigui F la foliaci´ ampliada amb el camp de Reeb Z i sigui α una forma de o contacte per la qual F ´s Legendriana i tal que Z ´s el seu camp de Reeb llavors e e existeix un difeomorfisme definit en un entorn de O que porta F a la foliaci´ o lineal, l’`rbita O al torus {xi = 0, yi = 0, pi = 0} i la forma de contacte a la forma o de contacte de Darboux α0 . Alguns dels resultats continguts en aquesta tesi estan continguts a les publicacions i prepublicacions que citem a continuaci´, o

xliv • Publicacions:

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1. C. Curr´s-Bosch i E. Miranda, Symplectic linearization of singular Laa grangian foliations in M 4 , Differential Geom. Appl. 18 (2003), no. 2 , 195-205. 2. E. Miranda, On the symplectic classification of singular Lagrangian foliations. Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltr´, 2000), 239–244, Publ. R. Soc. Mat. Esp., 3, R. u Soc. Mat. Esp., Madrid, 2001. • Prepublicacions: 1. E. Miranda i Nguyen Tien Zung, Equivariant normal forms for nondegenerate singular orbits of integrable Hamiltonian systems, preprint 2003, http://xxx.arxiv.org/abs/math.SG/0302287. 2. C. Curr´s-Bosch i E. Miranda, Symplectic germs of singular Lagrangian a Foliations, preprint 265 de la Facultat de Matem`tiques. Universitat de a Barcelona, 1999.

0.3

Conclusions

En aquesta tesi estudiem dos problemes de classificaci´ per a foliacions definides o a varietats simpl`ctiques i de contacte. e En quant al primer problema de classificaci´: Com s’ha detallat a la secci´ de o o resultats provem que una foliaci´ Lagrangiana definida pels gradients simpl`ctics o e d’una aplicaci´ moment pr`pia ´s equivalent a la foliaci´ linealitzada amb la forma o o e o simpl`ctica de Darboux en un entorn d’una `rbita compacta singular no degenee o rada. En relaci´ a aquest problema tamb´ provem un teorema de linealitzaci´ simpl`ctica o e o e per accions simpl`ctiques de grups compactes que preserven l’aplicaci´ moment. e o

0.3. Conclusions

xlv

En quant al segon problema de classificaci´: Provem que una foliaci´ Legeno o driana completament integrable amb camp de Reeb definit per una acci´ de una o S 1 ´s equivalent a la foliaci´ linealitzada amb la forma de Darboux en un entorn e o d’una `rbita compacta no degenerada de la foliaci´ ampliada amb el camp de Reo o eb. Tamb´ provem un teorema de linealitzaci´ per accions de grups compactes per e o contactomofismes a varietats de contacte.

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Chapter 1 Differentiable linearization
1.1 Introduction

The aim of this chapter is to recall results of differentiable linearization for foliations given by a certain class of singular integrable Hamiltonian systems. Throughout this thesis and otherwise stated all the objects considered will be C ∞. In this chapter and throughout the thesis, we will consider germ-like foliations in a symplectic manifold (M 2n , ω) defined by n first integrals in a neighbourhood of a compact submanifold L. Since we are considering germs-like objects, the foliation is defined in a neighbourhood U (L) of L. We denote by f1 , . . . fn the n-first integrals. The leaves of the foliation are L(c1 ,...,cn ) = {p ∈ U (L), f1 (p) = c1 , . . . fn (p) = cn }. We denote by F the function F = (f1 , . . . , fn ). We will require the following condition on the functions fi . We will assume that the functions fi are in involution with respect to the Poisson bracket associated to ω in the neighbourhood considered. That is to say, {fi , fj } = 0 for any pair i, j. When this condition is fulfilled we say that F defines a completely integrable Hamiltonian system on U (L) and the mapping F is called the moment map. Namely, 1

2

Chapter 1. Differentiable linearization

Definition 1.1.1 A completely integrable Hamiltonian system on a symplectic manifold (M 2n , ω) is a C ∞ Poisson Rn -action, generated by a moment map F : M 2n −→ Rn . The foliation defined by the level sets of F can also be considered as a pair (χ, A), where χ is an n-dimensional commutative Lie algebra of vector fields and A is a vector space of first integrals of the vector fields of χ. This presentation of the foliation is specially interesting when the foliation has singularities. Then the foliation obtained from χ is called the singular Lagrangian foliation. The compact submanifold L that we will consider is an orbit of χ through a singular point. Let us introduce the definition of singular point, Definition 1.1.2 A point x0 ∈ M 2n is a singular point of the integrable Hamiltonian systems if the rank of dx0 F = (dx0 f1 , . . . , dx0 fn ) is less than n. Remark: Since L is an orbit of χ, all the points in L are singular points for F and L is contained in a singular leaf of the foliation. On the other hand, observe that singular orbits do not necessary coincide with singular leaves of the foliation as the following example shows. Consider M = R2 endowed with coordinates (x, y).
∂ ∂ Let F =< x ∂y + y ∂x >. A first integral for F is f = x2 − y 2 . The only singular

point is (0, 0). The orbit through this point is just the point, but the leaf containing the singular point is L = {(x, y), through the origin. When we talk about differentiable linearization in a neighbourhood of L, we mean that there exists a diffeomorphism in a neighbourhood of L, fixing L and taking the given foliation to a simpler foliation defined by a linear model. Linearization is not always possible. We will need additional assumptions on the functions x2 − y 2 = 0} which consists of a pair of lines

1.1. Introduction

3

fi . Namely, the completely integrable Hamiltonian system considered fulfills the following three hypotheses: 1. The global moment map F : M2n −→ Rn is a proper map. 2. The singular orbits L of minimal rank are tori. 3. The singularities considered are of “non-degenerate” type. Remarks • As a consequence of Liouville-Mineur-Arnold theorem, if F is a regular foliation given by a completely integrable system and L is an n-dimensional compact leaf. Then this leaf is a torus and the foliation in a neighbourhood of L is a foliation by n-dimensional tori. Those tori are Lagrangian for the symplectic form considered. Now assume that the foliation is allowed to have singularities. The first example that comes to our mind is to create a singular foliation by collapsing some of the cycles of the regular tori L and leaving the rest of the foliation invariant. In this way, the resulting foliation will be a foliation by regular tori except for the singular one L which will be a torus with dimension r < n. When all the cycles of the initial torus L are collapsed we obtain an isolated singular leaf whose dimension has been decreased to 0, that is, a point. From a symplectic point of view, this torus is an isotropic submanifold, that is to say it preserves all the properties of Lagrangianity except for the maximal dimension. We will take this example as a starting point. The foliation that we will consider has a torus as an isolated singular leaf but the neighbouring orbits are not always tori. The example described above corresponds to the picture of a “completely elliptic” singularity of corank n − r. As we will see this

4

Chapter 1. Differentiable linearization example is one of the “differentiable models” for our foliation. In fact we will see that our foliation is differentiably equivalent in a finite covering of a neighbourhood of the singular leaf to a direct product type foliation of a regular Lagrangian foliation by tori with a singular foliation which is also a direct type foliation of ke components of elliptic type, kh components of hyperbolic type and kf components of focus-focus type. • The third condition (non-degeneracy) is a condition on the quadratic parts of the components of the moment map. Its role in the linearization process is similar to that of non-degeneracy for Morse-like theorem for single functions. In fact the differentiable linearization that we prove is a kind of “multiple Morse” theorem. That is, we can find a diffeomorphism, in a finite covering of the initial neighbourhood considered, such that the foliation determined by the moment map can be taken to the foliation determined by the quadratic parts of the components of the moment map. This is the main difference with the result of Morse for non-degenerate singularities of differentiable functions. The involution of the components of the moment map make this simultaneous linearization possible. In this chapter, the symplectic properties of the foliation will be temporarily

left aside and our attention will be focused on the differentiable side of the story. In any case we will need some facts from symplectic geometry which we introduce in the section called “Preliminaries” of this chapter. The linearization will be carried out in a neighbourhood of the singular orbit.

1.2

Preliminaries.

Let us recall some notations and definitions:

1.2. Preliminaries.

5

1.2.1

Hamiltonian vector fields and the Poisson bracket

Let us start with the definition of symplectic manifold. Definition 1.2.1 A symplectic manifold is a pair (M, ω) where M is a differentiable manifold and ω is a closed non-degenerate 2-form. Remarks: • As a consequence of the definition all symplectic manifolds are even dimensional. • In contrast to Riemannian manifolds, symplectic manifolds have no local invariants. This is due to the Theorem of Darboux which establishes the uniqueness of a local model. Theorem 1.2.1 Let (M, ω) be a 2n-dimensional symplectic manifold and let p be a point in M then there exists local coordinates (x1 , y1 , . . . , xn , yn ) in a neighbourhood U of p such that,

ω|U = dx1 ∧ dy1 + · · · + dxn ∧ dyn In the spirit of this theorem the main goal of this thesis is to establish models for symplectic manifolds with Lagrangian foliations in a neighbourhood of a singular orbit of the foliation. Let (M, ω) be a symplectic manifold. Consider the set of differentiable functions on M , C = C ∞ (M ). Let us introduce the notion of Hamiltonian vector field associated to a function f ∈ C and the notion of Poisson bracket associated to a pair of functions f and g contained in C.

6

Chapter 1. Differentiable linearization

Definition 1.2.2 Let f ∈ C, we define the Hamiltonian vector field associated to f as the unique vector field Xf satisfying, iXf ω = −df. Definition 1.2.3 Let f, g ∈ C we define the Poisson bracket of f and g as {f, g} = ω(Xf , Xg ). Remarks: • The following formula can be derived from the definition of Poisson bracket, X{f,g} = [Xf , Xg ] • Take M = R2n endowed with coordinates (x1 , y1 , . . . , xn , yn ) and let ω be the Darboux symplectic form ω = Poisson bracket {f, g} equals
n i

dxi ∧ dyi . The standard Poisson bracket is

the one associated to ω. Given two functions f, g ∈ C ∞ (R2n ), the standard

(
i=1 ∞ 2n

∂f ∂g ∂f ∂g − ). ∂xi ∂yi ∂yi ∂xi

The pair (C (R ), {., .}) is a Lie algebra. Now consider Q(2n, R) the set of quadratic forms in the variables x1 , y1 , . . . , xn , yn then the standard Poisson bracket of two quadratic forms is again a quadratic form. Therefore the pair (Q(2n, R), {., .}) is a Lie subalgebra of (C ∞ (R2n ), {., .}).

1.2.2

Completely integrable systems and regular Lagrangian foliations

Recall that a system is completely integrable if it is defined by n first integrals in involution with respect to the Poisson bracket. The following proposition relates completely integrable Hamiltonian systems to Lagrangian foliations,

1.2. Preliminaries.

7

Proposition 1.2.2 Let f1 , . . . , fn be n functions such that {fi , fj } = 0, ∀i, j. Assume that p ∈ M is a point for which dp f1 ∧ · · · ∧ dp fn = 0. Then the distribution generated by the Hamiltonian vector fields D =< Xf1 , . . . , Xfn > is involutive and the leaf through p is a Lagrangian submanifold. Proof: Since [Xfi , Xfj ] = X{fi ,fj } , the condition {fi , fj } = 0 implies [Xfi , Xfj ] = 0, ∀i, j and the distribution is involutive. On the other hand, from the definition of Poisson bracket {fi , fj } = ω(Xfi , Xfj ). So the foliation defined by D is isotropic. The condition dp f1 ∧ · · · ∧ dp fn = 0 implies that the Hamiltonian vector fields Xfi span an n-dimensional vector space at the point p. Therefore the leaf through p is Lagrangian. Remark From the definition of Hamiltonian vector fields iXfi ω = −dfi and since ω(Xfi , Xfj ) = 0 for any pair of vector fields tangent to the Lagrangian foliation then Xfi (fj ) = 0, ∀i, j. Those conditions imply that the functions fi are first integrals for the foliation defined by the distribution D.

1.2.3

Orbit versus leaf

In this thesis we will deal with problems of equivalence for symplectic structures in the neighbourhood of an orbit of a foliation F. Observe that for foliations given by a completely integrable systems there are two ways of describing the foliation: the set of orbits and the set of levelsets of the moment map F . An orbit of the foliation is the orbit of the distribution XFi , where Fi is the ith component of the moment map. A leaf of the foliation is a levelset of the moment map F .

8

Chapter 1. Differentiable linearization

1.2.4

Transversal linearization at a singular point

Let x0 be as singular point of the foliation defined by F , we start by defining the rank and corank of a singular point. Definition 1.2.4 Let x0 ∈ M 2n be a singular point of the integrable Hamiltonian system we say that the rank of x0 is k if the rank of the moment map at x0 is k, that is to say if rank dx0 F = rank (dx0 f1 , . . . , dx0 fn ) = k. We say that a singular point of rank k has corank n − k. Recall that the foliation can be thought as a pair (χ, A) , where χ is an ndimensional commutative Lie algebra of vector fields and A is a vector space of first integrals of the vector fields of χ. The foliation obtained from χ is called the singular Lagrangian foliation. We follow Nguyen Tien Zung [61] for the definitions concerning the notion of transversal linearization at a singular point. Let x0 be a singular point and let χx0 be the subspace of Tx0 M generated by Xx0 , ∀X ∈ χ. Let Kx0 = ∩f ∈A Kerdx0 f , and let Bx0 the set of f ∈ A such that dx0 f = 0. Then ∀f ∈ Bx0 the 2-order jet of f − f (x0 ) gives a quadratic form on Kx0 , such that its kernel contains χx0 , so it gives a quadratic form fxT on 0 Kx0 /χx0 , the set of quadratic forms obtained in this way which we denote by AxT , 0 is a commutative subalgebra under the Poisson bracket, which is often called the transversal linearization of F. Notice that Kx0 /χx0 carries a natural symplectic structure ω x0 , and it is symplectomorphic to a subspace Rx0 ⊂ Tx0 M 2n . We are going to introduce the notion of nondegenerate point but first we need to recall the definition of Cartan subalgebra. Definition 1.2.5 A Cartan subalgebra is a maximal self-centralizing abelian subalgebra.

1.2. Preliminaries.

9

Definition 1.2.6 A singular point of corank k is called non-degenerate if AxT is 0 a Cartan subalgebra of the algebra of quadratic forms on Rx0 .

1.2.5

The linear model

Let us recall the following classical result of Williamson [60], which will be the starting point for the linearization.

Theorem 1.2.3 (Williamson) For any Cartan subalgebra C of Q(2n, R) there is a symplectic system of coordinates (x1 , . . . , xn , y1 , . . . , yn ) in R2n and a basis f1 , . . . , fn of C such that each fi is one of the following:

2 fi = x2 + yi i

for 1 ≤ i ≤ ke ,

(elliptic) (hyperbolic)

f i = xi y i for ke + 1 ≤ i ≤ ke + kh ,   fi = xi yi+1 − xi+1 yi , (focus-focus pair)  fi+1 = xi yi + xi+1 yi+1 for

i = ke + kh + 2j − 1, 1 ≤ j ≤ kf (1.2.1)

This result ensures the existence of a transversal linear model on Kx0 /χx0 . This basis is often called Williamson basis. The Williamson type of an orbit In order to prove the existence of a linear model in a whole neighbourhood one must consider in general a finite normal covering of the initial neighbourhood. First let us introduce the notion of Williamson type of an orbit of the integrable system.

10

Chapter 1. Differentiable linearization Observe that because of theorem 1.2.3 the triple (ke , kh , kf ) at p is an invariant

of the point. This triple is called the Williamson type of the singular point p. As it has been shown by Nguyen Tien Zung in [61], this triple is also an invariant of the orbit. That is why it is also called the Williamson type of the orbit L. A Hamiltonian free action of Tk in a covering In order to introduce the linear model we need to recall a result of Nguyen Tien Zung [61] which ensures the existence of a locally free action of Tk (k is the dimension of the orbit) in a neighbourhood of the orbit which preserves the foliation. There exists a normal finite covering of a neighbourhood of the orbit such that this action can be lifted to a free action in the covering. In ([61]) Nguyen Tien Zung proves the following results concerning the existence of Hamiltonian actions of tori in a neighbourhood of a singular leaf of a Hamiltonian system. Let N be a singular leaf (not a singular orbit of the foliation). In [61] the pair (U (N ), F) stands for a foliated neighbourhood of a singular leaf (F is the singular Lagrangian foliation). Theorem 1.2.4 ( Nguyen Tien Zung ) Let (U (N ), F) be a nondegenerate singularity of Williamson type (ke , kh , kf ) and of corank n − k = ke + kh + 2kf of an integrable system with n degrees of freedom. Then there exists an effective Hamiltonian action of a torus Tk+ke +kf in U (N ) which preserves the moment map. This action is unique up to automorphisms of the torus. As observed in [61] in order for this action to be free one must consider a finite covering of (U (N ), L) and choose a subtorus Tk of Tk+ke +kf . Then the following theorem is proved in [61],

1.2. Preliminaries. Theorem 1.2.5 ( Nguyen Tien Zung )

11

Let (U (N ), F) be a strongly nondegenerate singularity of corank n − k = ke + kh + 2kf of an integrable system with n degrees of freedom. Then there exists a normal finite covering (U (N ), F) of (U (N ), F) and a free Hamiltonian action of the torus Tk in the covering (U (N ), F) which preserves the moment map. Remark: In [61] a nondegenerate singularity (U (N ), F) of an integrable system is called strongly nondegenerate singularity if the set of singular values of the moment map when restricted to U (N ) coincides with the set of singular values of a singular point of maximal corank in N . In the case N coincides with an orbit this condition is automatically satisfied. Therefore we may apply this result to a neighbourhood of a nondegenerate orbit of an integrable Hamiltonian system. Namely, since the dimension of the orbit equals k, the isotropy group of the action is a finite abelian group so there exists a finite covering U (L) of the neighbourhood of the orbit such that the foliation, the symplectic form and the action of Tk can be lifted to U (L). And if L is an orbit of an integrable Hamiltonian system we may restate the theorem above as, Theorem 1.2.6 Let U (L) be a neighbourhood of a nondegenerate singular orbit of an integrable system with n degrees of freedom. Assume the corank of the orbit is n − k = ke + kh + 2kf . Let F be the singular Lagrangian foliation defined by the integrable system. Then there exists a normal finite covering U (L) of U (L) such that the foliation can be lifted to F and a free Hamiltonian action of the torus Tk in the covering U (L) which preserves the moment map. Now we can introduce the linear model associated to the orbit L. Later, we will see that the invariants associated to the linear model are the Williamson type of the orbit and a twisting group Γ attached to it. First we introduce the linear model in the covering,

12 The linear model in the covering

Chapter 1. Differentiable linearization

Denote by (p1 , ..., pk ) a linear coordinate system of a small ball Dk of dimension k, (θ1 , . . . , θk ) is a standard periodic coordinate system of the torus Tk , and (x1 , y1 , ..., xn−k , yn−k ) a linear coordinate system of a small ball D2(n−k) of dimension 2(n − k). Now we consider the manifold V = Dk × Tk × D2(n−k) with the standard symplectic form moment map: F = (p1 , ..., pk , f1 , ..., fn−k ) : V → Rn where
2 fi = x2 + yi for 1 ≤ i ≤ ke , i

(1.2.2) dxj ∧ dyj , and the following (1.2.3)

dpi ∧ dθi +

fi = xi yi for ke + 1 ≤ i ≤ ke + kh , fi = xi yi+1 − xi+1 yi and fi+1 = xi yi + xi+1 yi+1 for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf

(1.2.4)

The linearized foliation in the covering is the foliation determined by the above moment map. This presentation of the foliation would be the one of A, that is, the above components of the moment map are the first integrals of the system. We can also look for generators of χ to define the linearized foliation in the covering. After performing a linear change of coordinates in such a way that the hyperbolic
2 functions can be written as fi = x2 − yi , the following vector fields form a basis of i

χ, Yi =
∂ ∂θi

for 1 ≤ i ≤ k ,

∂ ∂ Xi = −yi ∂xi + xi ∂yi for 1 ≤ i ≤ ke , ∂ ∂ Xi = yi ∂xi + xi ∂yi for ke + 1 ≤ i ≤ ke + kh , ∂ ∂ Xi = xi ∂x∂ − yi+1 ∂yi − xi+1 ∂xi + yi ∂y∂ and i+1 i+1 ∂ ∂ Xi+1 = −xi ∂xi + yi ∂yi − xi+1 ∂x∂ + yi+1 ∂y∂ for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf i+1 i+1

1.2. Preliminaries.

13

As a matter of notation, when we talk about rank r foliations or corank n − r foliations, we mean that k=r and that the foliation is defined in a neighbourhood of an r-dimensional torus. When we refer to completely elliptic foliations we mean that the Williamson type of the orbit L is (n − k, 0, 0). We denote by completely hyperbolic foliations in any corank those foliations for which the Williamson type of the orbit L is (0, n − k, 0). The linear model in the neighbourhood will be determined by the following two data: The linear model in the covering and a twisting group attached to the isotropy group of the Hamiltonian Tk -action along the singular orbit L. In any case the role of the twisting group in the linearization process will be clarified when we prove the equivariant version of the symplectic linearization. The linearized foliation in the initial neighbourhood considered U (L) is the linearized foliation in the covering quotiented by the action of the twisting group.

1.2.6

The parametrized Morse lemma

In this section we are going to recall the parametrized Morse lemma. The Morse lemma without parameters establishes the existence of a diffeomorphism in a neighbourhood of a nondegenerate singular point of a smooth function which takes the given function to its quadratic part. If the function depends on parameters then the above diffeomorphism also exists and depends smoothly on the parameters. We include here the proof of the Morse lemma without parameters provided by Richard S. Palais in [51], the Morse lemma with parameters will be a consequence of it. So let us recall the content and proof of the Morse lemma. But before let us outline the following: Palais also proved the theorem for Banach spaces; we will stick to the differentiable case. Theorem 1.2.7 Let f be a smooth function defined in convex neighbourhood W of the origin in a finite dimensional vector space V . Let 0 stand for the origin of the vector space. Suppose that f (0) = 0, d0 f = 0 and that 1 d2 f is a nonsingular 2 0

14

Chapter 1. Differentiable linearization

quadratic form Q. Then there exists a neighbourhood U of the origin and a diffeomorphism φ : U −→ W with φ(0) = 0 and d0 φ = Id such that for x ∈ U , f (φ(x)) = Q(x, x). Proof: The proof uses the path method. Put f = f1 and define f0 (x) = Q(x, x). We define the path ft = f0 + t(f1 − f0 ) for t ∈ [0, 1]. Notice that f˙t = dft = f1 − f0 .
dt

We look for a one parameter family of diffeomorphisms φt such that ft ◦ φt = f0 and satisfying the condition φ0 = Id . Once the one parameter family is found, the diffeomorphism φ that we are looking for will be φ1 . Now we introduce the t-parametric vector field associated to the family φt , Xt0 (φt0 (x)) = d (φt (x))|t=t0 , dt (1.2.5)

From this expression the following relation is obtained (lemma 1 in [51]), d (ft ◦ φt (x)) = (f˙t + Xt (ft )) ◦ φt (x) dt From this equation in particular if Xt (ft ) = −f˙t , ∀t ∈ [0, 1] then ft ◦ φt = f0 . So going back to the problem posed at the beginning, it is enough to find a dependent vector field Xt such that Xt (ft ) = −(f1 − f0 ). (1.2.6)

Given a smooth mapping g we will denote by dx g the differential of g at the point x. In order to see which is the convenient vector field Xt observe that since d0 ft = 0, we can write dx ft (v) = And this last term is
1

0 t Bx (x, v) where, t Bx (u, v)

d dsx ft (v)ds = ds

1 0

d2 ft (x, v)ds sx

1

=
0

d2 ft (u, v)ds sx

t 0 1 0 t Observe that Bx = Bx + t(Bx − Bx ). On the other hand, since B0 = 2Q and Q is a t nonsingular quadratic form for all t then Bx is nonsingular ∀x in a neighbourhood

1.2. Preliminaries. of 0 and ∀t ∈ [0, 1]. Now equation 1.2.6 can be rewritten as,
t Bx (x, Xt ) = f0 − f1 ,

15

so if we could identify g = f0 − f1 with a quadratic form then the equation 1.2.6 has a well-defined solution. We make similar computations to the ones we did with dx ft ,
1

f0 − f1 =
0

d g(sx)ds = ds

1

1

1 0

dsx g(x)ds =
0 0

d2 g(sx, x)drds rsx

This last expression equals the quadratic form Cx (x, x) being Cx the bilinear form,
1 1 0

Cx (u, v) =
0

d2 g(su, v)drds rsx

Finally, the Xt satisfying the equation is the unique solution of the equation,
t Bx (u, Xt ) = Cx (u, x),

∀u ∈ V.

Now the diffeomorphism φt defined by the equation 1.2.5 is such that φ1 is a solution. Observe also that because φ0 is the identity mapping then d0 φ0 is also the identity mapping and since d0 Xt = 0 then d0 φt is the identity mapping ∀t in particular for t = 1, as desired. This ends the proof of the theorem. Remark: As observed by Guillemin and Sternberg in [30], the proof provided by Palais allows to claim that if f depends smoothly on parameters then the diffeomorphism obtained φ will depend smoothly on the same parameters because the vector field Xt depends differentiably on them (all the operations performed to find Xt are differentiable). This is the content of the so-called parametrized Morse Lemma.

1.2.7

Our notion of equivalent symplectic germs

We say that a foliation is generically Lagrangian if its regular leaves are Lagrangian submanifolds and its singular leaves are isotropic. Let L be an orbit of the foliation. Let us introduce the notion of equivalence for the singular Lagrangian foliation in a neighbourhood of the orbit that we will use in the sequel.

16

Chapter 1. Differentiable linearization

Definition 1.2.7 Let (U1 , ω1 ) and (U2 , ω2 ) be two symplectic germs such that F is generically Lagrangian for both ω1 and ω2 . We say that ω1 and ω2 are equivalent if there exists a diffeomorphism φ : U1 −→ U2 such that: 1. φ∗ (ω2 ) = ω1 . 2. φ preserves F. 3. φ fixes L. Notation 1.2.8 We write ω1 ∼F ω2 to denote equivalent symplectic germs.

1.3

Differentiable equivalence in a finite normal covering

In this section we recall the result of linearization in a finite normal covering. That is, we will see that there exists a finite covering in which the foliation can be linearized in a neighbourhood of a singular orbit for the foliation. This result was proved by Eliasson. We give our own proof in the corank 1 case which uses Morse methods to linearize in the covering. The objective of this section is to provide the following main result, Theorem 1.3.1 In U (L) the singular Lagrangian foliation is diffeomorphic to the linearized one. Remarks: • The linearization result for analytical systems was proved by Vey. • The differentiable linearization for maximal corank singularities, that is when L is reduced to a point, was proved by Eliasson in [23].

1.3. Linearization in the covering

17

• The differentiable linearization for any corank was proved by Eliasson in [24] and [23]. Another proof in the completely elliptic case for any corank was provided by Dufour and Molino in [21]. Before proceeding to the proof of the theorem for rank n − 1 foliations, we recall the result for rank 0 foliations which was proved by Eliasson. We would like to remark that Eliasson’s theorem of linearization is also valid for a set of functions h1 , . . . , hk in involution with respect to the Poisson bracket attached to the corresponding symplectic form with k ≤ n. In any case, we state here the version for n commuting functions. In the statement of the theorem, the set q1 , . . . , qn stands for a Williamson basis of the Cartan subalgebra attached to the singularity as guaranteed by theorem 1.2.3 and {, }0 stands for the standard Poisson bracket. Theorem 1.3.2 ( Eliasson ) Let (M, ω) be a symplectic manifold and let h1 , . . . , hn be a set of functions in involution with respect to the Poisson bracket {, } attached to ω. Assume p is a non-degenerate critical point of rank 0. Then there exists a local chart φ : T0 M −→ M such that dφ(0) = Id and such that {hj ◦ φ, qi }0 = 0 for all i, j. If there are no hyperbolic elements among the qi then there exists germs of smooth functions ψ1 , . . . , ψn such that, hj ◦ φ = ψj (q1 , . . . , qn ). Remark: Although the exception made for the hyperbolic components in the Cartan subalgebra in the statement of the theorem above, the condition {hj ◦ φ, qi }0 = 0 is enough to guarantee that the foliation is linearizable. This is due to the fact that the condition {hj ◦ φ, qi }0 = 0 is equivalent to the condition Xi (hj ◦ φ) = 0, ∀i, j and this, in turn, implies that the foliation is generated, in the new coordinates provided by φ, by the vector fields of the linearized foliation. Finally we proceed to prove the linearization theorem for rank n − 1 foliations in a 2n-dimensional manifold, Proof of 1.3.1 in the corank 1 case:

18

Chapter 1. Differentiable linearization In U (L), the algebra A is generated by n functions g1 , . . . , gn−1 , f . Let Hgi ,

1 ≤ i ≤ n − 1 be the infinitesimal generators of the Hamiltonian free Tn−1 -action. Recall that L is diffeomorphic to a torus Tn−1 . Then one can take coordinates (θ1 , . . . , θn−1 , p1 , . . . , pn−1 , x, y) in U (L) such that, Hgi =
∂f ∂θi ∂ ∂θi

and gi = pi for 1 ≤ i ≤ n − 1.

Let f be a singular first integral. Since {f, gi } = 0, in particular we obtain = 0 and the function f does not depend on θi for any i. Let p be a point in L, since p is nondegenerate we may apply the result of Williamson and there exists coordinates on the vector space Kx0 /χx0 such that f is one of the following: • If the Williamson type of the orbit is (1, 0, 0) then f = x2 + y 2 . • If the Williamson type of the orbit is (0, 1, 0) then f = x2 − y 2 . Let us set the following simplifying notation which we will use throughout the proof, θ = (θ1 , . . . , θn−1 ) and p = (p1 , . . . , pn−1 ). The notation D (p) stands for a disk of radius in the coordinates p1 , . . . , pn−1 centered at the origin. In order to prove the theorem we need the following lemma:

Lemma 1.3.3 Let N ⊂ U (L) be defined as N = {(θ, p, x, y) ∈ U (L) | ∂f ∂f = 0, = 0}. ∂x ∂y

Then, under the non-degeneracy assumptions, there exist functions h1 : Tn−1 × D (p) −→ R and h2 : Tn−1 × D (p) −→ R and a tubular neighbourhood W (L) of L, W (L) ⊂ U (L) such that N ∩ W (L) = {(θ, p, x, y) ∈ W (L) | x = h1 (θ, p), y = h2 (θ, p)}.

1.3. Linearization in the covering Proof: ˆ Let N be the set {(θ, p, x, y) ∈ U (L) | x = y = 0}.

19

There exists a neighbourhood V (L) of L such that the differential of the mapping H: V (L) −→ ˆ N × R2

(θ, p, x, y) −→ (θ, p, ∂f , ∂f ) ∂x ∂y is non-singular along L. So there is an open neighbourhood W (L) of L in V (L), such that H|W (L) is a diffeomorphism. We use this diffeomorphism to define h1 and h2 . ˆ Notice that H(N ) = N × (0, 0). Finally defining h1 (θ, p) = π3 ◦ H −1 (θ, p, 0, 0) and h2 (θ, p) = π4 ◦ H −1 (θ, p, 0, 0) (where π3 and π4 stand for the projections on the x-axis and y-axis respectively) we have N ∩ W (L) = {(θ, p, x, y) ∈ W (L) | x = h1 (θ, p), y = h2 (θ, p)}. And this concludes the proof of the lemma.

The proof of the theorem continues as follows: The following diffeomorphism G: W (L) −→ G(W (L))

(θ, p, x, y) −→ (θ, p, x − h1 , y − h2 ) takes N ∩ W (L) to {(θ, p, 0, 0) ∈ G(W (L))}. Let (θ, p, x1 , y1 ) stand for coordinates on G(W (L)). After applying the parametrized Morse lemma we get coordinates (θ, p, x, y ) in ¯ ¯ a neighbourhood T (L) of L, T (L) ⊂ W (L), such that

20

Chapter 1. Differentiable linearization

f = x2 + y 2 . where = 1 if f is elliptic and = −1 in the hyperbolic case. So, the singular
∂ , ∂θi

Lagrangian foliation becomes differentiably equivalent to the one described by the orbits of the distribution generated by Yi and X being Yi =
∂ ∂ X = − y ∂x + x ∂y , where

1 ≤ i ≤ n − 1 and

= 1 if f is elliptic and

= −1 in the hyperbolic case.

This ends the proof of the theorem in the corank 1 case.

Chapter 2 Analytic tools and symplectic linearization in dimension 2
2.1 Introduction

In this chapter we will prove some results which will play an important role in the symplectic linearization process in any dimension concluding also the symplectic linearization in dimension 2. The chapter is organized as follows: In the first section we prove some results concerning two special decomposition for functions. The kind of tools are those of analysis. Some of this results have already been proved by Eliasson [23] and Colin de Verdi`re and Vey in dimension 2 [6]. In any case, we extend those results to e any dimension. This generalization will be needed in the chapters that follow. In the last section we prove a symplectic linearization result for singular Lagrangian foliations fulfilling the hypotheses posed in the first chapter in dimension 2. When we talk about symplectic linearization we mean the following: We consider a foliation given by a completely integrable system with singularities of nondegenerate type. In the first chapter we saw that those foliations are differentiably 21

22

Chapter 2. Analytic tools and dimension 2

linearizable in a finite covering of a neighbourhood of an orbit of the distribution generated by the Hamiltonian vector fields. We consider the linearized foliation in the covering and we pose the following problem. Given a symplectic form for which the foliation is Lagrangian, does there exist a diffeomorphism in the covering taking the given symplectic form to the Darboux symplectic form and preserving the foliation? When the answer to the question is affirmative we say that the foliation is symplectically linearizable in the covering. To attain the symplectic linearization in the initial manifold we will need to talk about an equivariant linearization result. This will be done in a further chapter. The aim of the second section is to give an affirmative answer to that matter in dimension 2.

2.2

Two special decompositions for functions

Let g be a smooth function if X is a smooth vector field on a manifold M and p ∈ M such that X(p) = 0, then it is a well-known result that g admits a local smooth decomposition of the following type: g = g1 + X(g2 ) , X(g1 ) = 0 (2.2.1)

In order to do that just take local coordinates (x1 , . . . , xn ) centered at a point p such that X =
∂ ∂x1

and apply the classical integration trick. That is, if we consider

the smooth function g1 (x1 , . . . , xn ) = g(0, x2 , . . . , xn ) and the smooth function
1

g2 =
0

g(tx1 , . . . , xn )dt

we obtain the desired decomposition 2.2.1. Now the question arises: Can we obtain similar local decomposition for singular vector fields?

2.2. Two special decompositions

23

In this section we are going to prove that similar decompositions can be obtai∂ ∂ ∂ ∂ ned for the following vector fields X = x1 ∂x2 − x2 ∂x1 or Y = −x1 ∂x1 + x2 ∂x2 .

This special decomposition for functions are going to become a key point in the proof of the local uniqueness theorem for the elliptic-elliptic, elliptic-hyperbolic and hyperbolic-hyperbolic cases. Let us state and proof the following propositions. The first proposition is proved by Eliasson in [24] and [23] in any dimension whereas a proof for the second proposition is proved by Eliasson when n = 2 in [23]. Let us point out that when the manifold is M = R2 a proof of this decomposition had been formerly given by Guillemin and Schaeffer [28] and by Colin de Verdi`re and Vey [6]. This generalie zation to any dimension seems to be new in the non-elliptic case. In any case the techniques used here are fairly inspired in those of the paper of Colin de Verdi`re e and Vey. Proposition 2.2.1 Let M be a differentiable manifold and let g be a germ of smooth function in a neighbourhood of a point p. Consider X a vector field which in
∂ ∂ local coordinates can be written as X = x1 ∂x2 −x2 ∂x1 then there exist differentiable

functions g1 and g2 such that: g = g1 (x2 + x2 , x3 . . . , xn ) + X(g2 ) 1 2 Proof: We follow Eliasson’s recipe [23] for this proof: Let φt be the flow of the vector field X. Since the orbits of X are circles, after shrinking the neighourhood U of the point p if necessary we can assume that φt (U ) ⊂ U . On the other hand, the orbits of X are periodic of period 2π. Thus we can consider the following well-defined function, g1 (x1 , . . . , xn ) = 1 2π
2π

g(φt (x1 , . . . , xn ))dt
0

24 and

Chapter 2. Analytic tools and dimension 2
2π 1 (tg(φt (x1 , . . . , xn )) − g1 (x1 , . . . , xn ))dt. 2π 0 Clearly, these functions are differentiable. Let us check that these g1 and g2

g2 (x1 , . . . , xn ) =

give the decomposition sought. First we check X(g1 ) = 0 and since the orbits of the vector field are connected this implies that g1 = g1 (x2 + x2 , x3 , . . . , xn ). 2 1 g1 (φs (x1 , . . . , xn )) − g1 (x1 , . . . , xn ) , s→0 s we compute this derivative: X(g1 ) = lim
2π 1 g(φt (φs (x1 , . . . , xn )))dt 2π 0 Since φs is a one-parameter subgroup we get:

Since

g1 (φs (x1 , . . . , xn )) =

1 g1 (φs (x1 , . . . , xn )) = 2π becomes:

2π

g(φt+s (x1 , . . . , xn ))dt
0

Now we perform the change of variable t = t + s and the right hand side

2π+s 1 g(φt (x1 , . . . , xn ))dt 2π s Now we differentiate this expression with respect to s to get:

1 (g(φ2π+s (x1 , . . . , xn )) − g(φ2π (x1 , . . . , xn ))) 2π Since φt is 2π-periodic this expression equals 0 for all s, in particular, for s = 0 and this proves X(g1 ) = 0. Now we perform the same kind of calculations for g2 . We have to check that X(g2 ) = g − g1 . We have,

2.2. Two special decompositions

25

g2 (φs (x1 , . . . , xn )) =

1 2π
1 2π

2π

((t+s)g(φt (φs (x1 , . . . , xn )))−(t+s)g1 (φs (x1 , . . . , xn )))dt
0

We split this into two integrals. The first integral
2π (t 0

+ s)g(φt (φs (x1 , . . . , xn )))dt becomes
2π+s s

1 2π

(tg(φt (x1 , . . . , xn )))dt

under the change of variable t = t + s. Now differentiating in s we obtain 1 ((2π + s)g(φ2π+s (x1 , . . . , xn )) − sg(φs (x1 , . . . , xn ))) 2π Again, since φt is 2π-periodic this expression equals 1 (2πg(φs (x1 , . . . , xn ))) 2π Finally, put s = 0; since φ0 = Id we get g(x1 , . . . , xn ). As for the second integral,
2π 2π

1 2π

(t + s)g1 (φs (x1 , . . . , xn ))dt = g1 (φs (x1 , . . . , xn ))
0 0

(t + s)dt

= g(φs (x1 , . . . , xn ))(

(2π)2 + 2πs) 2

Finally differentiating in s and setting s = 0 this expression equals, 1 (2π)2 ( X(g1 (x1 , . . . , xn )) + 2πg1 (x1 , . . . , xn )) 2π 2 But since X(g1 ) = 0 this integral is g1 (x1 , . . . , xn ). This proves X(g2 ) = g − g1 and we are done.

26

Chapter 2. Analytic tools and dimension 2

∂ Now let us prove a similar result but for a vector field of type Y = −x1 ∂x1 + ∂ x2 ∂x2 .

We will prove, Proposition 2.2.2 Let M be a differentiable manifold and let g be a germ of smooth function in a neighbourhood of a point p. Consider X a vector field which
∂ ∂ in local coordinates can be written as Y = −x1 ∂x1 + x2 ∂x2 then there exist diffe-

rentiable functions g1 and g2 such that g = g1 (x1 x2 , x3 . . . , xn ) + Y (g2 ) Before we will need some lemmas concerning the smooth resolution of the equation Y (f ) = g for a given smooth g. Lemma 2.2.1 Let g be a smooth function, the equation Y (f ) = g admits a formal solution along the subspace S = {(0, 0, x3 , . . . , xn )} if and only if ∂ 2k g (0, 0, x3 , . . . , xn ) = 0. ∂xk ∂xk 1 2 Proof: Let us construct a solution considering the (x1 , x2 )-jets. That is, assume the (x1 , x2 )-jet of f along S = {(0, 0, x3 , . . . , xn )} is of g along S = {(0, 0, x3 , . . . , xn )}. Then the condition X(f ) = g implies the following conditions for the coefficient functions (−i + j)fij = gij , ∀i, j
∂ 2k g (0, 0, x3 , . . . , xn ) ∂xk ∂xk 1 2 ij

fij xi xj , the coefficients fij 1 2
ij

being functions in the variables (x3 , . . . , xn ). Denote by

gij xi xj the (x1 , x2 )-jet 1 2

Particularizing i = j in this equation we obtain gii = 0; so in order to have a solution by jets of the equation Y (f ) = g, the terms to vanish necessarily. have

2.2. Two special decompositions

27

On the other hand if i = j from the above relation, the following relation is met fij =
gij . −i+j

Therefore, if the condition

∂ 2k g (0, 0, x3 , . . . , xn ) ∂xk ∂xk 1 2

= 0 is fulfilled

this gives a solution by jets to the equation Y (f ) = g. According to Borel’s theorem there exists a smooth function f with the (x1 , x2 )jets previously found. It remains to solve this equation for functions for which ∂ i+j g (0, 0, x3 , . . . , xn ) = 0. ∂xi ∂xj 1 2 We will refer to this functions as (x1 , x2 )-flat functions along the subspace S = {(0, 0, x3 , . . . , xn )}. Lemma 2.2.2 Let g be a (x1 , x2 )-flat function along the subspace S = {(0, 0, x3 , . . . , xn )} then there exists a smooth function f for which Y (f ) = g. Proof: Consider the function,   1 x1  ln
2 2 x2 x2

T (x1 , . . . , xn ) =

x1 x 2 > 0 x1 x2 < 0

 1 −x1  ln

∂ Denote by φt (x1 , . . . , xn ) the flow of the vector field Y , being Y = −x1 ∂x1 + ∂ x2 ∂x2 . Observe that φt (x1 , . . . , xn ) = (e−t x1 , et y1 , . . . , xn ).

Now we define
T (x1 ,...,xn )

f (x1 , . . . , xn ) = −
0

g(φt (x1 , . . . , xn ))dt.

(2.2.2)

This function is defined outside the set Ω = Ω1 ∪ Ω2 being Ω1 = {(x1 , . . . , xn ), x1 = 0} and Ω2 = {(x1 , . . . , xn ), x2 = 0}. Let us prove that f admits a smooth continuation in the whole neighbourhood considered and that it is a solution to our problem.

28

Chapter 2. Analytic tools and dimension 2 In order to check that it admits a smooth extension. We compute the deriva-

tives. Formally differentiating under the integral sign, the computation of the first derivatives reads, • If i = 1, 2
T (x1 ,...,xn ) 0

∂ ∂ f = −g(φT (x1 ,x2 ,...,xn ) ) T− ∂xi ∂xi • When i = 1 and i = 2, ∂ f =− ∂xi set Ω = Ω1 ∪ Ω2 .

∂ g(φt (x1 , . . . , xn ))dt (2.2.3) ∂xi

T (x1 ,...,xn ) 0

∂ g(φt (x1 , . . . , xn ))dt ∂xi

(2.2.4)

Observe that the set S equals S = Ω1 ∩ Ω2 . Observe that f is smooth outside the The first term in 2.2.3 is smooth outside the set Ω = Ω1 ∪ Ω2 . And observe that if p lies in Ω then from the definition of T , the point φT (p) lies in S. On the other hand, the function g is flat along the subspace S. Thus the first term
∂ in 2.2.3 −g(φT (x1 ,x2 ,...,xn ) ) ∂xi T is smooth in the whole neighbourhood of the origin

considered. As for the second term, we could reproduce word by word the proof supplied by Eliasson in [23] in the two dimensional case. The proof can be adapted because the function g is flat along S. In fact, it is just the parametric version of Eliasson’s result. In the same way, Eliasson’s proof yields that the integral 2.2.4 is a smooth function. The same arguments applied to the successive derivatives prove that f admits a C ∞ continuation. In fact in [23] it is proved that the integral defining f is absolutely integrable, thus we can differentiate with respect to s. This lets us prove that f is, in fact, a solution to the equation Y (f ) = g.

2.2. Two special decompositions Now let us check that this is a solution to the equation. First,
T (φs (x1 ,...,xn ))

29

f (φs (x1 , . . . , xn )) = −
0
−s

g(φs (φt (x1 , . . . , xn )))dt
−s

(2.2.5)

x1 The relations ln ees xx1 = ln x2 − 2s when x1 x2 ≥ 0 and ln −es x2x1 = ln −x1 − 2s e x2 2

when x1 x2 ≤ 0 imply T (φs (x1 , . . . , xn )) = T (x1 , . . . , xn ) − s. On the other hand, since φs is a one-parameter subgroup. Equation 2.2.5 can be written as,
T (x1 ,...,xn )−s

f (φs (x1 , . . . , xn )) = −
0

g(φt+s (x1 , . . . , xn ))dt

Now we perform the change of variable t = t + s and this equation reads,
T (x1 ,...,xn )

f (φs (x1 , . . . , xn )) = −
s

g(φt (x1 , . . . , xn ))dt

Now after differentiating in s this equation yields, df (φs (x1 , . . . , xn )) = g(φs (x1 , . . . , xn )) ds Finally, put s = 0 to obtain Y (f ) = g as we wanted. This ends the proof of the lemma. Let us go back to the proof of proposition 2.2.2. Given a differentiable function g, we want to find smooth functions g1 and g2 such that g = g1 (x1 x2 , x3 . . . , xn ) + Y (g2 ). The strategy for finding this decomposition will be to find a solution by (x1 , x2 )jets and then apply the second lemma to gather all the remaining (x1 , x2 )-flat terms as Y (f ) for a certain smooth f . So let
ij

gij xi xj be the (x1 , x2 )-Taylor expand for g at a point (0, 0, x3 , . . . , xn ) 1 2

lying in the subspace S = {(0, 0, x3 , . . . , xn )}.

30

Chapter 2. Analytic tools and dimension 2 Now we split this Taylor expand in two. The first one,
ii

gij xi xi , and the 1 2

second one

i=j

gij xi xj . Denote by r1 and r2 two smooth functions with the 1 2

previous jets. Then we can assert that r1 = g1 (x1 x2 , x3 , . . . , xn ) + φ(x1 , . . . , xn ), being φ(x1 , . . . , xn ) a (x1 , x2 )-flat function along S = {(0, 0, x3 , . . . , xn )}. Further, using the two above lemmas (2.2.1,2.2.2), the function r2 can be written as r2 = Y (R2 ). Now since φ is (x1 , y1 )-flat, according to lemma 2.2.2 we can write φ(x1 , . . . , xn ) = Y (R). Finally define g2 = R2 + R and g1 and g2 satisfy the decomposition sought g = g1 (x1 x2 , x3 . . . , xn ) + Y (g2 ). And this completes the proof of proposition 2.2.2.

Observation 2.2.1 Observe that the function defined by formula 2.2.2 is not smooth if g is not flat along the subspace S. If g is only flat at the origin then we can find examples which show that f does not admit a smooth continuation. For instance consider n = 4, the function g = e
1 −( x )2 3

is flat at the origin but

it is not flat along the subspace S = {(0, 0, x3 , x4 )}. Observe that the integral does not extend to a smooth function at points of the form (0, x0 , x0 , x0 ) with x0 = 0 2 3 4 2 and x0 = 0. 3 This integral has been used by some authors without the condition of flatness along the subspace and just the condition of flatness at the origin (see proposition 2.13 in [57]). Thus, the functions defined by those integrals in [57] do not always admit a smooth continuation unless the function g is flat along S.

2.3

Symplectic linearization in dimension 2

In this section we consider a foliation given a completely integrable system with singularities of non-degenerate type in a 2-dimensional manifold.

2.3. Symplectic linearization in dimension 2

31

As we observed in the first chapter, the foliation defined as above can be differentiably linearized because of Eliasson’s theorem for rank 0 singularities. Then we know that the foliation is differentiably equivalent to the foliation defined by the orbits of the vector field, X =x Where ∂ ∂ − y . ∂y ∂x = −1 in the hyperbolic case. That is,

= 1 in the elliptic case and

∂ ∂ the foliation that we will consider will be F =< x ∂y − y ∂x >.

The problem that we want to solve in this section is the following: Problem Given two symplectic structures ω1 and ω2 , we want to find a local foliation preserving diffeomorphism defined in a neighbourhood of the origin such that φ∗ (ω1 ) = ω2 This problem is not new. The affirmative answer was given by Vey [55] in the analytical case and by Colin de Verdi`re and Vey [6] in the smooth case. A proof for the smooth elliptic e case was given by Eliasson in [23]. In any case, we provide our own proof here. Observe that in dimension 2 this problem is equivalent to the problem of symplectic linearization of singular Lagrangian foliations. This second problem is more constraining in dimensions greater than 2. Let us recall what is the problem of symplectic linearization of singular Lagrangian foliations about, Problem Let F be a foliation given by a completely integrable system with singularities of non-degenerate type.

32

Chapter 2. Analytic tools and dimension 2 Given two symplectic structures ω1 and ω2 for which F is Lagrangian, we want

to find a local foliation preserving diffeomorphism defined in a neighbourhood of the origin such that φ∗ (ω1 ) = ω2 . This result has an utter importance in the forthcoming chapters. It can be considered as the first step of an inductive process valid for integrable systems without focus-focus components which will allow us to conclude the symplectic linearization. Now we can state and prove the following, Theorem 2.3.1 Let (M 2 , ω1 ) be a 2-dimensional symplectic manifold endowed with coordinates (x, y) and let F be a singular Lagrangian foliation with an elliptic or hyperbolic singularity at the origin (0, 0), then there exists a local diffeomorphism φ preserving F such that φ∗ (dx ∧ dy) = ω1 . Proof:
∂ ∂ Let X = x ∂y − y ∂x . We denote by f be the function f = x2 + y 2 . Now

assume ω1 = A(x, y)dx ∧ dy Then iX ω = −Adf . In the elliptic case ( = −1) lemma 2.2.1 shows that we can write A = A1 + X1 (A2 ) with A1 basic for convenient functions A1 and A2 . In the hyperbolic case ( = −1) we use lemma 2.2.2 to find a similar decomposition. But we have to perform a change of coordinates first, consider x = x+y, y = x − y and now apply lemma 2.2.2 which guarantees the existence of functions A1 and A2 such that A = A1 + X−1 (A2 ) with A1 basic. Now in both cases, we define α = A2 df . Observe that α is a basic 1-form. The next lemma shows that we can deform ω1 to an equivalent ω 1 with the coefficient function A basic for F. More exactly, we prove the following lemma which is a foliation-preserving version of the Moser path method. Lemma 2.3.2 Let α be an F-basic 1-form and let ω1 be a symplectic germ on a 2-dimensional manifold for which F is Lagrangian. Then:

2.3. Symplectic linearization in dimension 2

33

1. The 2-form ω0 = ω1 − dα is a symplectic structure in a neighbourhood of p. 2. There is a diffeomorphism η between two neighbourhoods of p preserving F and such that η ∗ (ω1 ) = ω0 . Proof: First, let us check that ω0 is a symplectic form in a neighbourhood of p. Clearly, ω0 is a closed 2-form. Let us see that it is non-degenerate; Since α is basic for the foliation, α = g · (df ). In particular α vanishes at p = (0, 0) and ω0|p equals ω1|p . Therefore, since ω1 is non-degenerate at p, the 2-form ω0 is non-degenerate in a neighbourhood of p. This ends the proof of the first assertion. In order to prove the second assertion we consider the following family of 2-forms: ωt = ω0 + t(ω1 − ω0 ), t ∈ [0, 1]

Let us see that these 2-forms are symplectic germs. Clearly, the 2-forms ωt are closed. And since ωt|p = ω0|p , we can repeat the argument above to see that the 2-forms are non-degenerate. Therefore, they are symplectic in a neighbourhood of the point p. Now we are going to use Moser’s path method to conclude. First, we consider the well-defined vector field Xt by the following equality: iXt ωt = −α. Recall that α vanishes at p, this guarantees [59] that the time-dependent vector field Xt is integrable. Let φt stand for the “flow” of the time dependent vector field Xt defined by the conditions: φ0 = Id, Xt =
dφs ds |s=t

We check that this field

is tangent to the foliation, in this way the flow of the time-dependent vector field will preserve the leaves of the foliation. The singular set for the foliation is reduced to the origin p. We will see that the vector field is tangent to the foliation in two steps:

34

Chapter 2. Analytic tools and dimension 2 • At points q = (0, 0): Since α is basic for the foliation Xt verifies ωt (Xt , X ) = −α(X ) = 0. Therefore, Xt belongs to its symplectic orthogonal and thus it has to be tangent to the foliation along the regular leaves. • At the point p = (0, 0): The vector field Xt is tangent to the foliation because α vanishes at p = (0, 0) and therefore Xt vanishes at p.

So, we conclude that its flow preserves the leaves of the foliation. Further, remember that we are looking for a symplectomorphism; this symplectomorphism will be given by the flow of the vector field Xt at time t = 1. Remember that the flow φt gives us a family of diffeomorphisms verifying: 1. φt (p) = p. 2. φ∗ ωt = ω0 ; that is to say, as a particular case, we have: φ∗ (ω1 ) = ω0 . t 1 3. φt preserves the leaves of the foliation. So φ1 is the symplectomorphism we are looking for and the two symplectic forms ω0 and ω1 define equivalent symplectic structures. This proves the second assertion of the lemma.

Now we continue with the proof of the theorem. We apply the lemma taking α = A2 df and the symplectic form ω 1 = ω1 − dα is equivalent to the initial ω1 and so far iX1 ω = −A1 df1 with A1 basic. The theorem will be proved once we prove the following lemma,

2.3. Symplectic linearization in dimension 2

35

Lemma 2.3.3 For any 2-form on R2 of the form ω = χ(f )dx ∧ dy verifying χ(0) = 0 , there is a germ of diffeomorphism ν preserving the origin and also preserving the foliation given by df = 0, such that ν ∗ (ω) = dx ∧ dy Proof of the lemma: To start with, observe that if ψ(f ) is any differentiable function of f such that ψ(0) = 0, the mapping G : (R2 , 0) −→ (x, y) (R2 , 0)

−→ (x · ψ(f ), y · ψ(f ))

defines a germ of diffeomorphism preserving the foliation df = 0. Moreover, observe that G∗ (dx ∧ dy) = (ψ 2 + 2ψψ f )dx ∧ dy. Consider the equation d 2 (ψ (u) · u) = χ(u), du where u = f . Observe that after integrating in u we obtain, ψ (u) = which is a smooth function. On the other hand since ψ (0) = lim
2 u (χ(u)) 0 2 u (χ(u)) 0

u

u→0

u

= χ(0)

and χ(0) = 0 (it is the coefficient of a symplectic 2-form) we can assert that ψ is a smooth function in a neighbourhood of the origin and that ψ(0) = 0. So taking as ψ the solution of the equation

36

Chapter 2. Analytic tools and dimension 2

d 2 (ψ (u) · u) = χ(u) du where u = f , we have the desired diffeomorphism ν and that finishes the proof of the lemma and therefore the proof of the theorem.

Chapter 3 Rank 1 singularities in dimension 4
3.1 Introduction

In the first chapter we attained the differentiable equivalence between the singular Lagrangian foliation and the linearized one. In this chapter we shall see that this equivalence becomes symplectic in a covering of a neighbourhood of a nondegenerate singular periodic orbit L. That is to say, we consider a foliation F given by a completely integrable system on a four dimensional manifold fulfilling the hypotheses of non-degeneracy established in the first chapter. Then we consider a symplectic germ in a neighbourhood of the non-degenerate singular periodic orbit for which the foliation is Lagrangian. We prove that there exists a diffeomorphism defined in a neighbourhood of L then the foliation can be symplectically linearized in a neighbourhood of a singular periodic orbit. Namely, we will prove the following theorem,
4 Theorem 3.1.1 Let M0 = S 1 × D3 , endowed with the coordinates (θ, p, x, y). Let

37

38 F0 be the foliation given by: Y1 = Y2 = y ∈ {−1, 1} ( = 1 elliptic case, ∂ ∂θ

Chapter 3. Rank 1 in dimension 4

∂ ∂ − x ∂x ∂y

= −1 hyperbolic case).

Let L = S 1 × (0, 0, 0). Then any two symplectic two forms ω1 and ω2 in M 4 such that F0 becomes Lagrangian are equivalent, i.e, there exists a diffeomorphism between two neighbourhood of S 1 such that φ preserves F0 and φ∗ (ω2 ) = ω1 .

Observation 3.1.1 Once this theorem has been proved, as we proved in the first chapter that the singular Lagrangian foliation is differentiable equivalent to the
4 linearized one then it is symplectically equivalent to a neighbourhood of L in M0 ,

with a certain symplectic structure on it, but as this symplectic form is equivalent to the standard one, we finally get the desired symplectic equivalence which we formulate as: Theorem 3.1.2 Given an integrable Hamiltonian system on a symplectic manifold (M 4 , ω) with a non-degenerate singular periodic orbit L. Let F be the singular Lagrangian foliation associated to it. Let ω0 be the canonical symplectic structure
4 on M0 given by: ω0 = dp ∧ dθ + dx ∧ dy.

• If the singularity on L is elliptic there are neighbourhoods of L in M 4 and
4 M0 and a diffeomorphism between them φ such that φ∗ (ω0 ) = ω, sending F

to F0 . • If the singularity on L is hyperbolic there are in general a double covering of a
4 neighbourhood of L in M 4 , a neighbourhood of L in M0 and a diffeomorphism

φ between them such that φ∗ (ω0 ) = ω, sending F to F0 .

3.1. A Hamiltonian S 1 -action

39

The results contained in this section have been obtained jointly with Carlos Curr´s-Bosch and are contained in the paper [13]. a In any case we include two proofs of this fact here one of them is different from the one contained in the publication. Let us outline how the chapter is organized. In the first section, we prove the existence of a Hamiltonian S 1 -action tangent to the foliation which will be a key point in the proof of linearization. Namely, according to the first chapter, we may assume that the foliation is generated by Y1 =
∂ ∂θ ∂ ∂ and Y2 = y ∂x − x ∂y

∈ {−1, 1} being = 1 in the elliptic case and = −1

in the hyperbolic case. Let ω be a symplectic structure such that F is Lagrangian. In order to achieve the symplectic linearization we prove first the existence of a Hamiltonian S 1 -action by translations which is tangent to the foliation. In the second section we give two proofs of the theorem: The first one uses some of the analytical tools contained in the first chapter and defines some diffeomorphisms ad hoc. The second proof is based on the idea of finding a sort of splitting which separates clearly the singular part from the regular part of the foliation. Namely, we define two symplectic orthogonal distributions D1 and D2 such that Y1 ∈ D1 and Y2 ∈ D2 . Then we show that these two distributions are integrable. In this way we obtain new coordinates in a neighbourhood of the singular circle. Finally the symplectic form ω may be written as ω = ω1 + ω2 being ω1 and ω2 be two symplectic forms in the 2-dimensional submanifolds integrating the distributions D1 and D2 respectively. Since each distribution contains a vector field of the foliation, once reached this point, the symplectic linearization results in dimension 2 obtained in chapter 2 let us conclude the symplectic linearization process. This second proof sets a precedent for induction which will be used later.

40

Chapter 3. Rank 1 in dimension 4

3.2

Recovering a Hamiltonian S 1-action

In order to prove theorem 3.1.1 we need first to find a Hamiltonian S 1 -action by translations preserving the foliation.
4 Let ω be a symplectic form in M0 such that F0 is Lagrangian. Let f be a funcω tion defined on S 1 × D3 , we denote by Hf the Hamiltonian vector field associated

to f . Lemma 3.2.1 There exist coordinates (θ, p, x, y) in a neighbourhood of L ∼ S 1 × =
ω (0, 0, 0), such that Hp |N = ∂ ∂θ

on N = {(θ, p, 0, 0)}.

4 4 Proof: Let us consider N ⊂ M0 , N = {(θ, p, 0, 0)} and let i : N −→ M0 be the 4 inclusion of N in M0 . One can easily check that i∗ ω endows N with a symplectic

structure and that dp = 0 defines a regular Lagrangian foliation on N by circles; by a simple continuity argument: i ∂ ω|N = λdp , λ = 0, so one can take coordinates
∂θ

4 (θ, p) in N such that i ∂ (i∗ ω) = dp. Considering (θ, p, x, y) as coordinates in M0
∂θ

4 (after shrinking M0 if necessary), we have

i ∂ ω|N = dp.
∂θ

4 To avoid unnecessary changes of notation, we will consider from now on M0 ω endowed with coordinates (θ, p, x, y) verifying Hp = ∂ ∂θ

on N .

By using the generalized Poincar´ Lemma one can write e

ω = d(Adθ + Bdp + Cdx + Ddy) where A, B, C, D are differentiable functions vanishing on L. Now we need the following

3.2. A Hamiltonian S 1 -action

41

Lemma 3.2.2 There exist coordinates (θ, p, x, y) in a neighbourhood of L such that ω = d(pdθ + Bdp + Cdx + Ddy) where B, C, D vanish along L. Proof: Considering the decomposition A(θ, p, x, y) = A0 (p, x, y) + We can write ω = d(A0 (p, x, y)dθ + d(A) + (B − ∂A ∂A ∂A )dp + (C − )dx + (D − )dy). ∂p ∂x ∂y ∂A . ∂θ

Now ω = d(A0 (p, x, y)dθ + Bdp + Cdx + Ddy). Let us see that A0 is basic for the foliation F0 . As F0 is Lagrangian for ω, we have ∂ ∂ ∂ + x )A0 + (yC − xD) = 0. ∂x ∂y ∂θ So this yields the following two equalities (−y (y ∂ ∂ − x )A0 = 0 ∂x ∂y

−yC + xD = f (p, x, y). The first condition together with
ω On the other hand, as Hp = ∂A0 = 0 implies that A0 is basic for the ∂θ ∂ on N , in particular we obtain ∂A0 = ∂θ ∂p

foliation. 1 on N .

So the following mapping ϕ:
4 M0

−→

4 M0

(θ, p, x, y) −→ (θ, A0 (p, x, y), x, y)

42 is a foliation preserving diffeomorphism. Finally,

Chapter 3. Rank 1 in dimension 4

ϕ∗ (d(pdθ + B2 dp + C2 dx + D2 dy)) = ω.
ω Notice that as on N , Hp = ∂ , ∂θ

the following functions

∂B2 ∂C2 , ∂θ ∂θ

and

∂D2 ∂θ

vanish

on N . So, in particular B2 , C2 , D2 are constant on L. As ω = d(pdθ +(B2 −B2 (θ, 0, 0, 0))dp+(C2 −C2 (θ, 0, 0, 0))dx+(D2 −D2 (θ, 0, 0, 0))dy), we can assume that the coefficients B, C, D are zero along L. We will need the following lemma which is an application of Moser’s path method: Lemma 3.2.3 Let α be a 1-form, vanishing on L, and F0 -basic and let ω1 be a
4 symplectic structure on M0 such that F0 is Lagrangian. Then:

1. The 2-form ωo = ω1 − dα is a symplectic structure in a neighbourhood of L and makes the foliation Lagrangian.
4 2. There is a diffeomorphism η between two neighbourhoods of L in M0 such

that it preserves F0 and η ∗ (ω1 ) = ω0 . Proof: Let ω1 = d(pdθ + Bdp + Cdx + Ddy). As α is basic for the foliation, α = F (xdx + ydy) + Gdp. Consider the following family of 2-forms ωt = ω0 + t(ω1 − ω0 ), t ∈ [0, 1]. So,

ωt = d(pdθ + Bdp + Cdx + Ddy + (t − 1)F (xdx + ydy) + (t − 1)Gdp), where B, C, D and G vanish along L. And therefore ωt |L is the 2-form

3.2. A Hamiltonian S 1 -action

43

ωt |L = dp ∧ dθ + ( +(

∂B ∂C ∂G − + (t − 1) )dx ∧ dp+ ∂x ∂p ∂x

∂B ∂D ∂G ∂D ∂C − + (t − 1) )dy ∧ dp + ( − )dx ∧ dy, ∂y ∂p ∂y ∂x ∂y
∂C )| ∂y L

ω1 |L non-degenerate implies that ( ∂D − ∂x

= 0 and one can check that this

implies that ωt is non-degenerate along L, for all t ∈ [0, 1]. Therefore, we may assume that ∀t ∈ [0, 1] ωt is symplectic in a tubular neighbourhood of L. Moreover, as i ∂ ωt = (
∂θ

∂B ∂G ∂C ∂D ∂F − 1 + (t − 1) )dp + dx + dy + (t − 1) (xdx + ydy) ∂θ ∂θ ∂θ ∂θ ∂θ

we conclude that the foliation given by Y1 , Y2 is Lagrangian for all ωt . In particular, taking t = 0 we have proved the first assertion claimed in the lemma. Now, using non-degeneracy, we have a well-defined vector field Xt by the following equality iXt ωt = −α. (I) Notice that as we have assumed that α|L = 0, this guarantees ([59]) that the time-dependent vector field Xt is integrable (as it is integrable on L). Let φt stand for the “flow” of the time dependent vector field Xt defined by the conditions dφs . ds |s=t

φ0 = Id,

Xt =

We check that this field is tangent to the foliation, in this way the flow of the time-dependent vector field will preserve the leaves of the Lagrangian foliation. Let B = {(θ, p, 0, 0)} be the singular set for the foliation. We will see that the vector field is tangent to the foliation in two steps:

44 • Outside B:

Chapter 3. Rank 1 in dimension 4

Notice that since the regular leaves of the foliation are Lagrangian submanifolds for all ωt , any vector field belonging to its symplectic orthogonal has to be tangent to the foliation; but as α is basic for the foliation Xt verifies ωt (Xt , Y1 ) = −α(Y1 ) = 0 and ωt (Xt , Y2 ) = −α(Y2 ) = 0. So Xt is tangent to the foliation along the regular leaves. • In the singular set B: Let c = (θ0 , c0 , 0, 0) ∈ B. A singular leaf for the foliation through c is the circle Lc = {(θ, c0 , 0, 0)}. So a time-dependent vector field tangent to Lc at
∂ c has the form γt (θ, p, x, y) ∂θ . Let us check that Xt has this form: On B, Xt

has to be tangent to B otherwise its flow would reach a regular Lagrangian
∂ ∂ leaf of the foliation. So Xt |B = αt ∂p + βt ∂θ .

Finally, as Xt has to verify (I), αt has to be zero, and therefore Xt is tangent to the foliation along B. So, we conclude that its flow preserves the leaves of the foliation. Further, remember that we are looking for a symplectomorphism; this symplectomorphism will be given by the flow of the vector field Xt at time t = 1. The flow φt gives us a family of diffeomorphisms verifying: 1. It is equal to the identity on L. 2. φ∗ ωt = ω0 ; that is to say, as a particular case, we have φ∗ (ω1 ) = ω0 . 1 t 3. φt preserves the leaves of the foliation. So η = φ1 is the symplectomorphism we are looking for and the two symplectic forms ω0 and ω1 define equivalent symplectic structures. This proves the second assertion of the lemma.

3.2. A Hamiltonian S 1 -action

45

Now we are going to apply this lemma to prove that there is a Hamiltonian S 1 -action tangent to the foliation: Proposition 3.2.1 There is a Hamiltonian S 1 -action tangent to the foliation. In fact, there exist coordinates (θ, p, x, y) in a neighbourhood of L such that ω = d(pdθ + C(p, x, y)dx + D(p, x, y)dy) and the Hamiltonian S 1 -action is performed by translations with respect to θ. Proof: We will apply lema 3.2.3 in two stages: First stage: Consider ω1 = d(pdθ + Bdp + Cdx + Ddy) and ω0 = ω1 − d(Bdp). As B vanishes along L, applying lemma 3.2.3, ω0 defines a symplectic structure in a neighbourhood of L such that makes F0 into a Lagrangian foliation and it is equivalent to ω1 . And so far, we can assume ω = ω0 = d(pdθ + C(θ, p, x, y)dx + D(θ, p, x, y)dy). Second stage: As i ∂ ω = −dp +
∂θ

∂C ∂D dx + dy, ∂θ ∂θ

then

∂D ∂C dx + dy = λ(xdx + ydy) ∂θ ∂θ

for a certain function λ. Therefore, ∂C = xλ, ∂θ ∂D = yλ. ∂θ

46

Chapter 3. Rank 1 in dimension 4 This leads us to the following decomposition C = C0 (p, x, y) + x ∂H ∂θ ∂H . ∂θ

D = D0 (p, x, y) + y Now

∂H (xdx + ydy)). ∂θ Let ω0 = d(pdθ + C0 (p, x, y)dx + D0 (p, x, y)dy). So, we can apply lemma 3.2.3 ω = d(pdθ + C0 (p, x, y)dx + D0 (p, x, y)dy + again and we can assume ω = d(pdθ + C(p, x, y)dx + D(p, x, y)dy) Notice that i ∂ ω = −dp.
∂θ

So, S acts on this neighbourhood in a Hamiltonian fashion and p is the moment map. This concludes the proof of the proposition.

1

3.3

Two proofs for theorem 3.1.1

In this section we present two different proofs of the symplectic linearization for rank 1 singularities in dimension 4 which use the Hamiltonian S 1 -action obtained in the previous section. The first one is the one which appears in our joint paper with Carlos Curr´sa Bosch [13] but we also include here a proof for the elliptic rank 1 case which was already proved in a different way by Eliasson and Molino and Dufour. The second proof is based on a geometrical idea of finding a decomposition which singles out the singular part and the regular part. This technique has turned

3.3. Two proofs for theorem 3.1.1

47

out to be very useful to give a general proof for the symplectic linearization in any dimension. We define two symplectic orthogonal integrable distributions which allow to reduce the proof to the 2-dimensional case.

3.3.1

First proof

Let us go on with the proof of 3.1.1.
4 Without loss of generality, we may assume that the symplectic form on M0 =

S 1 × D3 can be expressed as: ω = dp ∧ dθ + A(p, x, y)dp ∧ dx + B(p, x, y)dp ∧ dy + C(p, x, y)dx ∧ dy Recall that F0 is, F0 =<
∂ , ∂θ ∂ ∂ y ∂x + x ∂y >.

Observation 3.3.1 As ω is symplectic C(0, 0, 0) = 0 so after shrinking the neighbourhood, if necessary, we may assume C(p, 0, 0) = 0. Let us introduce the following notation: For any function f (p, x, y), dT f will stand for the 1-form: dT f =
∂f dx+ ∂f dy. ∂x ∂y

We use the notation f to refer to the function f = x2 + y 2 , = −1 in the hyperbolic case.

where = 1 in the elliptic case and

Lemma 3.3.1 Given C(p, x, y) there exist C ∞ -functions χ and f such that: C(p, x, y)dx ∧ dy = χ(p, f )dx ∧ dy + dT f ∧ d(f ) Proof of the lemma: According to lemma 2.2.1 in the elliptic case we can write the following decomposition, C(p, x, y) = C1 (p, x2 + y 2 ) + X1 (C2 ) In the hyperbolic case, after performing the change of coordinates x = x + y, y = x − y, we can apply lemma 2.2.2 to write a similar decomposition,

48

Chapter 3. Rank 1 in dimension 4

C(p, x, y) = C1 (p, x2 + y 2 ) + X−1 (C2 ) Now take χ = C1 and f = the desired decomposition, C(p, x, y)dx ∧ dy = χ(p, f )dx ∧ dy + dT f ∧ d(f ) And this ends the proof of the lemma. Once reached this point, we can assert that ω can be written as:
−H 2

and the function decompositions above yield

ω = dp ∧ dθ + A(p, x, y)dp ∧ dx + B(p, x, y)dp ∧ dy + χ(p, f )dx ∧ dy + d(f df ) Now applying lemma 3.2.3 we can eliminate d(f df ) by using a diffeomorphism which preserves the foliation. The next step will be to “normalize” the coefficient of dx ∧ dy, we achieve this applying lemma 2.3.3 with parameters. Namely, Observe that if ψ(p, f ) is a smooth function such that ψ(0, 0) = 0. Then the mapping: G:
4 (M0 , 0)

−→

4 (G(M0 ), 0)

(θ, p, x, y) −→ (θ, p, x · ψ(p, f ), y · ψ(p, f )) defines a germ of diffeomorphism preserving the foliation df = 0. As we saw on the proof of lemma 2.3.3 we can take as ψ the solution of the equation: d 2 (ψ (p, u) · u) = χ(p, u) du where u = f , we have the desired φ, and that finishes the proof of the lemma. Using the previous lemma we can find a diffeomorphism preserving F0 , such that the pull-back of ω is:

3.3. Two proofs for theorem 3.1.1

49

ω = dp ∧ dθ + A(p, x, y)dp ∧ dx + B(p, x, y)dp ∧ dy + dx ∧ dy Finally, as ω is closed, there exists g(p, x, y) such that, A= And using the diffeomorphism: φ:
4 M0

∂g ∂x

B=

∂g ∂y

−→

4 M0

(θ, p, x, y) −→ (θ + g(p, x, y), p, x, y) which preserves F0 , we can write ω = dp ∧ dθ + dx ∧ dy. And this ends the proof of the theorem.

3.3.2

Second proof

In this section we construct two symplectic orthogonal distributions. Those distributions will be 2-dimensional regular distributions and will allow us to reduce the proof to the 2-dimensional case.
4 Recall that we may assume that the symplectic form on M0 = S 1 × D3 can be

expressed as: ω = dp ∧ dθ + A(p, x, y)dp ∧ dx + B(p, x, y)dp ∧ dy + C(p, x, y)dx ∧ dy Lemma 3.3.2 The distribution D1 =< X, Y > defined by the relations: iX ω = dp iY ω = dθ is C ∞ , symplectic in a neighbourhood of p and involutive everywhere. Proof: First of all, since ω is symplectic and the forms dp and dθ are differentiable and independent, the distribution D1 is clearly C ∞ and regular. Now let us prove

50

Chapter 3. Rank 1 in dimension 4

that this distribution is symplectic. Observe that this distribution is symplectically orthogonal to the distribution D2 =< gonal is also symplectic. Now let us see that this distribution is involutive. We have to check that [X, Y ] ∈ D1 , ∀X, Y ∈ D1 . In fact, it is enough to prove that [X, Y ] ∈ D1 for vector fields which are independent on a dense set in the neighbourhood considered. So we can take X = Y1 = LY1 (ω(Y,
∂ . ∂θ ∂ , ∂ ∂x ∂y

>. As a consecuence of observation

3.3.1, D2 is symplectic in a neighbourhood of the origin then its symplectic ortho-

By Leibnitz’s rule:

∂ ∂ ∂ ∂ )) = LY1 (ω)(Y, ) + ω(LY1 Y, ) + ω(Y, LY1 ( )) ∂x ∂x ∂x ∂x

Now if we take any Y ∈ D1 then the left hand side of the equality above equals zero. As for the right hand side: The first term is zero because Y1 is Hamiltonian
∂ and, in particular, it is symplectic; the third term vanishes because LY1 ( ∂x ) = 0. ∂ ∂ So we are led to ω(LY1 Y, ∂x ) = 0. In the same way, we prove that ω(LY1 Y, ∂y ) = 0

and therefore the distribution is involutive.

Now consider two distributions D1 and D2 those distributions are symplectically orthogonal distributions and contain Y1 and Y2 , respectively. Since these regular distributions are involutive, there are regular foliations F1 and F2 integrating D1 and D2 respectively. Now take a point p in the singular orbit, Frobenius Theorem provides new coordinates (p, θ, x, y) in a neighbourhood of p such that the leaves of F1 are L1b = {(p, θ, b1 , b2 ), b1 , b2 ∈ R} and the leaves of F2 are L2a = {(a1 , a2 , x, y), a1 ∈ R a2 ∈ [0, 2π)}. Since the vector field
∂ ∂θ

is Hamiltonian,

in particular its flow is a symplectomorphism and preserves the symplectic orthogonal decomposition. Thus, sliding along the singular circle, we can extend these coordinates to a whole neighbourhood of the singular orbit. Now let us see what the expression of ω is in these coordinates; Since D1 and D2 are symplectically ortogonal and since dω = 0, in these new coordinates the symplectic form can be

3.3. Two proofs for theorem 3.1.1 written as: ω 2 = A(p, θ)dp ∧ dθ + B(x, y)dx ∧ dy.

51

Observe on the other hand, that Y1 defines a foliation by circles on the submanifolds L1b and Y2 defines a foliation on the submanifolds L2a with a singularity of elliptic type when = 1 and hyperbolic type when = −1. It remains to apply the known results of symplectic uniqueness in dimension 2. Namely, Y1 defines a regular Lagrangian foliation by circles on L1b . Thus after the theorem of LiouvilleMineur-Arnold, there exists a local diffeomorphism defined in a neighbourhood of the singular circle such that φ∗ (dp∧dθ) = A(p, θ)dp∧dθ, being p, θ the coordinates 1 defined by φ1 . The vector field Y2 defines a nondegenerate singularity in each 2dimensional submanifold L2a . So we may apply the symplectic linearization result in dimension 2 ( Theorem 2.3.1) to find a local diffeomorphism in a neighbourhood of the origin φ2 such that φ∗ (dx ∧ dy) = B(x, θ)dx ∧ dy, being x, y the coordina2 tes provided by the diffeomorphism φ2 . Finally, we define a diffeomorphism in a neighbourhood of L, φ(p, θ, x, y) = (φ1 (p, θ), φ2 (x, y)). This diffeomorphism preserves the foliation F and satisfies that (φ∗ )(dp ∧ dθ + dx ∧ dy) = ω 2 . This ends the second proof of the theorem.

52

Chapter 3. Rank 1 in dimension 4

Chapter 4 Rank 0 singularities in dimension 4
4.1 Introduction

In this section we consider the symplectic linearization problem for foliations defined by a completely integrable system with rank 0 singularities in dimension 4. According to the first chapter we will assume that the foliations are already linear. That is, F =< X1 , X2 >. Recall that there are 4 possible cases from a differentiable point of view: • Elliptic-elliptic case: In this case there are coordinates (x1 , y1 , x2 , y2 ) centered at the point p such that two first integrals of the Hamiltonian system are f1 =
2 x2 +y1 1 2

and f2 =

2 x2 +y2 2 . 2

• Elliptic-hyperbolic case: There are coordinates (x1 , y1 , x2 , y2 ) centered at the point p such that two first integrals of the Hamiltonian system are f1 =
2 x2 +y1 1 2

and f2 =

2 x2 −y2 2 . 2

• Hyperbolic-hyperbolic case: There are coordinates (x1 , y1 , x2 , y2 ) centered at the point p such that two first integrals of the Hamiltonian system are 53

54 f1 =
2 x2 −y1 1 2

Chapter 4. Rank 0 in dimension 4 and f2 =
2 x2 −y2 2 . 2

• The focus-focus case: There are coordinates (x1 , y1 , x2 , y2 ) centered at the point p such that two first integrals of the Hamiltonian system are f1 = x1 y1 + x2 y2 and f2 = x1 y2 − x2 y1 . For the sake of brevity, we will refer to the first three types as decomposable at p and we will say that F is non-decomposable at p if its Williamson type is of focus-focus type. Following the spirit of the preceding chapters, we look for a local symplectic classification of a foliation defined by the “linearized” model under the only constraint that the regular leaves of the foliation are Lagrangian. We give a complete proof for the symplectic uniqueness result in all the decomposable cases. Recall that this symplectic uniqueness for the “linearized” model is what we call symplectic linearization. We do not provide a proof for the focus-focus case. In fact, so far, two proofs for the symplectic linearization in the focus-focus case are known to the author: The one provided by Eliasson in his thesis [23] and another proof provided by Nguyen Tien Zung, recently [65]. We would like to point out that this symplectic uniqueness is a consequence of Eliasson’s Theorem. One of the main motivations for providing another proof for this fact is the following: the key point of Eliasson’s proof is Proposition 4 in his thesis [23] but this proposition is stated without proof. The analogous proposition when the system is of analytical nature was proved by Vey [54] (lemma 1). The transition from the analytical case to the smooth case entails a non-trivial work with flat functions which in our opinion cannot be neglected. A proof for this Proposition when the system is completely elliptic is contained in Eliasson’s paper [24]. In the case that the dimension of the manifold is equal to 2 or the dimension of this manifold is 4 and the foliation corresponds to a foliation

4.1. Introduction

55

of focus-focus type, a proof of this fact is contained in Eliasson’s thesis. In fact when the dimension of the manifold is equal to 2 and the foliation is of hyperbolic type this had been formerly proved by Colin de Verdi`re and Vey [6]. As far as the e remaining cases are concerned, to our knowledge the only attempt to provide a proof in the completely hyperbolic case is due to San Vu Ngoc [57]. Unfortunately, the proof provided by San Vu Ngoc in that paper is based on a construction of a solution to a differential equation which is only smooth under some additional conditions concerning the flat terms which are not necessarily fulfilled under the hypotheses considered as we showed in Observation 2.2.1. Thus, from out point of view the proof provided in [57] does not fill the gap. Therefore, in our opinion, the problem remains unsolved in the decomposable cases which are not completely elliptic. In this chapter we are going to provide a proof for the local symplectic uniqueness in dimension 4 for all the cases which does not rely on Proposition 4 of Eliasson’s theorem. We do not attempt to give a proof of this Proposition but to prove the local uniqueness result. In order to do that, we look for a symplectic orthogonal decomposition in the decomposable cases to reduce the problem of local uniqueness in dimension 4 to a problem in dimension 2 as we did in chapter 3. Although the proof is inspired by ideas coming from geometry, we had to plunge into analytical problems in order to find an orthogonal symplectic decomposition. Some of those analytical questions have already been presented in chapter 2. The chapter is organized as follows: In the second section we pose the problem for the decomposable cases and we outline the strategy of the proof. In the third section we prove three basic lemmas that will be used in the uniqueness proof. One of them is a foliation preserving version of Moser path’s method for corank 2 singularities. In the fourth section we prove a common proposition concerning basic 1-forms attached to the foliation and the symplectic form. In the fifth sec-

56

Chapter 4. Rank 0 in dimension 4

tion, we look for a special Hamitonian vector field in the elliptic-elliptic case and elliptic-hyperbolic cases that will lead us to the symplectic orthogonal decomposition. We give two proofs of this fact: one is based on computation with forms and the second one is a geometrical proof which uses Bott-Weinstein connection and results for rank 1 singularities in dimension 4 obtained in chapter 3. In section 6 we find a special Hamiltonian vector field for hyperbolic-hyperbolic singularities. Finally, in section 7 we prove the existence of a symplectic orthogonal decomposition which yields the symplectic uniqueness for the decomposable cases (elliptic-elliptic, elliptic-hyperbolic and hyperbolic-hyperbolic cases). As a side remark, let us point out that this technique of reduction to lower dimensional cases can be exported to dimension higher than 4 using induction to give a complete proof for the local uniqueness theorem in any dimension as we will see in the next chapter.

4.2

Strategy of the proof

Let M be a 4-dimensional manifold endowed with coordinates (x1 , y1 , x2 , y2 ) and let p be the point p = (0, 0, 0, 0). In the sequel we will deal with germ-like objects in
∂ ∂ a neighbourhood of the point p. Consider the vector fields X1, 1 = x1 ∂y1 − 1 y1 ∂x1 ∂ and X2, 2 = x2 ∂y2 − ∂ 2 y2 ∂x2

where

1

and

2

can be either +1 or −1. Throug2 1 y1

hout this section, F will stand for the germ of foliation given by the vector fields X1, 1 and X2, 2 and f1 and f2 will stand for the first integrals f1 = x2 + 1 f2 = x2 + 2
2 2 y2 .

and

The pair ( 1 , 2 ) labels the foliation. When ( 1 , 2 ) = (1, 1) we say

that F is of elliptic-elliptic type. If ( 1 , 2 ) is (1, −1) or (−1, 1), we talk about a foliation of elliptic-hyperbolic type. And finally when ( 1 , 2 ) = (−1, −1) the foliation is referred to as a foliation of hyperbolic-hyperbolic type. Our goal is to study the germs of symplectic structures at p for which the decomposable F is generically Lagrangian. We say that a foliation is generically Lagrangian if its regular

4.3. Three common lemmas

57

leaves are Lagrangian submanifolds and its singular leaves are isotropic. In this section we prove that if ω is such a symplectic germ then there exists a local diffeomorphism φ preserving F, fixing p and such that φ∗ (ω) = dx1 ∧ dy1 + dx2 ∧ dy2 . We organize the proof in two stages: In the first stage we find a symplectically orthogonal decomposition adapted to F. The proof of the construction of this symplectic orthogonal decomposition differs slightly in each of the cases (ellipticelliptic, elliptic-hyperbolic, hyperbolic-hyperbolic). In the second stage we use this decomposition and the known results in dimension 2 to prove that any two symplectic germs are equivalent. Let us state the main concluding results contained in this chapter: Theorem 4.2.1 (Symplectically orthogonal decomposition) Let ω be a symplectic germ for which F is generically Lagrangian. Then there exists a symplectic germ ω equivalent to ω and there exist two symplectic distributions D1 and D2 such that: 1. D1 and D2 are involutive and symplectically orthogonal with respect to ω. 2. X1, 1 ∈ D1 and X2, 2 ∈ D2 . Theorem 4.2.2 (Symplectic Uniqueness) Let ω be a symplectic germ at p for which F is generically Lagrangian then ω is equivalent to ω0 = dx1 ∧dy1 +dx2 ∧dy2 . In order to prove Theorem 4.2.1, we will find an equivalent ω for which the vector field X1 will be Hamiltonian with Hamiltonian function f1 . But first we need a few propositions and lemmas.

4.3

Three common lemmas

In this section we are going to prove several lemmas which will be used later to prove the local uniqueness of the symplectic germs in the decomposable cases.

58

Chapter 4. Rank 0 in dimension 4 When we write X1 , X2 we mean a basis of the Lagrangian foliation. Let us set

the particular basis we are going to use throughout the section:
∂ ∂ ∂ ∂ • In the elliptic-elliptic case, X1 = x1 ∂y1 − y1 ∂x1 and X2 = x2 ∂y2 − y2 ∂x2 . ∂ ∂ ∂ ∂ • In the elliptic-hyperbolic case, X1 = x1 ∂y1 − y1 ∂x1 and X2 = x2 ∂y2 + y2 ∂x2 . ∂ ∂ ∂ ∂ • In the hyperbolic-hyperbolic case, X1 = x1 ∂y1 +y1 ∂x1 and X2 = x2 ∂y2 +y2 ∂x2 . ∂ ∂ ∂ ∂ • In the focus-focus case, X1 = −x1 ∂x1 + y1 ∂y1 − x2 ∂x2 + y2 ∂y2 and X2, 2 = ∂ ∂ ∂ ∂ x1 ∂x2 − x2 ∂x1 + y1 ∂y2 − y2 ∂y1 .

The first lemma is a foliation-preserving version of the Moser path method. Lemma 4.3.1 Let α be an F-basic 1-form and let ω1 be a symplectic germ for which F is Lagrangian. Then: 1. The 2-form ω0 = ω1 − dα is a symplectic structure in a neighbourhood of p and makes the foliation Lagrangian. 2. There is a diffeomorphism η between two neighbourhoods of p preserving F and such that η ∗ (ω1 ) = ω0 . Proof: First, let us check that ω0 is a symplectic form in a neighbourhood of p. Clearly, ω0 is a closed 2-form. Let us see that it is non-degenerate; Since α is basic for the foliation, α = F (df1 ) + G(df2 ). In particular α vanishes at p = (0, 0, 0, 0) and ω0|p equals ω1|p . Therefore, since ω1 is non-degenerate at p, the 2-form ω0 is non-degenerate in a neighbourhood of p. In order to prove that the foliation is Lagrangian for ω0 we have to check that ω0 (X1 , X2 ) = 0. Since ω0 (X1 , X2 ) = ω1 (X1 , X2 ) − dα(X1 , X2 ) and F is Lagrangian for ω1 , we have to see that dα(X1 , X2 ) vanishes. Since α is F-basic and [X1 , X2 ] = 0, the formula dα(X1 , X2 ) = X1 α(X2 ) − X2 α(X1 ) − α([X1 , X2 ]) implies that dα(X1 , X2 ) = 0 .

4.3. Three common lemmas

59

This ends the proof of the first assertion. In order to prove the second assertion we consider the following family of 2-forms: ωt = ω0 + t(ω1 − ω0 ), t ∈ [0, 1]

Let us see that these 2-forms are symplectic germs for which the foliation F is Lagrangian. Clearly, the 2-forms ωt are closed. And since ωt|p = ω0|p , we can repeat the argument above to see that the 2-forms are non-degenerate. Therefore, they are symplectic in a neighbourhood of the point p. Further, since ω0 (X1 , X2 ) = 0 and ω1 (X1 , X2 ) = 0, the foliation F is Lagrangian for ωt , ∀t ∈ [0, 1]. Now we are going to use Moser’s path method to conclude. First, we consider the well-defined vector field Xt by the following equality: iXt ωt = −α. Recall that α vanishes at p, this guarantees [59] that the time-dependent vector field Xt is integrable. Let φt stand for the “flow” of the time dependent vector field Xt defined by the conditions: φ0 = Id, Xt =
dφs ds |s=t

We check that this

field is tangent to the foliation, in this way the flow of the time-dependent vector field will preserve the leaves of the Lagrangian foliation. Consider the sets: B1 = {(x1 , y1 , 0, 0)} \ {(0, 0, 0, 0)} and B2 = {(0, 0, x2 , y2 )} \ {(0, 0, 0, 0)}. Then the singular set for the foliation is B = {(0, 0, 0, 0)} ∪ B1 ∪ B2 if the foliation is decomposable. In the focus-focus case the singular set for the foliation reduces to (0, 0, 0, 0). We will see that the vector field is tangent to the foliation in two steps: • Outside B: Notice that since the regular leaves of the foliation are Lagrangian submanifolds for all ωt , any vector field belonging to its symplectic orthogonal has to be tangent to the foliation; but as α is basic for the foliation Xt verifies ωt (Xt , X1 ) = −α(X1 ) = 0 and ωt (Xt , X2 ) = −α(X2 ) = 0. So Xt is tangent to the foliation along the regular leaves.

60

Chapter 4. Rank 0 in dimension 4 • Along B, the singular set: We check that the vector field Xt is tangent to the foliation in the following cases: – The focus-focus case: In this case the only singular point is the origin. The vector field Xt is tangent to the foliation because α vanishes at p = (0, 0, 0, 0) and therefore Xt vanishes at p. – The decomposable cases. In this case the singular set is B = {(0, 0, 0, 0)}∪ B1 ∪ B2 . Let us check that Xt is tangent to the foliation: 1. At the point p = (0, 0, 0, 0): The vector field Xt is tangent to the foliation because α vanishes at p = (0, 0, 0, 0) and therefore Xt vanishes at p. 2. At a point q1 = (a1 , b1 , 0, 0) ∈ B1 : Since α|p = F (q1 )(a1 dx1 + b1 dy1 ), the symplectic orthogonal to Xt at the point q1 is generated by the
∂ vector fields X1 , ∂x2 and ∂ ∂y2

and since Xt|q1 we can write Xt|q1 =

∂ ∂ ft (a1 , b1 , 0, 0)X1 +gt (a1 , b1 , 0, 0) ∂x2 +ht (a1 , b1 , 0, 0) ∂y2 . If Xt|q1 is not

a multiple of X1 , the flow of Xt starting at q1 would reach a regular Lagrangian leaf of the foliation. So Xt|q1 = ft (a1 , b1 , 0, 0)X1 and, in particular, it is tangent to the foliation. 3. At a point q2 = (0, 0, a2 , b2 ) ∈ B2 : We proceed as in the previous case to see that Xt|q2 = rt (0, 0, a2 , b2 )X2 , and again Xt is tangent to the foliation at the point q2 . So, we conclude that its flow preserves the leaves of the foliation. Further, remember that we are looking for a symplectomorphism; this symplectomorphism will be given by the flow of the vector field Xt at time t = 1. Remember that the flow φt gives us a family of diffeomorphisms verifying: 1. φt (p) = p. 2. φ∗ ωt = ω0 ; that is to say, as a particular case, we have: φ∗ (ω1 ) = ω0 . 1 t

4.3. Three common lemmas 3. φt preserves the leaves of the foliation.

61

So φ1 is the symplectomorphism we are looking for and the two symplectic forms ω0 and ω1 define equivalent symplectic structures. This proves the second assertion of the lemma.

The following lemma will be used in the proof of the symplectic uniqueness for the decomposable cases. It also holds in the focus-focus case but we have included the proof only for the decomposable cases. The lemma proves that the forms iX1 ω and iX2 ω are basic for the foliation. That is, they are a combination of dfi for fi defining the foliation. Under these assumptions, Lemma 4.3.2 There exists C ∞ -functions h1 , h2 , g1 and g2 such that: iX1 ω = h1 df1 + h2 df2 iX2 ω = g1 df1 + g2 df2 Proof: Let us check that iX1 ω = H1 df1 + H2 df2 and that iX2 ω = G1 df1 + G2 df2 for certain differentiable functions. Let the symplectic form ω be ω = Adx1 ∧ dy1 + Bdx1 ∧ dx2 + Cdx1 ∧ dy2 + Ddy1 ∧ dx2 + Edy1 ∧ dy2 + F dx2 ∧ dy2 .
∂ ∂ In the decomposable cases, the foliation is generated by X1, 1 = x1 ∂y1 − 1 y1 ∂x1 ∂ and X2, 2 = x2 ∂y2 − ∂ 2 y2 ∂x2

.

where

1

and

2

can be either +1 or −1. If

1

and

2

have different sign, we say that the foliation is of elliptic-hyperbolic type. If the pair ( 1 , 2 ) = (1, 1), we say that the foliation is elliptic-elliptic. Finally, if the pair ( 1 , 2 ) = (−1, −1), we say that the foliation is hyperbolic-hyperbolic. Now let us look at the contractions iX1, 1 ω and iX2, 2 ω: iX1, 1 ω = −A(x1 dx1 +
1 y1 dy1 )

+ (Dx1 −

1 By1 )dx2

+ (Ex1 − Cy1 )dy2

62

Chapter 4. Rank 0 in dimension 4 iX2, 2 ω = ( 2 By2 − Cx2 )dx1 + ( 2 Dy2 − Ex2 )dy1 − F (x2 dx2 +
2 y2 dy2 )

But since F is Lagrangian, we have iX1 ω(X2 ) = 0; and so we are led to the equality: y2 (Dx1 −
1 By1 )

= ( 1 Cy1 − Ex1 )x2
Dx1 − 1 By1 x2

(I).
1 Cy1 −Ex1

From here it is clear that if we take H2 = equalities hold: Dx1 − 1 By1 = H2 x2 and y2 (Dx1 −
1 By1 ).

=

y2

, the following

1 Cy1 −Ex1

= H2 y2 . To check that this H2

is a C ∞ -function, we apply a classical integration trick: Consider φ(x1 , y1 , x2 , y2 ) = Then we can write the following decomposition:
1

φ(x1 , y1 , x2 , y2 ) = φ(x1 , y1 , 0, y2 ) + x2
0

∂φ (x1 , y1 , tx2 , y2 )dt. ∂x2

Due to (I) the function φ vanishes on (x1 , y1 , 0, y2 ), this implies that H2 equals the function
Dx1 − 1 By1 x2 1 ∂φ (x1 , y1 , tx2 , y2 )dt 0 ∂x2

which is C ∞ . So taking H1 = −A and H2 =

we have proven that iX1, 1 ω = H1 df1 + H2 df2 . The condition of La-

grangianity can also be written as: iX2, 2 ω(X1, 1 ) = 0; and this leads us now to the equality: y1 ( 2 By2 − Cx2 ) = ( 2 Dy2 − Ex2 )x1 . As before, we can prove that G1 =
2 By2 −Cx2

x1

is C ∞ . And taking G2 = −F we have that: iX2, 2 ω = G1 df1 + G2 df2 .

Finally, let us state and proof a very simple lemma which is a consequence of Cartan’s formula and the Lagrangianity condition. Lemma 4.3.3 The following equality holds LX2 iX1 ω = LX1 iX2 ω. Proof: First, since i[X1 ,X2 ] ω = LX1 iX2 ω − iX2 LX1 ω and [X1 , X2 ] = 0, then LX1 iX2 ω = iX2 LX1 ω. Now we compute: iX2 LX1 ω = iX2 diX1 ω
dω=0 LX =diX +iX d

=

LX2 iX1 ω − diX2 iX1 ω

iX2 iX1 ω=0

=

LX2 iX1 ω

4.4. A common proposition And this proves the formula.

63

Observe that this lemma holds for all the cases, decomposable or non-decomposable and also in any dimension.

4.4

A common proposition

We will assume throughout the section that the foliation is decomposable since the proofs are supplied just in the decomposable cases. As a matter of fact, as we saw on the preceeding sections the proposition holds also for the focus-focus case. But we do not give a proof for this fact. In subsequent sections we will try to identify the Hamiltonian vector field associated to f1 . The main goal will be to find new coordinates in such a way that X1 can be identified with the Hamiltonian vector field of the function f1 in convenient coordinates. The first step is given by the following proposition, Proposition 4.4.1 There exists a symplectic germ ω 1 equivalent to ω such that, iX1, 1 ω 1 = H1 df1 + H2 df2 . for F-basic functions H1 and H2 . Proof: First by lemma 4.3.2 we can write iX1, 1 ω = H1 df1 + H2 df2 . We distinguish the following cases:

64

Chapter 4. Rank 0 in dimension 4

4.4.1

Proof of proposition 4.4.1 in the non-completely hyperbolic cases

We prove 4.4.1 in the elliptic-hyperbolic case and the elliptic-elliptic case: In this case, we can assume
1

= 1. For the sake of simplicity, we write X1

instead of X1,1 . Applying proposition 2.2.1 we can write:
2 H1 (x1 , y1 , x2 , y2 ) = h1 (x2 + y1 , x2 , y2 ) + X1 (h1 ) 1 2 H2 (x1 , y1 , x2 , y2 ) = h2 (x2 + y1 , x2 , y2 ) + X1 (h2 ) 1

.

Now consider the 1-form α = h1 df1 +h2 df2 . Since α is F-basic, we can apply lemma 4.3.1 taking ω1 = ω. As a consequence, ω 1 = ω − dα will be a symplectic germ equivalent to the initial ω. Let us check that for this ω 1 the conditions stated in the proposition are fulfilled. First, we calculate iX1 ω 1 . We have iX1 ω 1 = iX1 ω − iX1 dα. Due to Cartan’s formula, we have iX1 dα = diX1 α + LX1 α. But since α is F-basic, in particular iX1 α = 0. So in the end, iX1 dα = X1 (h1 )df1 + X1 (h2 )df2 . Finally, we
2 2 have that iX1 ω 1 = h1 (x2 + y1 , x2 , y2 )df1 + h2 (x2 + y1 , x2 , y2 )df2 . 1 1

Now iX1 ω 1 = h1 df1 + h2 df2 iX2, 2 ω 1 = g1 df1 + g2 df2 for certain differentiable functions g1 and g2 . For the sake of simplicity we write X2 instead of X2, 2 . According to lemma 4.3.3 the following formula (5.1) takes place LX2 iX1 ω 1 = LX1 iX2 ω 1 . From this formula we obtain the following relations, X2 (h1 ) = X1 (g1 ), X2 (h2 ) = X1 (g2 ), X1 (h1 ) = 0 and X1 (h2 ) = 0 to get (IIa) (IIb)

Now, apply LX1 to these equalities and use that [X1 , X2 ] = 0 and the fact that

4.4. A common proposition

65

X1 (X1 (g1 )) = 0, X1 (X1 (g2 )) = 0,

(A) (B)

Now we want to prove that this implies X1 (g1 ) = 0 and X1 (g2 ) = 0. In order to do this, we consider the change to polar coordinates given by the equalities x1 = r cos θ, y1 = r sin θ, x2 = x2 , y2 = y2 . This change of coordinates is valid outside the meagre set (0, 0, x2 , y2 ). In these new coordinates, equations (A) and (B) are written as
∂2 (g ) ∂θ2 1

= 0 and

∂2 (g ) ∂θ2 2

= 0, respectively. So from these equations, the

functions g1 and g2 are affine functions in the θ-coordinate. Since they are 2πperiodic in the coordinate θ, they have to be constant in the coordinate θ. And as a consequence the conditions X1 (g1 ) = 0 and X1 (g2 ) = 0 are satisfied in the whole neighbourhood of p considered. Finally, turning back to (II a) and (II b), we are led to the equalities X2 (h1 ) = 0 and X2 (h2 ) = 0. This completes the proof of the proposition in the elliptic-hyperbolic and the elliptic-elliptic cases.

4.4.2

Proof of proposition 4.4.1 in the completely hyperbolic cases

We prove proposition 4.4.1 in the hyperbolic-hyperbolic case. For the sake of simplicity, we write X1 instead of X1,−1 . We consider the change of coordinates, x1 = Applying proposition 2.2.2 we can write: H1 (u1 , v1 , x2 , y2 ) = h1 (u1 .v1 , x2 , y2 ) + X1 (h1 ) H2 (u1 , v1 , x2 , y2 ) = h2 (u1 .v1 , x2 , y2 ) + X1 (h2 ) .
u1 +v1 ,y 2

=

And X1 in the new coordinates can be written as, X1 =

u1 −v1 , x 2 = x2 , y 2 2 ∂ ∂ −u1 ∂u1 + v1 ∂v1 .

= y2 .

Now consider the 1-form α = h1 df1 +h2 df2 . Since α is F-basic, we can apply lemma 4.3.1 taking ω1 = ω. As a consequence, ω 1 = ω − dα will be a symplectic germ

66

Chapter 4. Rank 0 in dimension 4

equivalent to the initial ω. Let us check that for this ω 1 satisfies the conditions stated in the proposition. First, we compute iX1 ω 1 . We have iX1 ω 1 = iX1 ω − iX1 dα. Due to Cartan’s formula, we have iX1 dα = diX1 α + LX1 α. But since α is F-basic, in particular iX1 α = 0. So, iX1 dα = X1 (h1 )df1 + X1 (h2 )df2 . Finally, we have that iX1 ω 1 = h1 (u1 .v1 , x2 , y2 )df1 + h2 (u1 .v1 , x2 , y2 )df2 . Now iX1 ω 1 = h1 df1 + h2 df2 iX2, 2 ω 1 = g1 df1 + g2 df2 for certain differentiable functions g1 and g2 . For the sake of simplicity we write X2 instead of X2, 2 . According to lemma 4.3.3 the following formula (5.1) takes place LX2 iX1 ω 1 = LX1 iX2 ω 1 . From this formula we obtain the following relations, X2 (h1 ) = X1 (g1 ), X2 (h2 ) = X1 (g2 ), (IIa) (IIb)

.

We are going to use this relations to draw conclusions about the (u1 , v1 )-jets of h1 and g1 along {(0, 0, x2 , y2 )} using lemma 2.2.1. By construction and as it was seen in the proof of lemma 2.2.2 the (u1 , v1 )-jet of h1 along {(0, 0, x2 , y2 )} is of the form
i hii ui v1 . 1 i

On the other hand as it was seen in the proof of lemma 2.2.1, the (u1 , v1 )-jet of X1 (g1 ) along {(0, 0, x2 , y2 )} has the form
j rij ui v1 1 i=j

4.4. A common proposition

67

for certain differentiable functions rij (x2 , y2 ). With all this information at hand, we can look at the equation (IIA), X2 (h1 ) = X1 (g1 ) at the level of (u1 , v1 )-jets along {(0, 0, x2 , y2 )}. We obtain
i i X2 (hii )ui v1 = 1 i=j j rij ui v1 . In particular, X2 (hii ) = 1

0, ∀i. And from this relations h1 = S1 + φ1 where S1 satisfies X2 (S1 ) = 0 (and X1 (S1 ) = 0) and φ1 is an (u1 , v1 )-flat function along {(0, 0, x2 , y2 )}. Finally, apply lemma 2.2.2 to ensure that we can write φ1 = X1 (R1 ) for a smooth R1 . Therefore, so far we have h1 = S1 + X1 (R1 ), h2 = S2 + X1 (R2 ), Now, iX1 ω 1 = (S1 + X1 (R1 ))df1 + (S2 + X1 (R2 ))df2 for basic S1 and S2 . Finally, we apply Moser again 4.3.1 with α = R1 df1 + R2 df2 to obtain a new symplectic form ω 2 equivalent to ω such that iX1 ω 1 = S1 df1 + S2 df2 for basic functions S1 and S2 . This ends the proof of the proposition in the hyperbolichyperbolic case and therefore the proof of the proposition. X1 (S2 ) = 0, X1 (S1 ) = 0, X2 (S2 ) = 0. X2 (S1 ) = 0. We may proceed in the same way for h2 to write the following decomposition

4.4.3

A normalization result

Observe that h1 = −A, that is to say −h1 coincides with the coefficient function of dx1 ∧ dy1 . If we could “normalize” our symplectic form in the (x1 , y1 )-direction ( that is to say, find a foliation preserving symplectomorphism such that A = 1) we would be closer to our result. The following lemmas will ensure that we can “normalize” our symplectic form in a foliation preserving way. Lemma 4.4.1 Let ω be a symplectic form such that the foliation F is Lagrangian. Let D1 and D2 stand for the distributions D1 =<
∂ , ∂ ∂x1 ∂y1

> and D2 =<

∂ , ∂ ∂x2 ∂y2

>.

68

Chapter 4. Rank 0 in dimension 4

Let ω|D1 and ω|D2 be the restriction of ω to the planes integrating the distributions D1 and D2 , respectively. Then ω|D1 and ω|D2 are symplectic forms in a neighbourhood of the point p. Proof: The condition ω(X1, 1 , X2, 2 ) = 0 for all q in the neighbourhood considered, implies in particular the following relations: w( w( ∂ ∂ , )|q = 0, ∂x1 ∂y2 (y1 ,x2 ) q(y1 ,x2 )=(0,y1 ,x2 ,0) (I)

∂ ∂ , )|q = 0, q(y1 ,y2 )=(0,y1 ,0,y2 ) (II) ∂x1 ∂x2 (y1 ,y2 ) ∂ ∂ w( , )|q = 0, q(x1 ,x2 )=(x1 ,0,x2 ,0) (III) ∂y1 ∂y2 (x1 ,x2 ) w( ∂ ∂ , )|q = 0, ∂y1 ∂x2 (y1 ,x2 ) q(y1 ,x2 )=(x1 ,0,0,y2 ) (IV )

In particular all these relations are fulfilled at the point p = (0, 0, 0, 0). If ω|D1
∂ ∂ was not symplectic at p, then the w( ∂x1 , ∂y1 )|p = 0. But this condition together

with conditions (I) and (III) would imply that ω|p would vanish on a 3-dimensional vector space, which is not possible because the initial ω is symplectic. In the same way, we can prove that ω|D2 is symplectic at p. Since we have proved that ω|D1 and ω|D2 are symplectic at p and the condition of being symplectic is an open condition, they are also symplectic in a neighbourhood of p.

Lemma 4.4.2 There exists a symplectic germ ω 2 equivalent to ω 1 such that: iX1 ω 2 = −df1 + h2 df2 . Proof: Due to 4.4.1, the plane Π = (x1 , y1 , 0, 0) is symplectic and this implies that A = 0 in a neighbourhood of p. We apply the same trick as in the proof of lemma 2.3.3 in chapter 2 but with two variables u and v corresponding to the first integrals

4.4. A special Hamiltonian for the non-hyperbolic cases

69

of the foliation f1 and f2 . That is, observe that if ψ(f1 , f2 ) is any differentiable function of f1 and f2 such that ψ(0, 0) = 0 and U stands for a neighbourhood of the origin where everything is defined, the mapping G: (U, 0) −→ (G(U ), 0)

(x1 , y1 , x2 , y2 ) −→ (x1 · ψ(f1 , f2 ), y1 · ψ(f1 , f2 ), x2 , y2 ) defines a germ of diffeomorphism preserving the foliation defined by f1 and f2 . Consider the equation d 2 (ψ (u, v) · u) = A(u, v), du where u = f1 and v = f2 . As we saw in the proof of 2.3.3 ψ is smooth and normalizes the coefficient function of dx1 ∧ dy1 . Furthermore, since the function A is basic, this diffeomorphism is foliation preserving. Now if we define ω 2 = (φ−1 )∗ ω 1 . Then ω 2 is equivalent to ω 1 and satisfies iX1 ω 2 = −df1 + h2 df2 for a certain differentiable function h2 .

4.5

A special Hamiltonian for the non-completely hyperbolic cases

The aim of this section is to prove the existence of a diffeomorphism taking the initial symplectic form to a symplectic form for which the vector field X1 is the Hamiltonian vector field associated to f1 . The proof uses the preceeding lemmas but is shorter if the singularity is non-completely hyperbolic. The hyperbolic-hyperbolic case will be treated separately in the next section. The result is summed up in the following proposition,

70

Chapter 4. Rank 0 in dimension 4

Proposition 4.5.1 Let F be a foliation of elliptic-elliptic type or elliptic-hyperbolic
∂ ∂ type. Let X1 be the vector field X1 = x1 ∂y1 − y1 ∂x1 belonging to F. Under the as-

sumptions of lemma 4.4.2, we have: iX1 ω 2 = −df1 .

We present two proofs of this proposition. The first one uses decompositions of symplectic 2-forms while the second one is based on a geometrical argument.

4.5.1
Proof:

First proof of proposition 4.5.1

Let us recap information on the symplectic form. Since ω 2 is locally exact, we can write: ω 2 = dx1 ∧ dy1 + d(A1 dx1 + B1 dy1 + A2 dx2 + B2 dy2 ). Given a smooth function f , we denote by d(1) (h) =
∂h dx2 ∂x2 ∂h dx1 ∂x1

+

∂h dy1 ∂y1

and d(2) (h) =

+

∂h dy2 . ∂y2

By lemma 4.4.2, iX1 ω 2 = −df1 + h2 df2 .

This yields d(1) (A1 dx1 + B1 dy1 ) = 0 and therefore, A1 dx1 + B1 dy1 = d(1) (g1 ) for a certain differentiable function. On the other hand, taking into account that d(g1 ) = d(1) (g1 ) + d(2) (g1 ), the above symplectic form becomes, ω 2 = dx1 ∧ dy1 + d(d(g1 ) − d(2) (g1 ) + A2 dx2 + B2 dy2 ). So after gathering coefficients of the terms dx2 and dy2 we obtain, ω 2 = dx1 ∧ dy1 + d(C2 dx2 + D2 dy2 ),

4.5. A special Hamiltonian for the non-hyperbolic cases for certain smooth functions C2 and D2 . Now, we compute the contraction iX1 ω 2 again with this expression to get: iX1 ω 2 = −df1 + X1 (C2 )dx2 + X1 (D2 )dy2 . The Lagrangian condition yields in the elliptic-elliptic case, X1 (C2 ) = h2 (x2 ) X1 (D2 ) = h2 (y2 ) And X1 (C2 ) = in the elliptic-hyperbolic case. h2 (x2 )

71

X1 (D2 ) = −h2 (y2 ) In both cases, since X1 (h2 ) = 0, we can apply LX1 in these relations to obtain, X1 (X1 (C2 )) = 0 , X1 (X1 (D2 )) = 0.

Using these equations, we want to deduce that X1 (C2 ) = 0 and X1 (D2 ) = 0. In order to do this, we consider the change to polar coordinates given by the equalities x1 = r cos θ, y1 = r sin θ, x2 = x2 , y2 = y2 . This change of coordinates is valid outside the meagre set (0, 0, x2 , y2 ). In these new coordinates the equations above are written as
∂2 (C2 ) ∂θ2

= 0 and

∂2 (D2 ) ∂θ2

= 0, respectively. So from these

equations,C2 and D2 are affine functions in the α-coordinate. Since they are 2πperiodic in the coordinate α, they have to be constant in the coordinate α. And as a consequence the conditions X1 (C2 ) = 0 and X1 (D2 ) = 0 are satisfied in the whole neighbourhood of p considered. Finally, from this equations we obtain h2 = 0, this proves that X1 is a Hamiltonian vector field with Hamiltonian function f1 and this ends the proof of the proposition.

72

Chapter 4. Rank 0 in dimension 4

4.5.2

The Bott-Weinstein connection and a geometrical proof of proposition 4.5.1

In this short section we propose a digression. We will give another proof of proposition 4.5.1 based on geometrical arguments concerning the Bott-Weinstein connection. Observe that, a posteriori, the vector field X1 is Hamiltonian. Hamiltonian vector fields are a special class of parallel vector fields with respect to the BottWeinstein connection defined in the neighbouring regular leaves of the Lagrangian foliation. Let us introduce the notion of Bott-Weinstein connection for a regular Lagrangian foliation. The Bott-Weinstein connection associated to a Lagrangian foliation. Let F be a regular n-dimensional Lagrangian foliation. We denote by functions which locally define F are parallel with respect to the Bott-Weinstein . Now the question [59] connection associated to F. We recall that the Hamiltonian vector fields of the arises: Is the converse true? That is to say, Can we assert that a parallel vector field is locally Hamiltonian? The following innocuous example can help us to see that this is false in general. For example take ω = dx1 ∧dy1 +dx2 ∧dy2 , consider the regular Lagrangian foliation generated by the vector fields X =
∂ ∂x1

and Y =

∂ . ∂x2

Those vector fields define in turn a basis of parallel vector fields with respect to the
∂ Bott-Weinstein connection. Consider now the vector field Z = ey2 ∂x1 . This vector

field is parallel but since iZ ω = ey2 dy1 , it is not locally Hamiltonian. As this example shows the affirmative answer is far from being true; But in the case the foliation is given by the vector fields X1 and X2 of elliptic-elliptic type or hyperbolic-elliptic type, we use the existence of a Hamiltonian S 1 -action tangent to the regular leaves of the foliation to prove that the parallel vector field X1 is indeed Hamiltonian. Namely, recall that in the second section of chapter 3 we recovered a Hamiltonian S 1 -action for the elliptic-hyperbolic case and the elliptic-elliptic case. We will use the existence of this action in the elliptic-hyperbolic case to prove the result. In

4.5. A special Hamiltonian for the non-hyperbolic cases

73

the completely elliptic case we will use the Liouville-Mineur-Arnold theorem applied to the compact regular leaves. This, in particular, leads to another proof of proposition 4.5.1. Second proof of 4.5.1 Let S be the singular set for the foliation S = {(0, 0, x2 , y2 )} ∪ {(x1 , y1 , 0, 0)}. We denote by B the dense set B = M \S. Then F = F ∩B is a regular Lagrangian foliation. Let be the Bott-Weinstein connection associated to F . We are going to prove that X1 is Hamiltonian in B. We are going to distinguish cases: • The Elliptic-elliptic case. The foliation F is a regular foliation by tori on B and the functions f1 and f2 are regular functions. According to Liouville-Mineur-Arnold, there exist a basis of Hamiltonian vector fields Z1 and Z2 which are periodic of constant period 2π and which are tangent to the foliation by tori. Those vector fields form a basis of parallel vector fields, so we may write the vector field X1 = g1 Z1 + g2 Z2 for basic functions g1 and g2 . Now the vector fields X1 , Z1 and Z2 are periodic vector fields of constant period, hence values in Q therefore using continuity the quotient
g1 g2 g1 g2

takes
p q

is a rational number

with (p, q) = 1. Summing up, we can write X1 = g2 ( p Z1 + Z2 ). If we proof q that g2 is constant then as a consequence X1 will be a Hamiltonian vector field. In order to do this we need the following well-known sublemma which will be useful later. Sublemma 4.5.1 Let XG1 and XG2 be two Hamiltonian vector fields tangent to F. Denote by φs G and φs G the time-s-map of XG1 and XG2 respectively. X X
1 2

74 Then φs G X
1

Chapter 4. Rank 0 in dimension 4

+XG2

= φs G ◦ φs G . X X
1 2

Proof Since {G1 , G2 } = ω(XG1 , XG2 ) and XG1 and XG2 are tangent to the Lagrangian fibration F then {G1 , G2 }L = 0 for any regular fiber L of F. On the other hand, since the set of regular fibers is dense and XG1 and XG2 are also tangent along the singular fibers, the bracket {G1 , G2 } vanishes everywhere. This implies in turn that [XG1 , XG2 ] = 0 and therefore φs G ◦ φt G = φt G ◦ φs G , X X X X
1 2 2 1

∀s, t.

(4.5.1)

Now consider αs = φs G ◦φs G . Due to 4.5.1, αs is a one-parameter subgroup. X X
1 2

It remains to compute the infinitesimal generator. Since d(φs G ◦ φs G ) X X
1 2

ds XG1 + XG2 .

=

d(φt G ◦ φs G ) X X
1 2

dt

|t=s

+

d(φs G ◦ φr G ) X X
1 2

dr

|r=s

,

setting s = 0 this expression implies that the infinitesimal generator of αs is

In particular this proves that φs G X
+XG2

1

= φs G ◦ φs G , X X
1 2

∀s.

Now going back to the vector field X1 applying this sublemma the period of X1 is
2πq . g2

But X1 is a periodic vector field with constant period. Therefore

g2 is constant and the vector field is Hamiltonian on B. Observe that since X1 is Hamiltonian on B we obtain d(iX1 ω 2 ) = 0 on B. This implies that

4.5. A special Hamiltonian for the non-hyperbolic cases
∂ h ∂x1 2

75

and

∂ h ∂y1 2

vanish on B and hence, using the density of the set B, they

vanish everywhere. So h2 does not depend on the variables x1 and y1 . But since iX1 ω 2 |(0,0,x2 ,y2 ) = 0, the function h2 vanishes. Therefore, iX1 ω 2 = −df1 as we wanted. • The elliptic-hyperbolic case In this case we cannot go straight to the regular foliation F (LiouvilleMineur-Arnold only works when we consider regular foliations with compact leaves and in the elliptic-hyperbolic case those leaves are cylinders). So let us consider the auxiliary foliation F = F ∩ (M \ {(0, 0, x2 , y2 )}, By virtue of proposition in chapter 3 , we know that there exists a unique Hamiltonian S 1 action which is tangent to the leaves of the B . Consider now Y1 , a vector field generated by this Hamiltonian S 1 -action with constant period 2πk. Complete Y1 to a basis Y1 , Y2 of Hamiltonian vector fields tangent to the leaves of F . Observe that Y2 cannot be periodic because the leaves of F ∩ B are cilinders. Using lemma 4.4.2, the vector field X1 can be expressed as X1 = Xf1 − h2 Xf2 for a basic function h2 . As a consequence, the vector field X1 is parallel with respect to and therefore we can write X1 = α1 Y1 + α2 Y2 for basic functions
2πk . α1

α1 and α2 . Since X1 and Y1 have periodic orbits but Y2 does not, α2 has to vanish. Now as X1 = α1 Y1 , the period of X1 has to be But the period
∂ h ∂x1 2

of X1 is 2π, so α1 = k and therefore, X1 is Hamiltonian on B. Finally, X1 is Hamiltonian on B yields d(iX1 ω 2 ) = 0 on B. This implies that
∂ h ∂y1 2

and

vanish on B and hence, using the density of the set B, they vanish

everywhere. So h2 does not depend on the variables x1 and y1 . But since iX1 ω 2 |(0,0,x2 ,y2 ) = 0, the function h2 vanishes. And in the end, the condition iX1 ω 2 = −df1 is met. This ends the second proof of proposition 4.5.1.

76

Chapter 4. Rank 0 in dimension 4

4.6

A special Hamiltonian for the completely hyperbolic cases

It remains to prove an equivalent proposition for the hyperbolic-hyperbolic case. In the hyperbolic-hyperbolic case we do not have a privileged S 1 -action. This means that to reach a similar result we need to apply our Moser type lemma again. Proposition 4.6.1 Let F be a foliation of hyperbolic-hyperbolic type. Let X1 be
∂ ∂ the vector field X1 = x1 ∂y1 + y1 ∂x1 belonging to F. Under the assumptions of

lemma 4.4.2, there exists an equivalent symplectic form ω 3 for which we have: iX1 ω 3 = −df1 . Proof: Since ω 2 is locally exact, we can write: ω 2 = dx1 ∧ dy1 + d(A1 dx1 + B1 dy1 + A2 dx2 + B2 dy2 ). Using lemma 4.4.2, iX1 ω 2 = −df1 + h2 df2 . This yields d(1) (A1 dx1 + B1 dy1 ) = 0 and therefore, A1 dx1 + B1 dy1 = d(1) (g1 ) for a certain differentiable function. And collecting the coefficients of the terms dx2 and dy2 as it was seen in the first proof of 4.5.1, we can write ω 2 = dx1 ∧ dy1 + d(C2 dx2 + D2 dy2 ) for certain smooth functions C2 and D2 . Now, we compute the contraction iX1 ω 2 again with this expression to get: iX1 ω 2 = −df1 + X1 (C2 )dx2 + X1 (D2 )dy2 . After making a change of coordinates in such a way that f1 = x1 y1 and f2 = x2 y2 , the Lagrangian condition implies,

4.6. Symplectic linearization for the decomposable cases

77

X1 (C2 ) = h2 (y2 ) X1 (D2 ) = h2 (x2 ) Let us use the first equation, for instance, to draw conclusions about the (x1 , y1 )-jets of the function h2 along {(0, 0, x2 , y2 )}. As observed in the proof of 2.2.2, the (x1 , y1 )-Taylor expand of the function X1 (C2 ) along {(0, 0, x2 , y2 )} has the form of the form
i=j j cij xi y1 where cij are smooth func1

tions in the variables (x2 , y2 ). Therefore the funtion h2 has a (x1 , y1 )-Taylor expand
i=j j hij xi y1 for certain differentiable hij . 1

But, since the function h2 is basic, X1 (h2 ) = 0. This implies that h2 is (x1 , y1 )flat along {(0, 0, x2 , y2 )}. Now we apply 2.2.2 to ensure that there exists a smooth h such that h2 = X1 (h). So far, iX1 ω 2 = −df1 + (X1 (h))df2 . Finally, we can apply lemma 4.3.1 with the basic 1-form α = hdf2 to obtain a new symplectic form ω 3 , equivalent to ω 2 for which, iX1 ω 3 = −df1 . This ends the proof of the proposition.

4.7

Two symplectic orthogonal distributions and symplectic linearization

For the sake of simplicity, let us unify the notation in all the cases and denote ω the symplectic form equivalent to the initial ω for which X1 is Hamiltonian with

78

Chapter 4. Rank 0 in dimension 4

Hamiltonian function f1 . Once reached this point, we are close to the symplectic orthogonal decomposition. Let us proof the following lemma before: Lemma 4.7.1 The distribution D1 =< X, Y > defined by the relations: iX ω = dx1 iY ω = dy1 is C ∞ , symplectic in a neighbourhood of p and involutive everywhere. Proof: First of all, since ω is symplectic and the forms dx1 and dy1 are differentiable and independent, the distribution D1 is clearly C ∞ and regular. Now let us prove that this distribution is symplectic. Observe that this distribution is symplectically orthogonal to the distribution D2 =<
∂ , ∂ ∂x2 ∂y2

> defined in lemma 4.4.1.

Since D2 is symplectic in a neighbourhood of p (lemma 4.4.1), the distribution D1 is also symplectic in a neighbourhood of p. Now let us see that this distribution is involutive. We have to check that [X, Y ] ∈ D1 , ∀X, Y ∈ D1 . In fact, it is enough to prove that [X, Y ] ∈ D1 for vector fields which are independent on a dense set in the neighbourhood considered. So we can take X = X1 . By Leibnitz’s rule: LX1 (ω(Y, ∂ ∂ ∂ ∂ )) = LX1 (ω)(Y, ) + ω(LX1 Y, ) + ω(Y, LX1 ( )) ∂x2 ∂x2 ∂x2 ∂x2

Now if we take any Y ∈ D1 then the left hand side of the equality above equals zero. As for the right hand side: The first term is zero because X1 is Hamiltonian
∂ and, in particular, it is symplectic; the third term vanishes because LX1 ( ∂x2 ) = 0. ∂ ∂ So we are led to ω(LX1 Y, ∂x2 ) = 0. In the same way, we prove that ω(LX1 Y, ∂y2 ) = 0

and therefore the distribution is involutive.

Now let us use the distribution above to prove Theorem 4.2.1. We recall the precise statement of Theorem 4.2.1.

4.7. Symplectic linearization for the decomposable cases Theorem 4.2.1(Symplectically orthogonal decomposition)

79

Let ω be a symplectic germ for which F is generically Lagrangian. Then there exists a symplectic germ ω equivalent to ω and there exist two symplectic distributions D1 and D2 such that: 1. D1 and D2 are involutive and symplectically orthogonal with respect to ω. 2. X1, 1 ∈ D1 and X2, 2 ∈ D2 .

Proof of Theorem 4.2.1: Consider D1 the distribution defined in the above lemma. Observe that propositions 4.5.1 and 4.6.1 prove that this distribution contains the vector field X1 in the elliptic-elliptic, elliptic-hyperbolic and hyperbolichyperbolic cases. On the other hand, we consider the distribution D2 =<
∂ , ∂ ∂x2 ∂y2

>,

this distribution contains the vector field X2 . The distribution D2 is symplectic due to lemma 4.4.1 and trivially involutive. The distribution D1 is symplectically orthogonal to D2 by construction. This ends up proving Theorem 4.2.1 in all the cases. Next step, we use the symplectic orthogonal decomposition to prove that there is just one symplectic germ making the foliation F into a Lagrangian foliation. For the sake of clarity, we recall the precise statement of Theorem 4.2.2: Theorem 4.2.2 Let ω be a symplectic germ at p for which F is generically Lagrangian then ω is equivalent to ω0 = dx1 ∧ dy1 + dx2 ∧ dy2 . Proof: Firstly, by virtue of Theorem 4.2.1 there exist symplectically orthogonal distributions D1 and D2 containing X1 and X2 , respectively. Since these regular

80

Chapter 4. Rank 0 in dimension 4

distributions are involutive, there are regular foliations F1 and F2 integrating D1 and D2 respectively. Furthermore, Frobenius Theorem provides new coordinates (x1 , y 1 , x2 , y 2 ) in a neighbourhood of p such that the leaves of F1 are L1b = {(x1 , y 1 , b1 , b2 ), b1 , b2 ∈ R} and the leaves of F2 are L2a = {(a1 , a2 , x2 , y 2 ), a1 , a2 ∈ R}. Since D1 and D2 are symplectically ortogonal and since dω 2 = 0, in these new coordinates the symplectic form can be written as: ω 2 = A(x1 , y 1 )dx1 ∧ dy 1 + B(x2 , y 2 )dx2 ∧ dy 2 . Since X1 belongs to D1 and X2 belongs to D2 it remains to apply the known results of symplectic uniqueness in dimension 2 (theorem 2.3.1 in section 2) in the (x1 , y 1 )-coordinates and in the (x2 , y 2 )-coordinates separately. More exactly, let us recall this results in dimension 2 and then we perform a composition of symplectomorphisms. Theorem 2.3.1 Let (M 2 , ω1 ) be a 2-dimensional symplectic manifold endowed with coordinates (x, y) and let F be a singular Lagrangian foliation with an elliptic or hyperbolic singularity at the origin (0, 0), then there exists a local diffeomorphism φ preserving F such that φ∗ (dx ∧ dy) = ω1 . Let φ1 and φ2 be the diffeomorphisms provided by the above theorem, attached to D1 and D2 respectively. We define a local diffeomorphism φ(x1 , y 1 , x2 , y 2 ) = (φ1 (x1 , y 1 ), φ2 (x2 , y 2 )). This diffeomorphism preserves the foliation F and satisfies that φ∗ (dx1 ∧ dy1 + dx2 ∧ dy2 ) = ω 2 .

Chapter 5 Higher dimensions
The aim of this chapter is to give a general result for symplectic linearization in arbitrary dimension and for foliations defined by completely integrable systems in a neighbourhood of a rank k-orbit of the system. In the preceeding chapter we set a precedent for induction. The idea of the proof relies on an inductive process and the symplectic orthogonal decomposition. Those tecniques were thoroughly studied in the last chapter. Most of the lemmas contained in this chapter will be claimed without proof. In fact, they are the higher dimensional counterparts to the lemmas contained in chapter 4. That is why we omit their proofs understanding that the dimension makes no difference in that matter. The chapter is organized as follows: In the first section we study the rank 0foliations and we prove the symplectic uniqueness for those foliations in the case there are no focus-focus components. In the second section we pose the problem for rank k-foliations and we prove that the linear foliation in the covering is symplectically linearizable. We do it by defining a splitting of the regular and singular parts. The splitting is again a symplectic orthogonal decomposition. For the regular part, we apply the classical Liouville-Mineur-Arnold theorem and we apply the result of symplectic uniqueness established in the first section for the singular part. 81

82

Chapter 5. Higher dimensions

5.1

Rank 0 foliations in any dimension

In this section we deal with the rank 0 case. That is assume that F is the linear foliation defined by F =< X1 , . . . , Xn > where the vector fields are the linear vector fields introduced in the first chapter. This foliation is a linear foliation on M 2n with a rank 0 singularity at the origin p. Assume that the Williamson type of the singularity is (ke , kh , kf ). Recall that the foliation is then generated by the following vector fields,

∂ ∂ Xi = −yi ∂xi + xi ∂yi for 1 ≤ i ≤ ke , ∂ ∂ Xi = yi ∂xi + xi ∂yi for ke + 1 ≤ i ≤ ke + kh , ∂ ∂ Xi = xi ∂x∂ − yi+1 ∂yi − xi+1 ∂xi + yi ∂y∂ and i+1 i+1 ∂ ∂ Xi+1 = −xi ∂xi + yi ∂yi − xi+1 ∂x∂ + yi+1 ∂y∂ for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf i+1 i+1

We want to prove the following theorem, Theorem 5.1.1 Let ω be a symplectic form defined in a neighbourhood of the origin for which F is Lagrangian, then there exists a local diffeomorphism φ : (U, p) −→ (φ(U ), p) such that φ preserves the foliation and φ∗ ( being xi , yi local coordinates on (φ(U ), p). Remarks Observe that this theorem has already been proved in dimension 2 in chapter 2 and in dimension four in chapter 4. So we propose here to prove it by induction. Before starting the induction process we claim the following two lemmas (we omit the proof) which are the higher dimensional analogues of lemmas 4.3.2 and 4.3.1, respectively. We include the proof here just in the case that the Williamson type is (ke , kh , 0). Lemma 5.1.2 There exists C ∞ -functions hj , ∀i, j ∈ {1, n} such that i
i

dxi ∧ dyi ) = ω,

5.1. Rank 0 in any dimension
n

83

iXi ω =
j=1

hi dfj j

The following lemma is a foliation-preserving version of the Moser path method in the general case. Lemma 5.1.3 Let α be an F-basic 1-form and let ω1 be a symplectic germ for which F is Lagrangian. Then: 1. The 2-form ω0 = ω1 − dα is a symplectic structure in a neighbourhood of p and makes the foliation Lagrangian. 2. There is a diffeomorphism η between two neighbourhoods of p preserving F and such that η ∗ (ω1 ) = ω0 . We will also need the following lemma. We already proved this lemma in any dimension in the previous chapter.

Lemma 4.3.3 The following equality holds LXi iXj ω = LXj iXi ω, ∀i, j ∈ {1, n}.

Now we can start the proof of the theorem. Proof of theorem 5.1.1 We prove it by induction on n. Recall that dim M = 2n. • Case n = 1. This is nothing but the symplectic linearization result (theorem 2.3.1) in dimension 2 which is contained in chapter 2.

84

Chapter 5. Higher dimensions • Now let us assume that the theorem holds for r ≤ n − 1 let us prove that the result is true for r = n. Observe that when we pass from n − 1 to n we attach a component of the foliation which can be elliptic or hyperbolic. In the case we attach an elliptic or hyperbolic component we are adding a vector field (generically independent from the others) to the distribution. So we need to make out the following cases 1. Attaching an elliptic component. According to lemma 5.1.2 we can write,
n

iXi ω =
k=1

hi dfk , k

∀i ≤ n

On the other hand for each function hn we can apply the decomposition k of proposition 2.2.1 in page 23 (chapter 2) to write,
n n

n hn = hk + Xn (Hk ), k

Xn (hk ) = 0,

for convenient differentiable functions. Now consider the 1-form α =
n k=1 n Hk dfk . We can apply lemma 5.1.3 with this 1-form and ω will be

equivalent to ω = ω − dα. Now we compute iXn ω = iXn ω − iXn dα. We can compute the second term in the right hand side of the equality using Cartan’s formula to obtain,
n

iXn dα = LXn α − diXn α =
k=1

n Xn (Hk )dfk

where in the last equality we have used the fact that the 1-form is F-basic. In the end this yields,

5.1. Rank 0 in any dimension
n

85

iXn ω =
k=1

hk dfk ,

n

Xn (hk ) = 0.
n

n

For the sake of simplicity we will keep the notation hn instead of hk k formula 5.1 LXi iXj ω = LXj iXi ω of lemma 4.3.3 yields the relations, Xj (hi ) = Xi (hj ), k k ∀i, j, k

bearing in mind that we can assume that Xn (hn ) = 0. Observe that k

(5.1.1)

After taking i = n in the relation above and applying LXn both sides we obtain,
2 Xn (hj ) = 0 k

now since Xn is a periodic vector field we can reproduce the arguments exposed in the first proof of proposition 4.5.1 in page 69 to obtain Xn (hj ) = 0, now going back to equation 5.1.1 we get, k Xj (hn ) = 0, k ∀j.

Next step, we normalize in the (xn , yn )-direction, that is to say as we did in the proof of 2.3.3 in page 34 first we consider the smooth solution of the differential equation d 2 (ψ (u, xn , yn ) · u) = hn (u, xn , yn ), ˆ ˆ ˆ ˆ n du
2 ˆ ˆ where u = x2 + yn , xn = (x1 , . . . , xn−1 ) and yn = (y1 , . . . , yn−1 ) and n

then we define the foliation preserving diffeomorphism φ(ˆn , yn , xn , yn ) = (ˆn , yn , ψ · xn , ψ · yn ). x ˆ x ˆ This diffeomorphism takes the symplectic form to a new symplectic form ω 1 such that,

86

Chapter 5. Higher dimensions

n−1

iXn ω 1 =
k=1

hk dfk − dfn ,

n

Xn (hk ) = 0.

n

(5.1.2)

Taking into account the expression above and since ω 1 is locally exact, we can write: ω 1 = dxn ∧ dyn + d(
k<n

(Ak dxk + Bk dyk )).
∂h dxn ∂xn

We use the notation by d(1) (h) =
∂h k<n ∂xk dxk

+

∂h dyn ∂yn

and d(2) (h) =

+

∂h dyk ∂yk

with d(1) (

k<n (Ak dxk

+ Bk dyk )) = 0 and there-

fore,

k<n (Ak dxk

+ Bk dyk ) = d(1) (gn ) for a certain differentiable func-

tion. After gathering coefficients conveniently this yields,

ω 1 = dxn ∧ dyn + d(
k<n

(Ak dxk + B k dyk )),

for certain smooth functions Ak and B k . Now, we compute the contraction iXn ω 1 again with this expression to get:

iXn ω 1 = −dfn +
k<n

(Xn (Ak )dxk + Xn (B k )dyk ).

Comparing this expression with equation 5.1.3 we obtain, Xn (Ak ) = Xn (B k ) = for any k, where
k

hn (xk ) k
n k hk (yk ) k

= 1 if the function fk is elliptic and

= −1 for an

hyperbolic function fk . Since Xn (hn ) = 0, we can apply LXn to these relations to obtain, k
2 Xn (Ak ) = 0, 2 Xn (B k ) = 0,

5.1. Rank 0 in any dimension again since Xn is periodic these relations yield, Xn (Ak ) = 0, proving so far Xn (B k ) = 0,

87

iXn ω 1 = −dfn . In order to finish the proof it remains to define a symplectically orthogonal decomposition and apply the induction hypothesis. As we did in chapter 4 we use the lemma, Lemma 5.1.4 The distribution D1 =< X, Y > defined by the relations: iX ω 1 = dxn iY ω 1 = dyn is C ∞ , symplectic in a neighbourhood of the origin and involutive everywhere. The proof can be treated along the same lines that is why we omit it. This lemma allows us to talk about symplectically orthogonal decomposition. Notice that from the definition the 2-dimensional distribution D1 is symplectically orthogonal to the 2(n − 2) dimensional distribution D2 generated by the vector fields
∂ ∂xk

and

∂ , ∂yk

for k = n. Both distribu-

tions are involutive. The integral submanifolds Mc integrating the first one are symplectic whereas the integral submanifolds Nc integrating the second distribution are symplectic because they are symplectically orthogonal to the former. These two orthogonal distributions provide coordinates xi , y i . Such that the symplectic form may be expressed as ω 1 = ω1 + ω2 , where ω1 defines

88

Chapter 5. Higher dimensions a symplectic 2-form on Tp (Mc ) for each p ∈ Mc and, in the same way, ω2 defines a symplectic 2-form on Tp (Nc ) for each p ∈ Nc . The condition dω 1 = 0 implies that ω1 depends only on the xn , y n variables and ω2 depends only on the xk , y k variables (for k = n). Observe that (Mc , ω1 ) is a 2-dimensional manifold endowed with the foliation F1 defined by Xn . In the same way, (Nc , ω2 ) is a 2(n − 1)dimensional manifold endowed with the foliation F2 generated by Xk for k = n. The lagrangian condition imposed on the initial foliation F implies the Lagrangian condition for F1 and F2 . So we may apply the hypothesis for induction to the symplectic submanifolds Nc and Mc to define foliation preserving diffeomorphisms φ1 and φ2 for which φ∗ (ω1 ) = dxn ∧ dyn and φ∗ (ω2 ) =
k<n

dxk ∧ dyk .

Finally define φ = (φ1 , φ2 ) to get the desired foliation preserving diffeomorphism. 2. Attaching an hyperbolic component. The proof follows the guidelines of the elliptic case with some differences. Let us point them out. First, using lemma 2.2.2 in page 26 and lemma 5.1.3 we can assume that,
n

iXn ω =
k=1

hk dfk ,

n

Xn (hk ) = 0.

n

As we observed in the proof for completely hyperbolic case in chapter 4, this implies

2 Xn (hi ) = 0, k

∀i, k

5.1. Rank 0 in any dimension

89

Since Xn is a vector field corresponding to an hyperbolic singularity, this does not imply necessarily that Xn (hi ) = 0. It remains to apply k Moser again exactly as we did on chapter 4 to obtain Xn (hi ) = 0, k As a consequence of the commutation relations this yields consider the foliation preserving diffeomorphism φ(ˆn , yn , xn , yn ) = (ˆn , yn , ψ · xn , ψ · yn ). x ˆ x ˆ where ψ is the smooth solution of the differential equation d 2 (ψ (u, xn , yn ) · u) = hn (u, xn , yn ), ˆ ˆ ˆ ˆ n du
2 and u = x2 − yn , xn = (x1 , . . . , xn−1 ) and yn = (y1 , . . . , yn−1 ) This ˆ ˆ n

∀i. =0

Xj (hn ) k

, ∀i, k and then we can proceed to normalization. That is to say we

diffeomorphims takes the symplectic form to a new symplectic form ω 1 such that,
n−1 n n

iXn ω 1 =
k=1

hk dfk − dfn ,

Xn (hk ) = 0.

(5.1.3)

Now in view of this expression and following the same arguments as in the elliptic case, the symplectic form can be written, ω 1 = dxn ∧ dyn + d(
k<n

(Ak dxk + B k dyk )),

for certain smooth functions Ak and B k . The contraction iXn ω 1 reads, iXn ω 1 = −dfn +
k<n

(Xn (Ak )dxk + Xn (B k )dyk ).

From here and the Lagrangian condition we can write Xn (Ak ) = Xn (B k ) = hn (xk ) k
n k hk (yk )

,

90 for any k, where
k

Chapter 5. Higher dimensions = 1 if the function fk is elliptic and
k

= −1 for an

hyperbolic function fk . Since Xn (hn ) = 0, we can apply LXn to these relations to obtain, k
2 Xn (Ak ) = 0, 2 Xn (B k ) = 0,

from this expressions we can draw conclusions about the (xn , yn )-jet of the function along the subspace xn = 0, yn = 0 and then apply Moser again as we did in the proof for the completely hyperbolic case in chapter 4. Then we will be taken to a new symplectic form and for new coefficients Ak and B k for which Xn (Ak ) = 0, proving so far Xn (B k ) = 0,

iXn ω 1 = −dfn . From this moment on, we can define the symplectically orthogonal decomposition as we did in the proof of the elliptic case and follow the same proof which takes the initial symplectic form to the standard one. Remark This theorem was proved with all the details by Eliasson just in the completely elliptic case. Here we have included a different proof for foliations of Williamson type (ke , kh , 0). We have not include here the more general case kf = 0. But let us add a few words about this case, when kf = 0 we can prove the theorem using induction as well. Observe that when we attach a focus-focus component we are adding two vector fields to the distribution. One of this vector fields Xn−1 is periodic. For this vector field, we can obtain a decomposition as the one obtained in

5.1. Rank 0 in any dimension

91

chapter 2 proposition 2.2.1, page 23. Then using Moser and tecniques similar to the elliptic case we can prove that Xn−1 is a Hamiltonian vector field with hamiltonian function fn−1 . This enables to define a symplectic distribution which obviously contains Xn−1 . Since this distribution is symplectically orthogonal to the distribution generated by
∂ , ∂ ∂xk ∂yk

k = n, k = n − 1. This distribution also contains Xn . Once re-

ached this point, we have an orthogonal symplectic decomposition and we can apply the induction hypothesis exactly as we did in the other cases.

92

Chapter 5. Higher dimensions

5.2

Rank k foliations in any dimension

The goal of this section is to give a symplectic uniqueness result in the neighbourhood of an orbit for the linearized foliation. This result is contained in the joint paper [48] with Nguyen Tien Zung. We provide a different proof here. Recall that the foliation is given by the vector fields,

Yi =

∂ , ∂θi

1≤i≤k

∂ ∂ Xi = −yi ∂xi + xi ∂yi

for 1 ≤ i ≤ ke ,
∂ ∂ Xi = yi ∂xi + xi ∂yi for ke + 1 ≤ i ≤ ke + kh , ∂ ∂ Xi = xi ∂x∂ − yi+1 ∂yi − xi+1 ∂xi + yi ∂y∂ and i+1 i+1 ∂ ∂ Xi+1 = −xi ∂xi + yi ∂yi − xi+1 ∂x∂ + yi+1 ∂y∂ for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf i+1 i+1

We denote this foliation by F. We want to prove the following theorem, Theorem 5.2.1 Let ω and ω0 two symplectic forms in a neighbourhood of a singular orbit for which the foliation F is Lagrangian then ω and ω0 are equivalent. Remarks: • Taking into account the notion of equivalence, we could also state this theorem in the following way, Theorem 5.2.2 Let ω be a symplectic form defined in a neighbourhood of a singular orbit L, then there exists coordinates (θ1 , . . . , θk , p1 , . . . , pk , x1 , y1 , . . . , xn−k , yn−k ), a diffeomorphism φ in a neighbourhood of L preserving the foliation F such that
k n−k

φ∗ (
i=1

dpi ∧ dθi +
i=1

dxi ∧ dyi ) = ω.

5.2. Rank k

93

• The idea of the proof is to define an splitting at each point of the symplectic manifold into two symplectic manifolds Mp and Np which are symplectically orthogonal and such that the regular distribution ( generated by the vector fields Yi ) is tangent to Mp and the singular distribution ( generated by the vector fields Xi ) is tangent to Np . Proof of theorem 5.2.1: Let ω be a symplectic form for which F is Lagrangian. In order to define the symplectic orthogonal decomposition we need to prove the existence of a Hamiltonian Tk action tangent to the leaves of the foliation. In section 3, proposition 4.5.2 in page 75 we proved the existence of a Hamiltonian S 1 -action tangent to the leaves of the foliation using Moser path method and the Lagrangian condition. Iterating this procedure and using 5.1.3 we have the following proposition, we omit the proof because it is a straightforward generalization of proposition 4.5.2. Let us introduce the following simplifying notation, θ = (θ1 , . . . , θk ), p = (p1 , . . . , pk ) x = (x1 , . . . , xn−k ) and y = (y1 , . . . , yn−k ) Proposition 5.2.1 There is a Hamiltonian Tk -action tangent to the foliation. In fact, there exist coordinates (θ, p, x, y) in a neighbourhood of L such that ω = d(
k i=1

pi dθi +

C(p, x, y)dx + D(p, x, y)dy) and the Hamiltonian Tk -action is

performed by translations with respect to θi , for 1 ≤ i ≤ k. Under these conditions we can prove the following lemma, Lemma 5.2.3 The distribution D =< Z1 , T1 , . . . , Zk , Tk > defined by the relations: iZi ω = dpi iTi ω = dθi is C ∞ , symplectic in a neighbourhood of L and involutive everywhere.

94 Proof:

Chapter 5. Higher dimensions

Clearly this distribution is smooth and regular. First we prove that the distribution is symplectic. From the definition, this distribution is the symplectic orthogonal to the distribution D generated by
∂ , ∂ ∂xi ∂yi

for 1 ≤ i ≤ n − k. If we

prove that D is symplectic in a neighbourhood of the orbit L then we will be done. In order to check that D is symplectic we use the same strategy as in the proof of lemma 4.4.1 in page 67. From the Lagrangian conditions at any point in p in a neighbourhood of L the following relations are fulfilled, ω(Yi , Xj ) = 0, ω(Yi , Yj ) = 0 and ω(Xi , Xj ) = 0. Now along L the coordinates xi and yi vanish so in the absence of focus-focus components, the relations above read, w( w( ∂ ∂ , )|q = 0, ∂θi ∂θj ∂ ∂ , )|q = 0, ∂θi ∂xj ∂ ∂ , )|q = 0, ∂θi ∂yj ∂ ∂ , )|q = 0, ∂xi ∂xj ∂ ∂ , )|q = 0, ∂yi ∂yj q∈L q∈L

w(

q∈L

w(

q∈L

w( w(

q∈L q∈L

∂ ∂ , )|q = 0, ∂xi ∂yj

i=j

Following the same arguments as in proposition 4.4.1 these relations imply that D is symplectic and therefore its symplectic orthogonal distribution D is also symplectic. In the case there are focus-focus components the last relation changes for j = i + 1 in a focus-focus pair fi , fi+1 and following similar arguments we conclude that D is also symplectic.

5.2. Rank k

95

In order to see that the distribution is involutive observe first that the vector fields Zi coincide with the vector fields Yi =
∂ . ∂θi

Therefore, [Zi , Zj ] = 0. On the

other hand according to proposition 5.2.1 the coefficients of the symplectic form do not depend on the angular coordinates θi . From here, [Zi , Tj ] = 0 because of the expression of the symplectic form obtained in Proposition 5.2.1. It remains to check that [Ti , Tj ] = 0. We use the formula, i[Ti ,Tj ] ω = LTi iTj ω − iTj LTi ω. ∗ The second term vanishes because from the definition of Ti the vector field Ti is locally Hamiltonian. As for the first term, applying the definition of Ti , we obtain the following chain of equalities, LTi iTj ω = LTi dθj = d(Ti (θj )) = 0, where in the last equality we have used again the explicit expression of ω in proposition 5.2.1. Now going back to ∗ we obtain, i[Ti ,Tj ] ω = 0 Finally this yields [Ti , Tj ] = 0 and the distribution is involutive. This ends the proof of the lemma. Once this key lemma has been proved we are already done. Because we have two symplectic orthogonal distributions D and D . Now through each point p in L there are two symplectic submanifolds, symplectically orthogonal to each other, Mp and Np integrating D and D respectively. Observe that the distribution generated by the singular vector fields Xi is tangent to Mp thus defining a foliation tangent to Mp . We call this foliation F1 . In the same way, the distribution generated by the regular vector fields Yi is tangent to

96

Chapter 5. Higher dimensions

Np thus defining a foliation tangent to Np . We call this foliation F2 . Note, as well, that the Lagrangian condition imposed on F implies that the foliations F1 and F2 are Lagrangian with respect to the symplectic forms ω1 and ω2 induced by restriction of ω to Mp and Np , respectively. Further, since Mp and Np are symplectically orthogonal to each other we may write, ω = ω1 + ω2 . Let us have a look at the expression of ω in local coordinates. Frobenius theorem provides coordinates (θ1 , . . . , θk , p1 , . . . , pk , x1 , y 1 . . . , xn−k , y n−k ) in a neighbourhood of p such that (θ1 , . . . , θk , p1 , . . . , pk ) are coordinates in Np and (x1 , y 1 . . . , xn−k , y n−k ) are coordinates in Mp . The condition dω = 0 implies that the coefficient functions of ω1 just depend on xi and y i and that the coefficient functions of ω2 just depend on pi and θi . Once reached this point, we can apply theorem 5.1.1 to the pair (Mp , ω1 ) and there exists a diffeomorphims φ1 preserving the foliation F1 and coordinates (xi , yi ) such that φ∗ (
i

dxi ∧ dyi ) = ω1 . We can apply Liouville-Mineur-Arnold theorem dpi ∧dθi ) = ω2 . Observe that these coordinates

to the pair (Np , ω2 ) to obtain a diffeomorphims φ1 preserving the foliation F2 and coordinates (pi , θi ) such that φ∗ (
i

can be extended to a whole neighbourhood of the orbit using the flow of the vector fields Yi which are symplectomorphisms. Finally the desired preserving foliation diffeomorphism is φ = (φ1 , φ2 ). This ends the proof of theorem 5.2.1.

Chapter 6 Equivariant linearization and symplectic equivalence
6.1 Introduction

In the previous chapters we have attained the symplectic linearization of the foliation F in a finite normal covering U (L) of the initial neighbourhood of the orbit L. As we observed in the first chapter, the group Γ of deck transformation attached to the covering preserves the symplectic structure and the fibration given by the mapping F = (f1 , . . . , fn ). Therefore, in order to prove the symplectic equivalence in the initial neighbourhood of the orbit we have to check that the symplectomorphism which provides the linearization result in the covering can be chosen to be Γ-equivariant. We can pose the problem in the following terms, Denote by α the initial symplectic action of the group Γ in the covering U (L) and we denote by ω the symplectic form in the covering. This action preserves the fibration given by F. Let φ : U (L) −→ φ(U (L)) be the symplectomorphism attained in the previous chapter. This symplectomorphism preserves the foliation 97

98

Chapter 6. Equivariant linearization and symplectic equivalence
k i=1

F and satisfies (φ∗ )−1 (ω) = ω0 , being ω0 = and the fibration given by F = (f1 , . . . , fn ).

dpi ∧ dθi +

n−k i=1

dxi ∧ dyi .

Then the initial action α of Γ becomes an action ρ on φ(U (L)) preserving ω0 Summing up, if we prove that the Γ-equivariant equivalence taking the initial action to a linear action can be attained in the linear model with the symplectic structure ω0 =
k i=1

dpi ∧ dθi +

n−k i=1

dxi ∧ dyi , then we will be done.

Observe that the twisting group Γ is a finite group. So it would be sufficient to prove that the Γ-equivariance holds for finite groups. We will prove a more general theorem which will yield the desired result as a corollary. The general result that we prove provides an equivariant version of the symplectic linearization. In other words, we will assume that there exists a symplectic action of a compact Lie group in a neighbourhood of L preserving F = (f1 , . . . , fn ) and we will show that the symplectic linearization can be carried out in an equivariant way. As a consequence of this result we end up proving the symplectic linearization result in the original neighbourhood of the orbit. Another byproduct of the proof of this theorem is that if the action is effective then the group G fulfilling all the above-mentioned hypotheses has to be abelian. As a matter of fact, the equivariant symplectic linearization result turns out to have interest by itself. It follows the general philosophy of the large list of linearization results for compact group actions preserving additional structures. From our point of view, the first one in this long list is the one for fixed points due to Bochner. Let us state Bochner’s linearization theorem. Theorem 6.1.1 (Bochner) Let α be a smooth action of a compact group G on a manifold M and let
1 x0 ∈ M be a fixed point for the action, i.e α(g, x0 ) = x0 , ∀g ∈ G. Denote by αg the

differential at x0 of the diffeomorphism αg : M −→ M induced by α. Then there

6.1. Introduction

99

exists a G-invariant neighbourhood U of x0 and a diffeomorphism φ from U onto an open neighbourhood V of the origin 0 in Tx0 M , such that, φ(x0 ) = 0, and
(1) φ ◦ αg = αg ◦ φ,

dx0 φ = Id

∀g ∈ G,

x∈U

The orbit-like version of this theorem was given by Koszul [35]. This theorem has been known in the literature as the slice theorem. It guarantees that for any smooth action of a compact Lie group on a manifold M there exists a slice S through every point x ∈ M . Furthermore one can choose coordinates on S so that S is an open invariant disk in a vector space upon which the isotropy group acts linearly. Namely, Theorem 6.1.2 (Koszul) Let G be a compact connected Lie group acting on a manifold M and let x ∈ M be a point. A neighbourhood of the orbit G·x through the point x, is G-equivariantly diffeomorphic to a neighbourhood of the zero section of the homogeneous vector bundle G ×Gx W where W = Tx (M )/Tx (G · x) and where the action of Gx on W is linear. An extension of this theorem to proper actions of groups was provided by R. Palais [50], [49]. Another G-equivariant result concerning also symplectic forms is the G-equivariant Darboux theorem. We state it below, Theorem 6.1.3 Let G be a compact Lie group acting smoothly on a manifold M , let Y be a G-invariant compact submanifold of M , and let ω1 and ω2 be two Ginvariant symplectic forms on M such that ω0p = ω1p for all p ∈ Y . Then there

100

Chapter 6. Equivariant linearization and symplectic equivalence

exists a G-invariant neighbourhood U of Y and a G-equivariant diffeomorphism f of U onto another G-invariant neighbourhood of Y such that f (y) = y for all y ∈ Y and f ∗ (ω1 ) = ω0 . The first two theorems can be considered as linearization theorems for actions of compact Lie groups whereas the last theorem is also concerned with an additional geometrical structure (the symplectic form). In this spirit we will prove similar linearization theorems in the neighbourhood of a point and in the neighbourhood of an orbit under the more constraining condition that the diffeomorphism preserves the map F and the symplectic structure. This chapter is organized as follows: in the first section we introduce the notion of the linear action on the linear model. In the second section we study the case of a fixed point and we prove that the action can be linearized. As a by-product, we prove that the group of symplectomorphisms preserving the system is abelian. In the third section we prove the G-linearization in the neighbourhood of an orbit. As a corollary we obtain the symplectic equivalence in a neighbourhood of an orbit, this result is included in the last section. Throughout the chapter there will be two different concepts that show up. The concept of foliation preserving symplectomorphism and the concept of system preserving diffeomorphism. They are slightly different concepts. Let us point out the difference in advance. When we say that a symplectomorphism is foliation preserving we mean that it preserves the foliation. That is, it sends leaves to to leaves. When we refer to a system preserving diffeomorphism we mean that the diffeomorphism preserves the symplectic form considered and F. In particular, a system preserving diffeomorphism is foliation preserving. The results contained in this section have been obtained jointly with Nguyen Tien Zung. The proofs provided here (with the only exception of the proof of the linearization theorem in the neighbourhood of an orbit and the parametric version

6.2. The linear action on the linear model

101

of theorem 6.3.2 and theorem 6.3.4 (corollaries 6.3.3 and 6.3.5)) are contained in the joint paper [48].

6.2

The linear action on the linear model

We are going to introduce the notion of linear action on the linear model associated to the orbit L for a given symplectic action preserving the system. Later, we will see that the invariants associated to the linear model are the Williamson type of the orbit and a twisting group Γ attached to it. We recall the notion of linear model. Denote by (p1 , ..., pk ) a linear coordinate system of a small ball Dk of dimension k, (θ1 (mod1), ..., θk (mod1)) a standard periodic coordinate system of the torus Tk , and (x1 , y1 , ..., xn−k , yn−k ) a linear coordinate system of a small ball D2(n−k) of dimension 2(n − k). Consider the manifold V = Dk × Tk × D2(n−k) with the standard symplectic form ω0 = moment map: F = (p1 , ..., pk , fk+1 , ..., fn ) : V → Rn where
2 fi+k = x2 + yi for 1 ≤ i ≤ ke , i

(6.2.1) dxj ∧dyj , and the following (6.2.2)

dpi ∧dθi +

fi+k = xi yi for ke + 1 ≤ i ≤ ke + kh , fi+k = xi yi+1 − xi+1 yi and fi+k+1 = xi yi + xi+1 yi+1 for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf

(6.2.3)

For the sake of simplicity we will denote by p the mapping whose components are the k regular first integrals pi and h will stand for the mapping whose components are the singular first integrals fi , i ≥ k; following this convention we will write F = (p, h). Let Γ be a group with a symplectic action ρ(Γ) on V , which preserves

102

Chapter 6. Equivariant linearization and symplectic equivalence

the moment map F. We will say that the action of Γ on V is linear if it satisfies the following property: Γ acts on the product V = Dk × Tk × D2(n−k) componentwise; the action of Γ on Dk is trivial, its action on Tk is by translations (with respect to the coordinate system (θ1 , ..., θk )), and its action on D2(n−k) is linear with respect to the coordinate system (x1 , y1 , ..., xn−k , yn−k ). Suppose now that Γ is a finite group with a free symplectic action ρ(Γ) on V , which preserves the moment map and which is linear. Then we can form the quotient symplectic manifold V /Γ, with an integrable system on it given by the induced moment map as above:

F = (p1 , ..., pk , fk+1 , ..., fn ) : V /Γ → Rn

(6.2.4)

The set {pi = xi = yi = 0} ⊂ V /Γ is a compact orbit of Williamson type (ke , kf , kh ) of the above system. We will call the above system on V /Γ, together with its associated singular Lagrangian foliation, the linear system (or linear model) of Williamson type (ke , kf , kh ) and twisting group Γ (or more precisely, twisting action ρ(Γ)). We will also say that it is a direct model if Γ is trivial, and a twisted model if Γ is nontrivial. A symplectic action of a compact group G on V /Γ which preserves the moment map (p1 , ..., pk , fk+1 , ..., fn ) will be called linear if it comes from a linear symplectic action of G on V which commutes with the action of Γ. In our case, let G denote the group of linear symplectic maps which preserve the moment map then this group is abelian and therefore this last condition is automatically satisfied. In fact G is isomorphic to Tm ×G1 ×G2 ×G3 being G1 the direct product of ke special orthogonal groups SO(2, R), G2 the direct product of kh components of type SO(1, 1, R) and G3 the direct product of kf components of type R × SO(2, R), respectively.

6.3. G-linearization for rank 0 foliations

103

6.3

G-linearization for rank 0 foliations

In this section we consider the action of a compact Lie group on the linear model corresponding to a rank 0 point of Williamson type (ke , kh , kf ). We assume that this action preserves the symplectic form and the mapping F = (f1 , . . . , fn ) where fi are of elliptic, hyperbolic or focus-focus type as specified in the section above (formula 6.2.3). We prove that the action can be linearized in a foliation preserving way. We will also provide the analytic version of the theorem. The proof of this linearization theorem resembles very much the proof of Bochner’s linearization theorem. We will use the averaging method. Nevertheless, we have to make sure that the linearization can be carried out using symplectomorphisms and that all the symplectomorphisms preserve the Lagrangian foliation. In order to do that, we will often consider flows of Hamiltonian vector fields which are tangent to the foliation. Let us fix some notation that we will use throughout the chapter. The vector field XΨ will stand for a Hamiltonian vector field with associated Hamiltonian function Ψ. We will denote by φs t the time-s-map of the vector field Xt . Let ψ X be a local diffeomorphism ψ : (R2n , 0) → (R2n , 0). In the sequel, we will denote by ψ (1) the linear part of ψ at 0. That is to say, ψ (1) (x) = d0 ψ(x). In this section we will linearize the action of G using the averaging method and a theorem about local automorphisms of the linear integrable system (R2n ,
n i=1

dxi ∧ dyi , h).

First we recall this well-known sublemma (sublemma 4.5.1) of chapter 4. Sublemma 6.3.1 Let XG1 and XG2 be two Hamiltonian vector fields tangent to F. Denote by φs G and φs G the time-s-map of XG1 and XG2 respectively. Then X X
1 2

φs G X

1

+XG2

= φs G ◦ φs G . X X
1 2

Now we can state and prove the following theorem,

104

Chapter 6. Equivariant linearization and symplectic equivalence

Theorem 6.3.2 Suppose that ψ : (R2n , 0) → (R2n , 0) is a local symplectic diffeomorphism of R2n which preserves the quadratic moment map h. Then the linear part ψ (1) is also a system-preserving symplectomorphism, and there is a unique local smooth function Ψ : (R2n , 0) → R vanishing at 0 which is a first integral for the linear system given by h and such that ψ (1) ◦ ψ −1 is the time-1 map of the Hamiltonian vector field XΨ of Ψ. If ψ is real analytic then Ψ is also real analytic. If ψ depends smoothly (resp analytically) on parameters so does Ψ. Proof We are going to construct a path connecting ψ to ψ (1) contained in G = {φ : (R2n , 0) → (R2n , 0), such that φ∗ (ω) = ω, we consider   ψ ◦ gt  (x) t Stψ (x) =   ψ (1) (x) t ∈ (0, 1] t=0 h ◦ φ = h}. Given a function ψ ∈ G,

being gt the homothecy gt (x1 , . . . , xn ) = t(x1 , . . . , xn ). Observe that in case ψ is smooth, this mapping Stψ is smooth and depends smoothly on t. In case ψ is real analytic, the corresponding Stψ is also real analytic and depends analytically on t. If ψ depends smoothly or analytically on parameters so does Stψ . First let us check that h ◦ Stψ = h when t = 0. We do it component-wise. Let x = (x1 , . . . , xn ) and let fj be one of the components of h, then fj ◦ ( ψ ◦ gt (fj ◦ ψ ◦ gt )(x) fj ◦ gt (x) )(x) = = = fj (x) t t2 t2

where in the first and the last equalities we have used the fact that each component fj of the moment map h is a quadratic polynomial whereas the condition h ◦ ψ = h yields the second equality. Now we check that (Stψ )∗ (ω0 ) = ω0 when t = 0. Since ω0 = dxi ∧ dyi , then

6.3. G-linearization for rank 0 foliations
∗ gt (ω0 ) = t2 ω0 . But since ψ preserves ω0 then

105

(Stψ )∗ (ω) = ( when t = 0.

ψ ◦ gt ∗ ) ω0 = ω0 t

So far we have checked the conditions h ◦ Stψ = h and (Stψ )∗ (ω0 ) = ω0 when t = 0 but since Stψ depends smoothly on t we also have that h ◦ S0 ψ = h and
ψ ψ (S0 )∗ (ω0 ) = ω0 . So, in particular, we obtain that S0 = ψ (1) preserves the moment

map and the symplectic structure and therefore ψ (1) is also contained in G. Now consider Rt = ψ (1) ◦ St
(ψ −1 )

with t ∈ [0, 1], this path connects the identity to ψ (1) ◦ ψ −1 and is contained in G. We are going to use this path to define a Hamiltonian vector field such that its time-1-map is ψ (1) ◦ ψ −1 . First, we consider the t-dependent vector field Xt d −1 (Rs (q))|s=t , q = Rt (p) (6.3.1) ds with t ∈ [0, 1] . Since Rs is a symplectomorphism for any s contained in [0, 1], the Xt (p) = vector field Xt is locally Hamiltonian. Then the vector field
1

satisfying

X=
0

Xt dt

is also locally Hamiltonian. Since the symplectic manifold considered is a neighbourhood U of the origin, the vector field X is indeed Hamiltonian in U . There is a unique local Hamiltonian function Ψ associated to X satisfying Ψ(0) = 0. This Hamiltonian function is a first integral for the system since {Ψ, hi } = 0. If ψ is real analytic (respectively differentiable) Xt is also real analytic (respectively differentiable) and the same property holds for Ψ. Observe that in the case ψ depends smoothly (resp. analytically) on parameters then by construction the t-dependent vector field Xt depends smoothly (resp. analytically) on that parameters and therefore so does the vector field X and its time one map. It remains to check that the time-1-map of X is ψ (1) ◦ ψ −1 .

106

Chapter 6. Equivariant linearization and symplectic equivalence

We are going to prove that the time-1-map of the vector field X coincides with R1 by performing a partition of the interval [0, 1] into n pieces and then approximating the integral by a sum. First observe that formula 6.3.1 shows that the vector X1− k+1 (p) is tangent to
n

the curve Rs ◦ So in fact,

flow the vector is also tangent to the curve φs X

−1 R1− k+1 (p) n

at the point p. On the other hand by the definition of
1− k+1 n

(p) at the point p.

X1− k+1 (p) = lim
n

R1− k+1 +s ◦ R1− k+1 −1 (p) − p
n n

s→0

s

(6.3.2)

and also φs X (p) − p (6.3.3)

X1− k+1 (p) = lim
n

1− k+1 n

s→0

s

Therefore,
−1 R1− k+1 +s ◦ R1− k+1 (p) − φs X
n n

s→0

lim
n

1− k+1 n

(p) =0 (6.3.4)

s
n 1− k+1 n

−1 In other words, R1− k+1 +s ◦R1− k+1 (p) = φs X

(p) + o(s1 ). After refining the initial
1 n

partition if necessary, we can particularize s =
−1 n R1− k ◦ R1− k+1 = φX
n n 1

to obtain 1 + o( ). n

1− k+1 n

(6.3.5)

Since R1 can be written as,
−1 −1 R1 = (R1 ◦ R1− 1 ) ◦ R1− 1 · · · ◦ (R1− n−1 ◦ R0 ).
n n n

(6.3.6)

we can perform the necessary substitutions of 6.3.5 in 6.3.6 and we are led to
1 1 1 1 1 1 n n + o( )) ◦ (φX 2 + o( )) ◦ · · · ◦ (φX0 + o( )) 1− n n n n

n R1 = (φX

1 1− n

(6.3.7)

6.3. G-linearization for rank 0 foliations Assuming that

107

here stands for the composition of diffeomorphisms, we can write

this expression in a reduced form as
k=n−1
1

R1 = (
k=0

n φX

k 1− n

) + o(1).

(6.3.8)

Observe that the vector field Xt is tangent to the fibration F for any t contained in [0, 1] because the diffeomorphisms Rs preserve the fibration F, ∀s. On the other hand, each vector field Xt is Hamiltonian. Therefore we may apply sublemma 4.5.1 for any pair t, t contained in [0, 1] and the following equation holds φs t +Xt = X φs t ◦ φs t . Now this expression reads, X X
n R1 = (φPk=n−1 X k=0 1

k 1− n

) + o(1).

(6.3.9)

Since the time-1-map of
n φX = φ1 , we obtain, X n 1

X n

1 and the time- n -map of X are related by the formula

R1 = lim (φ1 P
n→∞

k=n−1 k=0

X k 1− n n

)

(6.3.10)

But,
k=n−1 n→∞

lim

X1− k n

1

n

=
0

Xt dt.

k=0

This identity shows that R1 = φ1 and this ends the proof of the theorem. X As a corollary of the above theorem we obtain a local linearization result of symplectomorphism depending on parameters. The corollary below will be a key point in the proof of the linearization in a neighbourhood of the orbit. Corollary 6.3.3 Let Dp stand for a disk centered at 0 in the parameters p1 , . . . , pk . We denote by p = (p1 , . . . , pk ). Assume that ψp : (R2n , 0) → (R2n , 0) is a local symplectic diffeomorphism of R2n which preserves the quadratic moment map h and which depends smoothly on the parameters p. Then there is a unique local

108

Chapter 6. Equivariant linearization and symplectic equivalence

smooth function Ψp : (R2n , 0) → R vanishing at 0 depending smoothly on p which
−1 is a first integral for the linear system given by h and such that ψ0 ◦ ψp is the (1)

time-1 map of the Hamiltonian vector field XΨp of Ψp . If ψp is real analytic and depends analytically on the parameters then Ψp is also real analytic and depends analytically on the parameters. Proof: According to theorem 6.3.2 there exists a first integral F1,p such that the time−1 one-map of the Hamiltonian vector field XF1,p is ψp ◦ ψp . We will apply the same (1)

trick that we applied to the path Rt in the proof of theorem 6.3.2 but applied to the path in G, Mt defined as follows,
1 Mt = ψ0 ◦ (ψgt (p) )−1 , (1)

where gt (p) = (tp1 , . . . , tpk ). Observe that the path is contained in G and is well defined (since the disk is convex). This path is smooth (resp. analytic) if ψ is smooth (resp. analytic) and depends analytically on t. Now we can apply the same reasoning as in the proof of theorem 6.3.2. Namely, we can consider the t-dependent vector field, Xt (p) = d (Ms (q))|s=t , ds q = Mt−1 (p) (6.3.11)

And also the averaged vector field
1

X=
0

Xt dt.

As we pointed out in the proof of theorem 6.3.2, this vector field is Hamiltonian and it is tangent to the foliation. Denote by F2,p the only Hamiltonian function attached to X such that F2,p (0) = 0. This function is a first integral of the system.
1 And the time-1-map of the vector field XF2,p coincides with ψ0 ◦ (ψp )−1 . (1)

6.3. G-linearization for rank 0 foliations

109

Now we consider the composition of the two time-1-maps associated to XF1,p and XF2,p respectively. The composition equals,
−1 (1) (1) (ψ0 ◦ (ψp )−1 ) ◦ (ψp ◦ ψp ) (1)

In the end we obtain the desired diffeomorphism,
−1 (ψ0 ◦ ψp ). (1)

This diffeomorphism has been presented as a composition of two time-1-maps. The time-1-map associated to XF1,p and the time-1-map associated to XF2,p . It remains to identify this composition as the time-1-map of a Hamiltonian vector field tangent to the foliation. On the one hand, according to sublemma 4.5.1 with s = 1 φ1 F X = φ1 F X ◦ φ1 F X

2,p

+XF1,p

2,p

1,p

On the other hand, XF1,p + XF2,p = XF1,p +F2,p ; in view of this decomposition the Hamiltonian vector field to consider is G = F1,p + F2,p . Since F1,p and F2,p are first integrals for the system so is G. This ends the proof of the corollary. By abuse of language, we will denote the local (a priori nonlinear) action of our compact group G on (R2n ,
n i=1

dxi ∧ dyi , h) by ρ. For each element g ∈ G,

denote by XΨ(g) the Hamiltonian vector field whose time-1 map is ρ(g)(1) ◦ ρ(g)−1 , where ρ(g)(1) denotes the linear part of ρ(g), as provided by the previous lemma. Consider the averaging of the family of vector fields XΨ(g) over G with respect to the Haar measure dµ on G. That is to say, XG (x) =
G

XΨ(g) (x)dµ,

x ∈ R2n

(6.3.12)

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Chapter 6. Equivariant linearization and symplectic equivalence
G

This vector field is Hamiltonian with Hamiltonian function Observe that this mapping preserves the system. Finally we can prove the local linearization theorem,

Ψ(g)dµ. It is

also tangent to the foliation. Denote by ΦG the time-1 map of this vector field XG .

Theorem 6.3.4 ΦG is a local symplectic variable transformation of R2n which preserves the system (R2n , becomes linear. Proof. Since ΦG is the time-1 map of a Hamiltonian vector field then it is a diffeomorphism satisfying Φ∗ (ω) = ω. Therefore it defines a local symplectic variable transforG mation. Let us check that this transformation linearizes the action of G. We want to show that for any h ∈ G the following relation is fulfilled ΦG ◦ρ(h) = ρ(h)(1) ◦ΦG . From the definition of ΦG and formula 6.3.12, ΦG (x) = φ1 G (x) = X But since, φ1 Ψ(g) = ρ(g)(1) ◦ ρ(g)−1 we have, X ΦG (x) =
G n i=1

dxi ∧ dyi , h) and under which the action of G

G

φ1 Ψ(g) (x)dµ X

ρ(g)(1) ◦ ρ(g)−1 (x)dµ

Now we write, (ρ(h)(1) ◦ ΦG ◦ ρ(h)−1 )(x) = ρ(h)(1) ◦
G

(ρ(g)(1) ◦ ρ(g)−1 )(ρ(h)−1 (x))dµ.

Using the linearity of ρ(h)(1) and the fact that ρ stands for an action, the expression above can be written as, (ρ(h) ◦ ρ(g))(1) ◦ (ρ(h) ◦ ρ(g))−1 (x)dµ
G

Finally this expression equals ΦG due to the left invariance property of averaging. So ΦG ◦ ρ(h) = ρ(h)(1) ◦ ΦG and this ends the proof of the lemma.

6.3. G-linearization for rank 0 foliations

111

As a consequence of this theorem and corollary 6.3.3 we obtain the following parametric version of the theorem, Corollary 6.3.5 In the case the action ρp depends smoothly (resp. analytically) on parameters there exists a local symplectic variable transformation of R2n , Φp which preserves the system and which satisfies, Φp ◦ ρp (h) = ρ0 (h)(1) ◦ Φp

Let G = {φ : (R2n , 0) → (R2n , 0), such that φ∗ (ω) = ω, h ◦ φ = h}

be the group of germs of smooth symplectomorphisms that preserve ω and h, i.e the symmetry group for the system. We will denote by G the subgroup of linear transformations contained in G. As we have observed in the introduction, this group is abelian. A direct consequence of theorem 6.3.2 is that any two diffeomorphisms contained in G whose linear part is the identity commute because they are the time-1-map of hamiltonian vector fields tangent to the foliation and those in turn commute by virtue of sublemma 4.5.1. In fact, this property extends to the whole G. This is the content of the theorem below. Before stating the theorem, let us point our some general observations which we will apply later. Let f be a diffeomorphism and let X be a vector field, we denote by f∗ and f ∗ the push-forward and pullback associated to f . Recall that the following formula holds i(f∗ X) ω = (f −1 )∗ (iX (f ∗ (ω))).

112

Chapter 6. Equivariant linearization and symplectic equivalence

Now let f be a diffeomorphism contained in G . In particular f ∗ (ω) = ω. Therefore the above formula shows that the pushforward of a Hamiltonian vector field is also a Hamiltonian vector field. In fact, f∗ (XΨ ) = XΨ◦f −1 . Theorem 6.3.6 The group G is abelian. Proof We will prove that any two diffeomorphisms contained in G commute by stages. First we prove that any diffeomorphism ψ contained in G commutes with any linear transformation A contained in G . We will denote by G0 the connected component of the identity of G . Observe that if the system does not contain any hyperbolic component G = G0 . Let XΨ be the Hamiltonian vector field with associated Hamiltonian Ψ whose time-1-map is ψ (1) ◦ψ −1 . The existence of this vector field is guaranteed by theorem 6.3.2. Now due to the above observation, the vector field Y = (A)∗ (XΨ ) is a Hamiltonian vector field with Hamiltonian function Ψ ◦ A−1 . Assume first that A is contained in G0 . We are going to show that for such a linear transformation, Ψ ◦ A−1 = Ψ. In order to do that we start by observing that since the vector field XΨ is tangent to the fibration F then {Ψ, hi } = 0. Let us assume first that there are no hyperbolic components among the hi . Then according to Vey [54] for an analytical h and Eliasson [23] (Proposition 1) in the differentiable case, we can assert that Ψ = φ(h1 , . . . , hn ) being φ an analytical (respectively differentiable) function. In the case there are hyperbolic components this assertion is no longer true for differentiable functions as was observed by Eliasson in [23]. But the property remains true if we restrict Ψ to each orthant. We label each orthant with an ntuple of signs ( 1 , . . ., and
i 2n ), i

∈ {−1, +1} following the convention

i

= +1 if xi ≥ 0

= −1 if xi ≤ 0. Then we can assert that Ψ restricted to each orthant is a
2n )

function φ( 1 ,...,

(h1 , . . . , hn ).

6.3. G-linearization for rank 0 foliations

113

After making this distinction, observe that in both cases since A−1 belongs to the connected component of the identity G, then it leaves the connected components of the fibration F invariant. Therefore, the transformation A−1 leaves the function Ψ invariant when restricted to each orthant. So in fact, Ψ ◦ A−1 = Ψ and the vector fields Y and XΨ coincide. As a consequence their flows coincide as well. Recall that if X is a vector field whose flow is φX then for any diffeomorphism the flow of (f∗ )X is f ◦ φX ◦ f −1 . The same is true replacing flows by time-1-maps. Therefore, since Y = XΨ we obtain the following relation ψ (1) ◦ ψ −1 = A ◦ ψ (1) ◦ ψ −1 ◦ A−1 . But since A and ψ (1) commute this expression reads ψ (1) ◦ ψ −1 = ψ (1) ◦ A ◦ ψ −1 ◦ A−1 which leads to the commutation of A with ψ. If A does not belong to G0 then we can write A = I2k,2k+1 ◦ · · · ◦ I2l,2l+1 ◦ B (6.3.13)

with B belonging to G0 and for certain diagonal matrices I2r,2r+1 (corresponding to hyperbolic involutions) whose entries aij satisfy the following relations a2r,2r = −1, a2r+1,2r+1 = −1 and ai,i = 1 i = 2r , i = 2r + 1. It can be checked that the linear transformations of type I2k,2k+1 commute with any diffeomorphism contained in G. Now since G is abelian the expression 6.3.13 shows that the linear transformation A commutes with any ψ contained in G. In particular, taking A = ψ (1)
(1) −1

we obtain that ψ commutes with ψ (1) . And
(1)

as a consequence, given two diffeomorphisms ψ1 and ψ2 contained in G the diffeomorphisms f = (ψ1 )−1 ◦ ψ1 and g = ψ2 ◦ (ψ2 )−1 , commute because they are the time-1-map of Hamiltonian vector fields.

114

Chapter 6. Equivariant linearization and symplectic equivalence

Finally we are going to show that any two diffeomorphisms ψ1 and ψ2 contained in G commute. We can write, ψ1 ◦ ψ2 = ψ1 ◦ ((ψ1 )−1 ◦ ψ1 ) ◦ (ψ2 ◦ (ψ2 )−1 ) ◦ ψ2
(1) (1) (1) (1)

(6.3.14)

As we have explained before, the diffeomorphisms within brackets commute and this expression reads, ψ1 ◦ (ψ2 ◦ (ψ2 )−1 ) ◦ ((ψ1 )−1 ◦ ψ1 ) ◦ ψ2
(1) (1) (1) (1)

Due to the commutation of any linear transformation contained in G with any diffeomorphims contained in G, we can write this expression as, ψ1 ◦ (ψ2 )−1 ◦ (ψ1 )−1 ◦ ψ2 ◦ (ψ2 ◦ ψ1 ) But since G is abelian the expression above equals ψ2 ◦ ψ1 and therefore coming back to equation 6.3.14, ψ1 and ψ2 commute. This completes the proof of the theorem. A smooth action of a group G on a manifold M is called effective if the condition ρ(h, p) = p, ∀p ∈ M implies that h = e. Now assume that we are given an effective action of a group ρ preserving the system. If the action of the group is effective the abelianity of G implies the abelianity of G. As a direct corollary we obtain, Corollary 6.3.7 Let G be a compact Lie group which acts effectively on the linear model of a rank 0 singularity preserving the system (R2n , is abelian. Proof: Consider the group of diffeomorphisms ρ(h), h ∈ G. According to theorem
i (1) (1) (1) (1)

dxi ∧ dyi , h) then G

6.4. Linearization in the neighbourhood of an orbit

115

6.3.6 this group of diffeomorphisms is abelian. Therefore ρ(h1 ) ◦ ρ(h2 ) = ρ(h2 ) ◦ ρ(h1 ), ∀h1 , h2 ∈ G. Since ρ is an action, this relation yields, ρ(h1 h2 h−1 h−1 ) = ρ(e), 1 2 h1 h2 h−1 h−1 = e, 1 2 ∀h1 , h2 ∈ G

But ρ(e) is the identity mapping and since the action is effective this implies ∀h1 , h2 ∈ G and therefore the group G is abelian.

6.4

Linearization in the neighbourhood of an orbit

In this section we prove a linearization theorem in the neighbourhood of an orbit. Recall the meaning of linear action of a group G in the neighbourhood of an orbit, G acts on the product V = Dk × Tk × D2(n−k) componentwise; the action of Γ on Dk is trivial, its action on Tk is by translations (with respect to the coordinate system (θ1 , ..., θk )), and its action on D2(n−k) is linear with respect to the coordinate system (x1 , y1 , ..., xn−k , yn−k ) Now we are ready to state and prove the linearization theorem in the neighbourhood of an orbit. Theorem 6.4.1 Let G be a compact Lie group preserving the system (Dk × Tk × D2(n−k) ,
k i=1

dpi ∧ dθi +
k i=1

n−k i=1

dxi ∧ dyi , F) then there exists ΦG a diffeomorphism
n−k i=1

defined in a tubular neighbourhood of the orbit L = Tk which preserves the system (Dk × Tk × D2(n−k) , of G becomes linear. Proof: After shrinking the original neighbourhood if necessary, we may assume without loss of generality that we are considering a G-invariant neighbourhood of L. First dpi ∧ dθi + dxi ∧ dyi , F) and under which the action

116

Chapter 6. Equivariant linearization and symplectic equivalence

of all, let us express in local coordinates how the action looks like. We denote by ρ the action of G. For convenience, we use the simplifying notation p = (p1 , . . . , pk ) and (x, y) = (x1 , y1 , . . . , xn−k , yn−k ). Since G preserves the system, in particular ρ preserves p and sends
∂ ∂θi

to

∂ ∂θi

because it preserves the symplectic form and it

sends the Hamiltonian vector fields associated to pi to the same vector fields. After all these considerations, for each h ∈ G the diffeomorphism ρ(h) can be written as,

h h ρ(h)(p, θ1 , . . . , θk , x, y) = (p, θ1 + g1 (p, x, y), . . . , θk + gk (p, x, y), αh (x, y, p)) h where the functions gi and αh are subdued to more constraints given by the

preservation of the system. Before considering these constraints, it will be most convenient to simplify the expression of αh first. This will be done using the local linearization theorem with parameters (corollary 6.3.5). In order to do that, we restrict our attention to the induced mapping, ρ(h)(p, x, y) = (p, αh (p, x, y)) and we consider the family of diffeomorphisms ρ(h)p : D2(n−k) −→ D2(n−k) defined as follows, ρ(h)p (x, y) = αh (p, x, y). We may look at p = (p1 , . . . , pk ) as parameters. For each p the mapping ρ(h)p (x, y) induces an action of G on the disk D2(n−k) which preserves the induced system (D2(n−k) ,
n i=1

dxi ∧ dyi , h). Observe that the preservation of the

induced system implies, in particular, that the action fixes the origin. According to corollary 6.3.5 we can linearize the action ρ(h)p in such a way that it is taken to the parametric-free linear action ρ(h)0 . We can extend trivially the diffeomorphism Φ in the disk provided by the corollary 6.3.5 to a diffeomorphism Ψ in the whole neighbourhood considered, simply by declaring,
(1)

6.4. Linearization in the neighbourhood of an orbit

117

Ψ(p, θ1 , . . . , θk , x, y) = (p, θ1 , . . . , θk , Φ(x, y)). After this linearization in the (x, y)direction the initial expression of ρ(h) looks like,

h h ρ(h)(p, θ1 , . . . , θk , x, y) = (p, θ1 + g1 (p, x, y), . . . , θk + gk (p, x, y), ρ(h)0 (x, y)),

(1)

Since the action preserves the symplectic form on the parameters (p1 , . . . , pk ). That is,

i=1

dpi ∧ dθi +

n i=1

dxi ∧ dyi

h we conclude that the functions gi do not depend on (x, y) and so far just depend

h h ρ(h)(p, θ1 , . . . , θk , x, y) = (p, θ1 + g1 (p), . . . , θk + gk (p), ρ(h)0 (x, y)), h Observe that if we prove that these functions gi do not depend on p then we will

(1)

be done because then the induced action on Tk will be performed by translations. And, in all, the action will be linear. Consider H = {ρ(h), h ∈ G}, we are going to prove that this group is abelian. We have to check that ρ(h1 ) ◦ ρ(h2 ) = ρ(h2 ) ◦ ρ(h1 ) We compute ρ(h1 ) ◦ ρ(h2 )(p, θ1 , . . . , θk , x, y) =
h h h h (p, θ1 + g1 2 (p) + g1 1 (p), . . . , θk + gk 2 (p) + gk 1 (p), ρ(h1 )0 (x, y) ◦ ρ(h2 )0 (x, y)) (1) (1)

on the other hand, ρ(h2 ) ◦ ρ(h1 )(p, θ1 , . . . , θk , x, y) =
h h h h (p, θ1 + g1 1 (p) + g1 2 (p), . . . , θk + gk 1 (p) + gk 2 (p), ρ(h2 )0 (x, y) ◦ ρ(h1 )0 (x, y)) (1) (1)

Clearly, the first 2k components coincide. As for the 2(n − k) last components, we can use theorem 6.3.6 to conclude the commutation. So far we know that the group H is abelian. It is also compact, therefore it is a direct product of a torus Tr with finite groups Z/mr Z. We are going to check

118

Chapter 6. Equivariant linearization and symplectic equivalence

h that for each ρ(h) ∈ H the functions gi do not depend on p. It is enough to check

it for ρ(h) in one of the components Z/nZ and Tr . So we distinguish two cases, • ρ(h) belongs to Z/nZ.
h Then ρ(h)n = Id this condition yields, ngi (p) = 2πmi , for all 1 ≤ i ≤ k and h mi ∈ Z. In particular, gi (p) = 2πmi n h and gi does not depend on p.

• ρ(h) belongs to Tr . We can consider a sequence ρ(hn ) lying on the torus which belong to a finite group Z/kn Z and which converge to ρ(h). For each of these points ρ(hn ) we can apply the same reasoning as before to obtain,
h gi n (p) = 2πmi . kn

Now for each n, the diffeomorphism ρ(hn ) does not depend on p, we may write this condition as, ∂ρ(hn ) = 0, ∂pi

1≤i≤k

Now since the action is smooth we can take limits in this expression to obtain that ∂ρ(h) = 0, ∂pi 1≤i≤k

h and finally gi (p) does not depend on p.

And this ends the proof of the theorem. We may look at this theorem as a slice theorem for integrable systems. In order to announce this result as a slice theorem we need some notation.

6.4.1

The classical slice theorem

Let G be a compact Lie group and let H be a closed subgroup. Let ϕ be a representation of H on a vector space E. Then we have an induced action of H on

6.4. Linearization in the neighbourhood of an orbit

119

the product G × E. The action is given by ρ(h, (g, u)) = (gh−1 , ϕ(h, u)). We can consider the space of orbits by this action; it is the quotient G × E/H. Observe that this quotient is a vector bundle over G/H with typical fiber E. Classically, this quotient in denoted by G ×H E. As a matter of notation a class in the quotient is denoted by [g, u]. Observe that the action of G on G × E defined as α(a, (g, u)) = (ag, u), G ×H E −→ G/H is G-equivariant. Example Assume that β stands for an action of a compact Lie group on a manifold M . Let p be a point in M . We denote by G·p the orbit through the point p. We denote by Gp the isotropy group for the action at the point p, Gp = {g ∈ G, α(g, p) = p}. It is a well-known fact that the isotropy group is a compact subgroup of G. Now consider a Gp -invariant Riemannian metric in a Gp invariant neighbourhood of the orbit G · p. Define E as the subspace of the tangent space at the point p which is orthogonal to the tangent space to the orbit. Since the metric chosen is Gp -invariant. The action of Gp induces an action of Gp on E. Denote by αp the differential of the action of Gp at the point p. As before, αp defines a representation when restricted to E. So if we take H = Gp and ϕ = αp the vector bundle defined above becomes, G ×Gp E. This example is more than an example. It is the standard model for the action of a compact Lie group on a manifold. The classical slice theorem [35] asserts that a neighbourhood of the orbit is diffeomorphic to G ×Gp E. The linear representation of Gp on E induced by the action of G is called the slice representation. In the case the action of the manifold preserves the fibration defined by F and the symplectic structure, we have a similar “slice theorem” in the neighbourhood
(1) (1) (1)

a ∈ G commutes

with the action of H and hence descends to the quotient and the projection π :

120

Chapter 6. Equivariant linearization and symplectic equivalence

of an orbit whenever the orbit L coincides with the orbit of the action of the group.

6.4.2

The slice statement of the linearization

Now, let us go back to our situation. Let p be a point lying on the orbit L. Observe that the preservation of the system yields that the orbit of the action through a point p ∈ L is contained in L but it does not have to coincide with L. From now on we will assume that L coincides with an orbit of the action. We take coordinates centered at p. We consider the isotropy group at p, Gp . Since Gp preserves the symplectic structure leaves the symplectic orthogonal to the orbit invariant. On the other hand, the isotropy group Gp fixes L thus it induces an action of Gp on D2(n−k) , where D2(n−k) is endowed with the (xi , yi ) coordinates. On D2(n−k) we have the induced system (U (L),
i

dxi ∧ dyi , h) being

h = (fk+1 , . . . , fn ). By virtue of theorem 6.3.4 we can linearize the induced action by the isotropy group in a foliation preserving way. Observe that this linear action can be extended trivially to a linear action βp on a vector space E1 foliation F on E1
2(n−k) 2(n−k)

containing

the disk D2(n−k) . In the same way the foliation defined by h can be extended to a
k . We denote this extension by βp . Now let E2 a vector space

containing the disk Dk endowed with coordinates (p1 , . . . , pk ). Finally, we define the linear representation of Gp on E1 Now form the bundle G×Gp (E1 G · p × F on G × (E1
2(n−k) 2(n−k) k × E2 as γp (u, v) = (βp (u), v). 2(n−k) k ×E2 ) attached to this linear representation.

On this bundle we can consider the foliation induced by the product foliation
k × E2 ).

With all this notations, now we are ready to give another presentation of theorem 6.4.1 “` la slice”. We do it in form of corollary, a Corollary 6.4.2 Let G be a compact group preserving the system (Dk × Tk × D2(n−k) ,
k i=1

dpi ∧ dθi +

n−k i=1

dxi ∧ dyi , F) and let p be a point in L. Assume that

L = G · p, then a neighbourhood of the orbit L is G-equivariantly diffeomorphic to

6.5. Equivariant symplectic equivalence a neighbourhood of the zero section of the bundle G ×Gp (E1 this diffeomorphism can be chosen to be foliation preserving.
2(n−k)

121
k × E2 ). Further,

6.5

Equivariant symplectic equivalence

As a corollary of the G-linearization results in the linear model obtained in the last section and the symplectic linearization in the covering obtained in the last chapter we obtain the equivariant symplectic equivalence. This equivariant symplectic equivalence is valid also for analytical systems (since the results for G-linearization are valid for analytical systems and the results of symplectic equivalence in the covering are valid for analytical systems [54]). Now we can formulate the equivariant symplectic linearization theorem for nondegenerate singular orbits of integrable Hamiltonian systems, that we have been envisaging for chapters: Theorem 6.5.1 Consider F the foliation defined by a completely integrable system and consider L, a compact orbit of Williamson type (ke , kh , kf ). Let ω be a symplectic for which the foliation F is Lagrangian. Then there exists a finite group Γ and a diffeomorphism taking the foliation to the linear foliation on V /Γ given by (6.2.1,6.2.2,6.2.3,6.2.4), and taking ω to ω0 , which sends L to the torus {pi = xi = yi = 0}. The smooth symplectomorphism φ can be chosen so that via φ, the system-preserving action of the compact group G near L becomes a linear system-preserving action of G on V /Γ. If the moment map F is real analytic and the action of G near L is analytic, then the symplectomorphism φ can also be chosen to be real analytic. If the system depends smoothly (resp., analytically) on a local parameter (i.e. we have a local family of systems), then φ can also be chosen to depend smoothly (resp., analytically) on that parameter. Observation 6.5.1 A proof for the twisted hyperbolic case when n = 2 and k = 1 was provided by Curr´s-Bosch in [11]. a

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Chapter 6. Equivariant linearization and symplectic equivalence

Chapter 7 Contact linearization of singular Legendrian foliations
7.1 Introduction

The aim of this chapter is to prove an analogue to the linearization result for singular Lagrangian foliations which was studied in the previous chapters but in the case of singular Legendrian foliations in contact manifolds. Consider a contact manifold M 2n+1 together with a contact form. We assume that the Reeb vector field associated to α coincides with the infinitesimal generator of an S 1 action. We assume further than there exists n-first integrals of the Reeb vector field which commute with respect to the Jacobi bracket. Then there are two foliations naturally attached to the situation. On the one hand, we can consider the foliation associated to the distribution generated by the contact vector fields. We call this foliation F . On the other hand we can consider a foliation F given by the horizontal parts of the contact vector fields. The functions determining the contact vector fields may have singularities. We will always assume that those singularities are of non-degenerate type. Observe that F is nothing but the enlarged foliation determined by the folia123

124

Chapter 7. Contact linearization of singular Legendrian foliations

tion F and the Reeb vector field. Let α be another contact form in a neighbourhood of a compact orbit O of F for which F is Legendrian and such that the Reeb vector field with respect to α coincides with the Reeb vector field associated to α. In this chapter we prove that then there exists a diffeomorphism from a neighbourhood of O to a model manifold taking the foliation F to a linear foliation in the model manifold with a finite group attached to it and taking the initial contact form to the Darboux contact form. As it was done in the last chapter for Lagrangian foliations determined by a completely integrable system, we also prove the G-equivariant version of this fact for Legendrian foliations. That is, we prove that in the case there exists a compact Lie group preserving the first integrals of the Legendrian foliation and preserving the contact form then the contactomorphism can be chosen to be G-equivariant. The problem of determining normal forms for foliations related to Legendrian foliations has its own story. P. Libermann in [38] established a local equivalence theorem for α-regular foliations. Loosely speaking, those foliations are regular foliations containing the Reeb vector field and a Legendrian foliation. The problem of classifying contact forms is different from the problem of classification of contact structures. As a example of this, if M is a compact manifold then any two contact structures are equivalent as Gray’s theorem asserts ([27]). Whereas one can find examples of two contact forms which are not equivalent (see for example [26]). The problem of classifying contact structures which are invariant under a Lie group was considered by Lutz in [41]. In particular he proves that two contact structures in a compact manifold M 2n+1 which are invariant by a locally free action of Rn+1 are equivalent in the sense that there exists an equivariant contactomorphism taking one to the other. The foliations studied by Libermann and Lutz are regular. The singular counterpart to the result of Lutz was proved by Banyaga and Molino in [4] but for contact forms.

7.1. Introduction

125

Namely, Banyaga and Molino study the problem of finding normal forms under the additional assumption of transversal ellipticity. The assumption of transversal ellipticity allows to relate the foliation F of generic dimension (n + 1) with the foliation given by the orbits of a torus action. This chapter pretends to extend these results for foliations which are related in the same sense to (n+1)-foliations but which are not necessarily identified with the orbits of a torus action. All our study of the problem is done in a neighbourhood of a compact orbit. Global results for contact manifolds admiting torus action have been obtained by Banyaga and Molino in [4] and recently by Lerman in [36]. Linearization results for contact vector fields in R2n with an hyperbolic zero were considered by Guillemin and Schaeffer in [28]. The chapter is organized as follows: In the first section we make a review of the basic facts in contact geometry that we will need later. In section 2 we define two foliations, F and F and we prove that we can find coordinates in a finite covering such that the foliations have a particularly simple form. In section 3 we prove that for any two contact forms for which F is Legendrian and having the same Reeb vector field, we can find a foliation preserving contactomorphism taking one to the other. It turns out that the Legendrian condition imposed on the foliation for the contact form α becomes a Lagrangian condition for the same foliation with the symplectic form dα defined in a convenient submanifold. The result appears then as an application of the symplectic equivalence results for Lagrangian foliations which we have been working out in the previous chapters. In the last Section we establish the G-equivariant version of contact equivalence. Applying this G-equivariant version to the particular case of the finite group attached to the finite covering, we obtain as a consequence the contact equivalence of any two contact forms fulfilling the above mentioned conditions.

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Chapter 7. Contact linearization of singular Legendrian foliations

7.2

Basics in contact geometry

In this section we recall some basic definitions in contact geometry. Definition 7.2.1 Let M 2n+1 be a 2n + 1-dimensional manifold. A 1-form on a manifold M 2n+1 is a contact form if the set E = {(p, u) ∈ T (M ), bundle E −→ M . When we talk about a contact pair we consider a pair (M, α) where α is a contact form on M . αp (u) = 0} is a smooth subbundle of T (M ) and dα|E is a symplectic structure on the vector

Remark: • The classical definition of contact manifold is the following. It is a pair (M, α) where α satisfies the condition α∧(dα)n = 0, ∀p ∈ M . In turn, this condition p implies the nonintegrability of the subbundle E = {(p, u) ∈ T (M ), T (S) = E. Suppose that α is a contact form on a manifold M . Then if f is a positive function the 1-form f α is also a contact form. This motivates the definition of contact structure, Definition 7.2.2 A contact structure on a manifold M is a subbundle E of the tangent bundle of the form E = {(p, u) ∈ T (M ), form α. The problem of classification of contact structures is different from that of contact forms. There are a lot of results in the literature concerning the classification of contact structures from a local, global or semilocal point of view. Finding their counterparts for contact forms is not always possible. αp (u) = 0} for some contact αp (u) = 0}. That is it is not possible to find a symplectic submanifold S such that

7.2. Basics in contact geometry Our problem of classification will always be focused on contact forms.

127

In contrast to symplectic manifolds (M, ω) where the condition iX (ω) = 0 implies X = 0, in a contact manifold we can find non-trivial solutions X to the equation iX (ω) = 0. A privileged solution of this equation has the particular name of Reeb vector field. It is a concept attached to the contact form rather than the contact structure. Definition 7.2.3 Given a contact pair (M, α), the Reeb vector field Z is the unique vector field satisfying the following two conditions, • iZ dα = 0. • α(Z) = 1. The Reeb vector field is a particular case of what we call contact vector field. Definition 7.2.4 Let f be a smooth function on the contact pair (M, α) the contact vector field associated to f is the unique vector field Xf fulfilling the following two conditions • iXf dα|E = −df|E . • α(Xf ) = f. Observe that the contact vector field associated to the function 1 is precisely the Reeb vector field. As it is proved in [38], we can express any vector field X in T (M ) as a sum of two vector fields X1 and X2 where the vector field X1 belongs to the subbundle E and its called the horizontal part of X and the vector field X2 is the component in the direction of the Reeb vector field. The standard notation for the horizontal vector field associated to X is X.

128

Chapter 7. Contact linearization of singular Legendrian foliations

We can now define the notion of Jacobi bracket of two functions, which is the contact counterpart to the Poisson bracket of two functions. Definition 7.2.5 Let f, g be two smooth functions on a contact pair (M, α), we define the Jacobi bracket as, [f, g] = α([Xf , Xg ]). The following relations are proved in [38], • X[f,g] = [Xf , Xg ] • [f, g] = dα(Xf , Xg ) + f (Z(g)) − g(Z(f )) (7.2.2) (7.2.1)

Definition 7.2.6 A submanifold N ⊂ M 2n+1 is Legendrian if dimN = n and α(X) = 0 for any X ∈ T (N ).

7.3

The foliation and its differentiable linearization

In this section we define the foliations that we will work with throughout the chapter and we will also define the linear model.

7.3.1

Posing the problem

Let (M 2n+1 , α) be a contact pair and let Z be its Reeb vector field. We make the following assumptions,

7.3. The foliation

129

• We assume Z coincides with the infinitesimal generator of an S 1 action. Let S be one of its orbits. • We assume that there are n first integrals f1 , . . . , fn of Z (that is Z(fi ) = 0) which fulfill the following additional hypotheses: 1. The first integrals are independent in an open dense set. That is, df1 ∧ · · · ∧ dfn = 0 in an open dense set. 2. The n-first integrals are in involution with respect to the Jacobi bracket associated to α. That is to say, [fi , fj ] = 0 , ∀i, j. 3. The minimum rank of the differential (df1 , . . . , dfn ) is k. Let p be a point in M 2n+1 such that the rank is exactly k. Let O be the orbit of the contact vector fields through p. We will assume the following, (a) O is diffeomorphic to a torus of dimension k + 1. (b) The first integrals f1 , . . . , fk are non-singular along O and the first integrals fk+1 , . . . , fn have a non-degenerate singularity in the MorseBott sense along O. Since [fi , fj ] = 0 then due to formula 7.2.1, [Xfi , Xfj ] = 0 and this implies that the distribution < Z, Xf1 , . . . , Xfn > is involutive because the functions fi are first integrals of the Reeb vector field. Thus, we can talk about the foliation generated by the contact vector fields of the functions 1, f1 , . . . , fn . This foliation will be denoted by F .

On the other hand, consider the horizontal parts of the contact vector fields. They have the form Xf = Xf − f Z. Thus the distribution < Xf1 , . . . Xfn > defines an involutive distribution. The foliation defined by this distribution will be denoted

130

Chapter 7. Contact linearization of singular Legendrian foliations

by F. Observe that since α(Xf ) = f and α(Z) = 0 then the regular leaves of this foliation are Legendrian submanifolds with respect to α. That is why this foliation will be called the singular Legendrian foliation. In fact we will work with germ-like foliations. That is, we will assume that the foliation is defined in a neighbourhood of O. Now let p ∈ M be a singular point. We will say that the point has rank r if the dimension of the orbit through p is r.

Once the two foliations F and F are defined we are ready to pose the following problem.

Problem Study the contact forms α defined in a neighbourhood of O for which F is Legendrian and such that the Reeb vector field with respect to α coincides with the Reeb vector field with respect to α.

As far as this problem is concerned we will prove the following. There exists a diffeomorphism φ defined in a neighbourhood of O such that φ∗ (α ) = α and φ preserves the foliations F and F .

In order to deal with this problem we will need to introduce coordinates in such a way that the foliations F and F are really simple. This judicious choice of coordinates leads us to the linear model.

7.3.2

Differentiable linearization

In this section we want to prove that under the above assumptions there exist coordinates in a neighbourhood of O such that the foliation can be linearized.

7.3. The foliation We prove the following theorem,

131

Theorem 7.3.1 There exist coordinates (θ0 , . . . , θk , p1 , . . . , pk , x1 , y1 , . . . , xn−k , yn−k ) in a finite covering of a tubular neighbourhood of O such that • The Reeb vector field is Z =
∂ . ∂θ0

• There exists a triple of natural numbers (ke , kh , kf ) with ke + kh + 2kf = n − k and such that the first integrals fi are of the following type, fi = pi , i ≤ k and
2 fi+k = x2 + yi for 1 ≤ i ≤ ke , i

1≤

fi+k = xi yi for ke + 1 ≤ i ≤ ke + kh , fi+k = xi yi+1 − xi+1 yi and fi+k+1 = xi yi + xi+1 yi+1 for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf • The foliation F is given by the orbits of the distribution D =< Y1 , . . . Yn > where Yi = Xi − fi Z being Xi the contact vector field of fi with respect to the contact form α = dθ0 + Proof: First of all, since Z is the infinitesimal generator of an S 1 -action, according to the Slice Theorem [50] a neighbourhood of O in M 2n+1 is diffeomorphic to the
1 1 bundle S 1 ×Sx W where Sx denotes the isotropy group at a point in the orbit. Thus n−k 1 i=1 2 (xi dyi

− yi dxi ) +

k i=1

pi dθi .

we can choose coordinates (θ0 , . . . , θk , p1 , . . . , pk , x1 , y1 , . . . , xn−k , yn−k ) in a finite covering of a neighbourhood of O such that the Reeb vector field has the form Z =
∂ . ∂θ0

Now the 1-form α can be written as α = dθ0 + α.

132

Chapter 7. Contact linearization of singular Legendrian foliations

Observe that since Z is the Reeb vector field in particular we obtain iZ dα = 0 Using Cartan’s formula LZ (dα) = diZ (α) + iZ dα we deduce that α does not depend on θ0 . Further, the condition on the contact form α ∧ dαn = 0 implies that dα is a symplectic form in the submanifold N0 = {p ∈ U (O), first integrals. The equation iY dα = −dfi has a unique well-defined solution when restricted to the symplectic submanifold
s N0 . We denote by Xfi the n Hamiltonian vector fields of the functions fi with c respect to the symplectic structure dα on N0 . We denote by Xfi the n contact

θ0 = 0}. Let fi be the n

vector fields of the functions fi with respect to the contact structure α. With all these information at hand we can write
c s Xfi = Xfi + gi Z

(7.3.1)

for certain smooth functions gi .

We are going to focus our attention in the symplectic submanifold N0 and in
s the Hamiltonian vector fields Xfi for a while.

First of all, we will check that {fi , fj } = 0 where {, } stands for the Poisson
s bracket attached to dα. Thus, the vector fields Xfi define a completely integrable

Hamiltonian system on N0 and the foliation they define is a singular Lagrangian foliation. We are going to check {fi , fj } = [fi , fj ]

7.3. The foliation Because of the definition of Poisson bracket,

133

s s {fi , fj } = dα(Xfi , Xfj ) s s Since dα = dα, we can write this last equality as, dα(Xfi , Xfj )

Taking into account this observation and due to 7.3.1 this equality can be written as,

c c {fi , fj } = dα(Xfi − gi Z, Xfj − gi Z)

But Z is the Reeb vector field and the last expression reads

c c dα(Xfi , Xfj )

which is, by definition, the Jacobi bracket of the functions fi and fj . Thus {fi , fj } = [fi , fj ] = 0

Denote by ON a singular compact orbit of minimal rank of the singular Lagrangian foliation in N0 . According to the symplectic linearization theorem (theorem 6.5.1) for Lagrangian foliations whose proof was concluded in the last chapter. There exists a diffeomorphism in a neighbourhood of a singular compact orbit which takes the foliation to the linearized one and the symplectic structure dα to the Darboux symplectic structure. Recall that the linearized foliation has a finite group attached to it. In particular, we can find a diffeomorphism in a covering of a tubular neighbourhood of ON , φ : (U (ON )) −→ φ((U (ON ))) such that in the new coordinates provided by the diffeomorphism the first integrals have the following simple form:

134

Chapter 7. Contact linearization of singular Legendrian foliations

2 fi+k = x2 + yi for 1 ≤ i ≤ ke , i

fi+k = xi yi for ke + 1 ≤ i ≤ ke + kh , fi+k = xi yi+1 − xi+1 yi and fi+k+1 = xi yi + xi+1 yi+1 for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf Now define, ϕ : S 1 × (U (ON )) −→ ϕ(S 1 × (U (ON ))) (θ0 , z) Observe that since
k n−k

−→

(θ0 , φ(z))

φ(
i=1

∗

dpi ∧ dθi +
i=1

dxi ∧ dyi ) = dα

Then φ∗ ( this yields,

1 (xi dyi 2

− yi dxi ) +

pi dθi + dH) = α

1 (xi dyi − yi dxi ) + pi dθi + dH) = dθ0 + α 2 Thus we may assume that in the new coordinates ϕ∗ (dθ0 +
n−k

α = dθ0 +
i=1

1 (xi dyi − yi dxi ) + 2

k

pi dθi + dH
i=1

Now consider the path of contact forms
n−k

αt = dθ0 +
i=1

1 (xi dyi − yi dxi ) + 2

k

pi dθi + tdH
i=1

Observe that α1 = α and α0 is the Darboux contact form. Let ψt be the flow of the vector field X = −HZ. Note that as a matter of fact, φ1 (θ0 , θ1 , . . . , θk , p1 , . . . , pk , x1 , . . . , yn−k ) = (θ0 −H, θ1 , . . . , θk , p1 , . . . , pk , x1 , . . . , yn−k ).
∗ So ψ1 (α1 ) = α0 . Thus, ψ1 (α1 ) = α0 and in the new coordinates provided by ψ1

we can assume that α is the Darboux contact form. That is to say, we can assume that α = dθ0 +
n−k 1 i=1 2 (xi dyi

− yi dxi ) +

k i=1

pi dθi .

7.3. The foliation Observe that L−HZ (fi ) = 0 this implies that the same form.
d ∗ ψ (fi ) dt t

135
∗ = 0 and therefore ψt (fi )

∗ ∗ does not depend on t thus ψ1 (fi ) = ψ0 (fi ) = fi . So in the new coordinates fi have

Finally the foliation we are considering is generated by the horizontal parts of Xfi which in the new coordinates are Yi = Xi −fi Z being Xi the contact vector field of fi with respect to the contact form α = dθ0 + This ends the proof of the theorem. This theorem establishes the existence of a linear foliation and a model manifold.
2n+1 The model manifold is the manifold M0 = Tk+1 × U k × V 2(n−k) , where n−k 1 i=1 2 (xi dyi −yi dxi )+ k i=1

pi dθi .

U k and V 2(n−k) are k-dimensional and 2(n − k) dimensional disks respectively. Now we introduce a contact form in this model manifold. We take coordinates (θ0 , . . . , θk ) on Tk+1 , (p1 , . . . , pk ) on U k and (x1 , . . . , xn−k , y1 , . . . yn−k ) on V 2(n−k) and we consider the following contact form
k (n−k)

α0 = dθ0 +
i=1

pi dθi +
i=1

1 (xi dyi − yi dxi ). 2

2n+1 The pair (M0 , α0 ) is called the contact model manifold. The Reeb vector

field in the contact model manifold is the vector field

∂ . ∂θ0

Now consider functions of the following type, fi = pi ,
2 fi+k = x2 + yi for 1 ≤ i ≤ ke , i

1 ≤ i ≤ k and

fi+k = xi yi for ke + 1 ≤ i ≤ ke + kh , fi+k = xi yi+1 − xi+1 yi and fi+k+1 = xi yi + xi+1 yi+1 for i = ke + kh + 2j − 1, 1 ≤ j ≤ kf The linear foliation is the foliation given by the orbits of the distribution D =< Y1 , . . . Yn > where Yi = Xi − fi Z being Xi the contact vector field of fi in the contact model manifold.

136

Chapter 7. Contact linearization of singular Legendrian foliations

In all, we have proved that there exists a finite covering of a neighbourhood U (O) of the compact orbit considered such that the lifted foliation in the covering is differentiably equivalent to the linear foliation in the contact model manifold. The linear model for the foliation F is the foliation expressed in the coordinates provided by the theorem together with a finite group attached to the finite covering. The different smooth submodels corresponding to the model manifold are labeled by a finite group which acts in a contact fashion and preserves the foliation in the model manifold. This is the only differentiable invariant. Therefore, our problem of contact equivalence will be studied in this model manifold and the equivalence will be established via the equivariant version equivalence which will be considered in the last section.

7.4

Contact linearization in the model manifold

The aim of this section is to prove the following theorem,
2n+1 Theorem 7.4.1 Let α be a contact form on the model manifold M0 for which

F is a Legendrian foliation and such that the Reeb vector field is (θ0 , . . . , θk , 0, . . . , 0) preserving F and taking α to α0 . Proof:

∂ . ∂θ0

Then there

exists a diffeomorphism φ defined in a neighbourhood of the singular orbit O =

We are going to solve the problem by adjusting the contact form to a point where we can apply our symplectic linearization result. Let us start by considering the contact 1-form α, α = Adθ0 + Bi dpi + Ci dθi + Di dxi +
∂ ∂θ0

Ei dyi

Observe that the fact that the Reeb vector field is conditions on α,

imposes the following two

7.4. Contact linearization
∂ • α( ∂θ0 ) = 1, that is to say A = 1.

137

So far we can write α = dθ0 + α , being α = Ei dyi . • i
∂ ∂θ0

Bi dpi +

Ci dθi +

Di dxi +

dα = 0,

Since dα = dα the condition becomes, i Now Cartan’s formula yields, 0=i dα = L α − di α dα = 0

∂ ∂θ0

∂ ∂θ0

∂ ∂θ0

∂ ∂θ0

Since the last term vanishes this chain of equalities give the condition, L α =0

∂ ∂θ0

Therefore, the coefficient functions do not depend on θ0 . Let us see that the submanifold θ0 = 0 equipped with the form dα is a symplectic submanifold of the model contact manifold. We denote this submanifold by N . Since α is a contact form dα has to be symplectic in the vector bundle E defined by E = {(p, u) ∈ T (M ), structure on N . Observe that the vector fields Xi = Xfi are tangent to the submanifold N . Next step, we check that the vector fields Xi are Lagrangian for N , observe that α(Xi ) = fi . Now since, dα (Xi , Xj ) = Xi α(Xj ) − Xj α(Xi ) − α([Xi , Xj ]) According to the computation above Xi α(Xj ) = Xi (fj ) but fi are first integrals for the foliation and therefore this term vanishes. Symmetrically, the second term vanishes. And since the Lie bracket of the vector fields are zero we obtain, dα (Xi , Xj ) = 0 αp (u) = 0} and dα = dα then dα defines a symplectic

138

Chapter 7. Contact linearization of singular Legendrian foliations

Therefore, the foliation F is Lagrangian for dα and we may apply the symplectic linearization result in a neighbourhood of L = Tk (theorem 5.2.1) to find a local diffeomorphism ϕ : U (L) −→ ϕ(U (L)) in a neighbourhood of the leaf L, preserving the foliation F and satisfying ϕ∗ (ω0 ) = dα , where ω0 =
i

dpi ∧ dθi +

dxi ∧ dyi .

After shrinking the initial neighbourhood if necessary, the neighbourhood of Tk+1 in the initial manifold M can be decomposed as a product, S1 × U (L). The S1 corresponds to an orbit of the Reeb vector field. We denote by z a point in U (L). Now we define a diffeomorphism in the following way, φ : S1 × U (L) −→ φ(S1 × U (L)) (θ0 , z) −→ (θ0 , ϕ(z))

Since ϕ preserves F it is clear that this diffeomorphism is foliation-preserving. Now consider φ(S1 × U (L)) endowed with the Darboux contact form. That is with the contact form α0 = dθ0 + First observe that since ϕ∗ (ω0 ) = dα and ω0 = d(β), being β = (
k i=1 k i=1

pi dθi +

(n−k) 1 (xi dyi i=1 2

− yi dxi ). It remains

to check that the diffeomorphism above is indeed a contactomorphism.

pi dθi +

(n−k) 1 (xi dyi i=1 2

− yi dxi )) we can assert

that ϕ∗ (β) = α + df for a smooth function f . Observe that since ϕ preserves the foliation the function f is a basic function for the foliation. Now consider the path αt = α0 + tdf being α0 the contact form α0 = dθ0 + α .
∂ Now, consider the vector field X = −f ∂θ0 . Denote by ψt its flow. Since

ψ1 (θ0 , θ1 , . . . , θk , p1 , . . . , pk , x1 , . . . , yn−k ) = (θ0 −H, θ1 , . . . , θk , p1 , . . . , pk , x1 , . . . , yn−k ),
∗ we obtain ψ1 (α1 ) = α0 .

Therefore φ is a contactomorphism. And clearly it preserves the foliation because [X, Xi ] = 0 and therefore the flow ψt preserves the foliation. And this ends the proof of the theorem.

7.5. Equivariant contact linearization

139

7.5

Equivariant contact linearization

In this section we consider a compact Lie group G acting on a contact model manifold in such a way that preserves the n first integrals of the Reeb vector field and preserves the contact form as well. We want to prove that there exists a diffeomorphism in a neighbourhood of O preserving the n first integrals , preserving the contact form and linearizing the action of the group. This result is a consequence of the equivariant symplectic linearization theorem of the last chapter. The notion of linear action of a Lie group on the contact model manifold is analogous to the equivalent notion for the symplectic model manifold.
2n+1 2n+1 2n+1 Let G be a group defining a smooth action ρ : G×M0 −→ M0 on M0 .

We assume that this action preserves the contact form α0 of the contact model manifold. That is to say ρ∗ (α0 ) = α0 . Assume further that it preserves the n-first g integrals (f1 , . . . , fn ), where fi = pi , 1 ≤ i ≤ k. For the sake of simplicity we denote by F the collective mapping F = (p1 , . . . , pk , fk+1 , . . . , fn ). We will say that
2n+1 the action of G on M0 is linear if it satisfies the following property: 2n+1 G acts on the product M0 = Dk × Tk+1 × D2(n−k) componentwise; the action

of G on Dk is trivial, its action on Tk+1 is by translations (with respect to the coordinate system (θ0 , . . . , θk )), and its action on D2(n−k) is linear with respect to the coordinate system (x1 , y1 , ..., xn−k , yn−k ). Under the above notations and assumptions. Now we can state and prove the following theorem, Theorem 7.5.1 There exists a diffeomorphism φ defined in a tubular neighbourhood of O such that, • it preserves the contact form α0 i.e φ∗ (α0 ) = α0 . • it preserves F . • it linearizes the action of G. That is to say φ ◦ ρg = ρg ◦ φ.
(1)

140 Proof:

Chapter 7. Contact linearization of singular Legendrian foliations

Recall that α0 = dθ0 + α0 being α0 the 1-form (

k i=1

pi dθi +

(n−k) 1 (xi dyi i=1 2

−

2n+1 yi dxi )). Consider the symplectic manifold S = M0 × (− , ) endowed with the

symplectic form ω0 = dt ∧ dθ0 + dα0 , where t stands for a coordinate function on
2n+1 (− , ). An action of G on M0 can be extended in a natural way to an action

of G on S as follows,
2n+1 2n+1 ρ : G × M0 × (− , )+ −→ M0 × (− , )

(g, z, t)

−→

(ρg (z), t)

On S we consider the moment mapping F = (F, t). We can apply the equivariant linearization theorem to obtain a symplectomorphism ϕ preserving F and linearizing the action ρ. From the definition of the action ρ and the definition of F ,
2n+1 this symplectomorphism clearly descends to a diffeomorphism ϕ on M0 which

linearizes the action ρ and which satisfies ϕ∗ (dα0 ) = dα0 . Therefore, ϕ∗ (α0 ) = α0 + dh Finally the diffeomorphism, φ(θ0 , . . . , θk , p1 , . . . , pk , x1 , . . . , yn−k ) = (θ0 − h, . . . , θk , p1 , . . . , pk , x1 , . . . , yn−k ) takes the form α0 + dh to α0 and provides new coordinates for which the action is linear. In the previous section we have attained the contact linearization in the covering. Now applying the theorem of equivariant linearization to the group of deck transformations we obtain as a corollary the following theorem, Theorem 7.5.2 Let F be a foliation fulfilling the hypotheses specified in section 7.3.1, let F be the enlarged foliation with the Reeb vector field Z and let α be a

7.5. The equivariant version

141

contact form for which F is Legendrian and such that Z is the Reeb vector field then there exists a diffeomorphism defined in a neighbourhood of O taking F to the linear foliation, the orbit O to the torus {xi = 0, yi = 0, pi = 0} and taking the contact form to the Darboux contact form α0 .

142

“Where am I? Metaphysics says No question can be asked unless It has an answer, so I can Assume this maze has got a plan.”

“The Maze” W.H Auden

143

144

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