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Geometry and Dynamics of Integrable Systems
 978-3-319-33503-2, 3319335030, 978-3-319-33502-5

Table of contents :
Front Matter ....Pages i-viii
Integrable Systems and Difierential Galois Theory (Juan J. Morales-Ruiz)....Pages 1-33
Singularities of bi-Hamiltonian Systems and Stability Analysis (Alexey Bolsinov)....Pages 35-84
Geometry of Integrable non-Hamiltonian Systems (Nguyen Tien Zung)....Pages 85-140

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Advanced Courses in Mathematics CRM Barcelona

Alexey Bolsinov Juan J. Morales-Ruiz Nguyen Tien Zung

Geometry and Dynamics of Integrable Systems

Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Enric Ventura

More information about this series at http://www.springer.com/series/5038

Alexey Bolsinov • Juan J. Morales-Ruiz Nguyen Tien Zung

Geometry and Dynamics of Integrable Systems Editors for this volume: Vladimir Matveev, Friedrich-Schiller-Universität Jena Eva Miranda, Universitat Politècnica de Catalunya

Alexey Bolsinov School of Mathematics Loughborough University Leicestershire, United Kingdom

Juan J. Morales-Ruiz Escuela Superior de Ingenieros de Caminos, Canales y Puertos Universidad Politécnica de Madrid Madrid, Spain

Nguyen Tien Zung Institut de Mathématiques Université Paul Sabatier Toulouse, France

ISSN 2297-0304 ISSN 2297-0312 (electronic) Advanced Courses in Mathematics - CRM Barcelona ISBN 978-3-319-33502-5 ISBN 978-3-319-33503-2 (eBook) DOI 10.1007/978-3-319-33503-2 Library of Congress Control Number: 2016957339 Mathematics Subject Classification (2010): 37J35, 37K10, 70H06, 12H05, 53D17 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword This book contains the elaborated version of lecture notes for the Advanced Course on Geometry and Dynamics of Integrable Systems which took place at the Centre de Recerca Matem`atica, CRM-Barcelona, in September 2013 during a CRM Special Research Program. Native to actual problem-solving problems in mechanics, the topic of Integrable Systems is currently on the crossroad of different disciplines in pure and applied mathematics and it has important interactions with physics. The study of integrable systems has had special impact and also actively uses methods of Differential Geometry, especially in the branches of Symplectic Geometry and Hamiltonian Dynamics, Mathematical Physics, Lie Theory and Algebraic Geometry (including Mirror Symmetry). This is why we expect that the present notes will attract the attention of experts from different backgrounds. Three aspects of finite dimensional integrable system are discussed in the book: obstructions to algebraic integrability coming from differential Galois Theory, connections to bi-Hamiltonian systems, and the study of Integrable Systems in the non-Hamiltonian settings. The first part of the book contains notes by Professor Juan Morales (Universidad Polit´ecnica de Madrid, Spain). The author introduces the Differential Galois Theory and the Picard–Vessiot Theory. These techniques are applied to the investigation of integrability of the Schr¨ odinger equation and of planar vector fields. The second part of the book is written by Professor Alexey Bolsinov (Loughborough University, UK). It is based on his lectures on Singularities of bi-Hamiltonian Systems and Stability Analysis. Bi-Hamiltonian Systems constitute one of the main tools used to producing interesting examples of integrable systems; most if not all classical integrable systems are bi-Hamiltonian. The author presents effective tools for describing the singularities of integrable bi-Hamiltonian Systems, and then uses them in stability analysis and the study of linearization. The third part of the book by Professor Nguyen Tien Zung (Universit´e de Toulouse, France) focuses on the notion of non-Hamiltonian Integrable Systems and, more concretely, on the study of local normal forms for these systems. This part of the book starts with the generalization of Liouville’s Theorem to the nonHamiltonian setting and the study of action-angle coordinates. The author also v

vi

Foreword

generalizes Birkhoff normal forms into this non-Hamiltonian context and ends up with the analysis of the geometry of systems of type (n, 0). An important tool and guiding light in this part of the book is the use of toric actions which have been very relevant for symplectic geometers. We hope that this book will be of interest to both specialists in this field and to scientists who start working on it. The authors have indeed included enough background material to make it available to a wide audience of mathematicians and physicists. The book contains many results which are published nowhere else, or were published elsewhere in a very reduced form. We would like to warmly thank the authors of these notes, Professor Morales, Professor Bolsinov, and Professor Zung, for their very active participation in the advanced course and for contributing with the elaboration of the chapters of this book. We are also indebted to the other members of the organizing and scientific committees, Professor Presas (ICMAT), and Professor Taimanov (Novosibirsk), as well as the participants of this activity for making the Advanced Course on the Geometry and Dynamics of Integrable Systems into such a success. Last but not least, we would like to thank the Centre de Recerca Matem` atica and the European Science Foundation via the research network CAST for making this special program possible with their financial support. Special thanks are due to the CRM staff for their efficiency and help in the Special Program on Geometry and Dynamics of Integrable Systems.

Barcelona, Jena, November 2015.

Eva Miranda, Vladimir Matveev

Contents Foreword 1

2

Integrable Systems and Differential Galois Theory Juan J. Morales-Ruiz 1.1 Differential Galois Theory . . . . . . . . . . . . . . . . . . 1.1.1 Algebraic Groups . . . . . . . . . . . . . . . . . . . 1.1.2 Picard–Vessiot Theory . . . . . . . . . . . . . . . . 1.1.3 Kovacic Algorithm . . . . . . . . . . . . . . . . . . 1.1.4 Algebrization . . . . . . . . . . . . . . . . . . . . . 1.1.5 Bessel, Whittaker, and Hypergeometric Equations 1.2 Non-Integrability of Hamiltonian Systems . . . . . . . . . 1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1.2.2 Homogeneous Potentials . . . . . . . . . . . . . . . 1.3 Integrability of the Schr¨ odinger Equation . . . . . . . . . 1.3.1 The Schr¨odinger Equation . . . . . . . . . . . . . . 1.3.2 Application of the Picard–Vessiot Theory . . . . . 1.4 Integrability of Fields on the Plane . . . . . . . . . . . . . 1.4.1 Riccati Fields . . . . . . . . . . . . . . . . . . . . . 1.4.2 Quadratic Polynomial Fields . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

Singularities of bi-Hamiltonian Systems and Stability Analysis Alexey Bolsinov 2.1 Integrable Systems: Singularities and Bifurcations . . . . . . . . . . 2.1.1 Integrable Systems, Lagrangian Fibrations and their Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Stability and Singularities for Integrable Systems . . . . . . 2.1.3 Non-Degenerate Singularities . . . . . . . . . . . . . . . . . 2.2 Jordan–Kronecker Decomposition . . . . . . . . . . . . . . . . . . . 2.2.1 Basic Poisson Geometry . . . . . . . . . . . . . . . . . . . . 2.2.2 Jordan–Kronecker Decomposition . . . . . . . . . . . . . . . 2.2.3 Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 10 14 16 18 18 20 23 24 24 27 27 28 31

35 35 35 39 40 44 44 47 50 vii

viii

Contents 2.2.4 Compatible Poisson Brackets and Commuting Casimirs . . 2.2.5 Properties of FΠ . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Linearisation of Poisson Pencils and a Criterion of Non-Degeneracy 2.3.1 Linearisation of a Poisson Structure . . . . . . . . . . . . . 2.3.2 Linearisation of a Poisson Pencil . . . . . . . . . . . . . . . 2.3.3 Linear Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Non-Degenerate Linear Pencils . . . . . . . . . . . . . . . . 2.3.5 Classification of Non-Degenerate Linear Pencils . . . . . . . 2.3.6 General Non-Degeneracy Criterion . . . . . . . . . . . . . . 2.4 How Does It Work? Examples and Applications . . . . . . . . . . . 2.4.1 Rubanovskii Case . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Mischenko–Fomenko Systems on Semisimple Lie Algebras . 2.4.3 Euler–Manakov Tops on so(n) . . . . . . . . . . . . . . . . 2.4.4 Periodic Toda Lattice . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 54 57 58 59 60 61 64 66 68 68 70 73 76 79

3 Geometry of Integrable non-Hamiltonian Systems Nguyen Tien Zung 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Normal forms, action-angle variables, and associated torus actions 3.2.1 Liouville torus actions and action-angle variables . . . . . . 3.2.2 Local normal forms of singular points . . . . . . . . . . . . 3.2.3 Geometric linearization of non-degenerate singular points . 3.2.4 Semi-local torus actions and normal forms . . . . . . . . . . 3.3 Geometry of integrable systems of type (n, 0) . . . . . . . . . . . . 3.3.1 Normal forms and automorphism groups . . . . . . . . . . . 3.3.2 Induced torus action and reduction . . . . . . . . . . . . . . 3.3.3 Systems of toric degree n − 1 and n − 2 . . . . . . . . . . . 3.3.4 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Totally hyperbolic actions . . . . . . . . . . . . . . . . . . . 3.3.6 Elbolic actions and toric manifolds . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 88 88 93 99 107 109 110 114 118 125 128 133 135

Chapter 1

Integrable Systems and Differential Galois Theory Juan J. Morales-Ruiz 1 1.1

Differential Galois Theory

At the end of the nineteenth century, Picard [25, 26], [27, Chapter XVII] and, in a clearer way, Vessiot in his PhD Thesis [30], created and developed a Galois theory for linear differential equations. This field of study, henceforth called Picard–Vessiot theory, was continued from the forties to the sixties of the twentieth century by Kolchin, through the introduction of the modern algebraic abstract terminology and the obtention of new important results, see [12] and references therein. Today, the standard reference of this theory is the monograph [29]. In the last years, a new revival of interest in the Differential Galois Theory is being observed. This is partially due to the connections and applications to other areas of mathematics: number theory [6, 10], asymptotic theory [18], etc., but also to the integrability of dynamical systems. As we shall see, within Differential Galois Theory there is a very nice concept of “integrability”, i.e., solutions in closed form. In this chapter we only review the necessary definitions and results of the Picard–Vessiot Theory in order to understand later the applications to Integrable Systems. For more information, see [29].

1I

would like to thank Sonja Hohloch for some corrections to a preliminary version of the present notes.

© Springer International Publishing Switzerland 2016 A. Bolsinov et al., Geometry and Dynamics of Integrable Systems, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-319-33503-2_1

1

2

Chapter 1. Integrable Systems and Differential Galois Theory

1.1.1 Algebraic Groups We present here the necessary results on linear algebraic groups. For more information, see the monographs [7, 8]. A linear algebraic group G (over C) is a subgroup of GL(m, C) whose matrix coefficients satisfy polynomial equations over C. It has both the structure of a nonsingular algebraic variety and that of a group, the two structures being compatible, i.e., the group operation and inversion are morphisms of algebraic varieties. We note that in a linear algebraic group there are two different topologies: the Zariski topology, where the closed sets are the algebraic sets, and the usual Hausdorff topology. In particular, an algebraic group is a complex analytic Lie group, and we can consider its Lie algebra. Therefore, the dimension of G is the dimension of the Lie algebra of G. Given a linear algebraic group, the maximal connected subgroup G0 which contains the identity is an algebraic group called the identity component of G. For those familiar to Lie theory, and since G is a complex analytic Lie group due to satisfying algebraic equations, the underlying group of G0 coincides with the identity component of G considered as a complex analytic Lie group. We remark that an algebraic linear (or affine) group G is usually defined as an affine algebraic variety with a group structure, with the above compatibility condition as to group multiplication and taking of inverses. Then, there is a rational faithful representation of G as a closed subgroup of GL(m, C), for some m, and we obtain the equivalence with our definition. It is clear that the classical linear complex groups are linear algebraic groups. For instance SL(n, C), SO(n, C) (rotation group) and Sp(n, C) ⊂ GL(2n, C) (symplectic group) are linear algebraic groups, since they are defined by polynomial identities. Proposition 1.1.1. The identity component G0 of a linear algebraic group G is a closed (with respect to the above two topologies) normal subgroup of G of finite index and it is connected (again, with respect to the two topologies). Furthermore, G/G0 is a finite group given by the classes of the irreducible connected components of G. We note that, by the above proposition, G0 is also a linear algebraic group and the Lie algebra of G, Lie(G) = G, coincides with the Lie algebra of G0 , Lie(G0 ) = G. As for every Lie group, G0 is solvable (resp. commutative) if and only if G is solvable (resp. commutative). Furthermore, G is connected if and only if G = G0 . The characterization of connected solvable linear algebraic groups is given by the Lie–Kolchin Theorem. Theorem 1.1.2 (Lie–Kolchin). A connected linear algebraic group is solvable if and only if it is conjugate to a triangular group. Given a subset S ⊂ GL(n, C), let M be the group generated by S and let G be the Zariski closure of M . By definition, the group G is a linear algebraic

1.1. Differential Galois Theory

3

group; we say G is topologically generated by M . Throughout the text, M will be typically the monodromy group of a Fuchsian linear differential equation and G will be its Galois group. Since most of the examples of irreducible equations appearing later on are second-order and symplectic, we end this section with a classification of the algebraic subgroups of SL(2, C) = Sp(1, C). We shall need two lemmas. Lemma 1.1.3 ([9]). Let G be an algebraic group contained in SL(2, C). Assume that the identity component G0 of G is solvable. Then G is conjugate to one of the following types: (i) G is finite;    0 λ 0 , (ii) G = β 0 λ−1

−β −1 0

   ∗  λ, β ∈ C ;

(iii) G is triangular. Lemma 1.1.4. Let G be an algebraic subgroup of SL(2, C) such that the identity component G0 is not solvable. Then G = SL(2, C). The last lemma is well-known and follows easily from consideration of the Lie algebra of G ⊂ SL(2, C). Indeed, if G0 is not solvable, then the dimension of G must be equal to 3, because all 2-dimensional Lie algebras are solvable. Proposition 1.1.5. Let G be an algebraic subgroup of SL(2, C). Then one of the following mutually excluding four cases must occur: (i) G is conjugate to a triangular group;    0 λ 0 , (ii) G is conjugate to the group β 0 λ−1

−β −1 0

    λ, β ∈ C∗ ;

(iii) G is a finite group and cases (i) and (ii) do not hold; (iv) G = SL(2, C). Proof. It is an easy exercise using the two lemmas above.



Proposition 1.1.6 ([24]). Any algebraic subgroup G of SL(2, C) is conjugate to one of the following types:   1 0 0 (i) finite, G = {1}, where 1 = ; 0 1    1 μ  (ii) G = G0 =  μ∈C ; 0 1    λ μ  (iii) Gk =  λ is a k-th root of unity, μ ∈ C , and −1   0 λ   1 μ  G0 =  μ∈C ; 0 1

4

Chapter 1. Integrable Systems and Differential Galois Theory 

  0  ∗ (iv) G = G =  λ∈C ; λ−1      0 −β −1  λ 0 ∗ , (v) G =  λ, β ∈ C , and −1 0  0 λ  β  λ 0  G0 =  λ ∈ C∗ ; 0 λ−1    λ μ  0 ∗ (vi) G = G =  λ∈C , μ∈C ; 0 λ−1 0

λ 0

(vii) G = G0 = SL(2, C). Proof. Assume G to be infinite and conjugated to a triangular group, i.e., contained in the total triangular group (isomorphic to the semidirect product of the additive group C and the multiplicative group C∗ ),    λ μ  ∗ λ ∈ C , μ ∈ C .  0 λ−1 Let ψ be the morphism of algebraic groups ψ : G → C∗ , defined by   λ μ ψ = λ. 0 λ−1 If ker ψ is trivial, then G must be the diagonal group    λ 0  ∗ λ ∈ C G= ,  0 λ−1 because then G is isomorphic to ψ(G), ψ(G) being an algebraic subgroup of the multiplicative group C∗ . But then ψ(G) must be equal to C∗ (the only possible non-trivial subgroups of C∗ are the cyclic finite groups). If ker ψ is non-trivial, then, as it is (isomorphic to) an algebraic subgroup of the additive group C, it is the total unipotent group    1 μ |μ∈C . 0 1 Now, as above, we have two possibilities: either ψ(G) is equal to the multiplicative group C∗ , or it is a finite cyclic group. The proposition follows from the two lemmas above.  We remark that the identity component G0 is commutative in the cases (i)–(v) and solvable in the cases (i)–(vi). The group G in the case (vi) is the maximal solvable algebraic subgroup of SL(2, C) and it is called the Borel subgroup of SL(2, C). The group G in the case (v) is called the infinite dihedral subgroup of SL(2, C).

1.1. Differential Galois Theory

5

1.1.2 Picard–Vessiot Theory Since for applications we will mainly be concerned with second-order differential equations, we specialize the theory to this case. The definitions for general ndimensional linear differential systems are straightforward from the above, and the results are also valid in this more general context. We say that (K,  ) —or, simply, K— is a differential field if K is a commutative field of characteristic zero and  is a derivation on K, i.e., (a + b) = a + b and (a · b) = a · b + a · b for all a, b ∈ K. We denote by C the field of constants of K, defined as C = {c ∈ K | c = 0}. From now on, we will restrict ourselves to the following families of differential fields: the field of meromorphic functions over some domain of the complex plane with the usual standard derivative or, more generally, the field of meromorphic functions over a Riemann surface. The field of constants is the complex field. As a concrete important example, we can consider the rational functions C(x), i.e., the field of meromorphic functions over the Riemann sphere. An extension of differential fields is an extension of fields such that the derivation is also extended to the bigger field. We will deal with second-order linear homogeneous differential equations, that is, equations of the form y  + py  + qy = 0,

p, q ∈ K,

(1.1)

and we will be concerned with the algebraic structure of their solutions. Let us suppose that y1 , y2 is a basis of solutions of equation (1.1), i.e., y1 , y2 are linearly independent over C and every solution is a linear combination over C of them. Let L = Ky1 , y2  = K(y1 , y2 , y1 , y2 ) be the differential extension of K such that C is the field of constants for K and L. Then we say that L, the smallest differential field containing K and {y1 , y2 }, is the Picard–Vessiot extension of K for the differential equation (1.1). Example. Two independent solutions of the equation y  +

1 =0 4x2

(1.2)

√ √ are y1 = x and y2 = x ln x. Then, considered over K = C(x), the above equation has a Picard–Vessiot extension √ √ C(x) ⊂ C(x)(y1 , y2 , y1 , y2 ) = C(x)( x, ln x) = C(x, x, ln x). We remark that equation (1.2) is a member of the well-known Cauchy–Euler family of equations. The group of all the differential automorphisms of L over K commuting with the derivation  is called the Galois group of L over K (or the Galois group of equation (1.1)) and is denoted by Gal(L/K). This means, in particular, that σ(a ) = (σ(a)) for all σ ∈ Gal(L/K) and a ∈ L, and that σ(a) = a for all a ∈ K.

6

Chapter 1. Integrable Systems and Differential Galois Theory

Thus, if {y1 , y2 } is a fundamental system of solutions of (1.1) and σ ∈ Gal(L/K), then {σy1 , σy2 } is also a fundamental system. This implies the existence of a nonsingular constant matrix   a b ∈ GL(2, C), Aσ = c d       such that σ y1 y2 = σ(y1 ) σ(y2 ) = y1 y2 Aσ . This fact can be extended in a natural way to a system       y1 y2 σ(y1 ) σ(y2 ) y y2 = = Aσ , σ 1 y1 y2 y1 y2 σ(y1 ) σ(y2 ) which leads to a faithful representation Gal(L/K) → GL(2, C), and makes possible to consider Gal(L/K) as a subgroup of GL(2, C) depending (up to conjugacy) on the choice of the fundamental system {y1 , y2 }. One of the fundamental results of the Picard–Vessiot Theory is the following. Theorem 1.1.7. The Galois group Gal(L/K) is an algebraic subgroup of GL(2, C). A differential extension K ⊂ L is normal if, any element b ∈ L invariant under Gal(L/K) (i.e., such that σ(b) = b for all σ ∈ Gal(L/K)) is, in fact, in K. Proposition 1.1.8. The Picard–Vessiot extension L = Ky1 , y2  = K(y1 , y2 , y1 , y2 ) is normal. Exercise. Prove that the Galois group of equation (1.1) is contained in SL(2, C) if and only if p = a /a, for some a ∈ K. Solution. Let



y Δ := det 1 y1

y2 y2

 ∈ L.

Then the Galois group acts on Δ by σ(Δ) = Δ det(Aσ ). Now, det(Aσ ) = 1 if and only if Δ ∈ K (one direction is obvious, for the other use normality). Then, by Liouville’s formula, p = −Δ /Δ = a /a, with a = 1/Δ. We say that equation (1.1) is integrable if the Picard–Vessiot extension L ⊃ K is obtained as a tower of differential fields K = L0 ⊂ L1 ⊂ · · · ⊂ Lm = L such that Li = Li−1 (η) for i = 1, . . . , m, where either (i) η is algebraic over Li−1 , that is, η satisfies a polynomial equation with coefficients in Li−1 ; or (ii) η is primitive over Li−1 , that is η  ∈ Li−1 ; or (iii) η is exponential over Li−1 , that is, η  /η ∈ Li−1 . In Differential Algebra the extensions L ⊃ K satisfying (i)–(iii) above are called Liouvillian extensions. Moreover, the following theorem holds.

1.1. Differential Galois Theory

7

Theorem 1.1.9 (Kolchin). Equation (1.1) is integrable if and only if the identity component of the Galois group (Gal(L/K))0 is solvable. Example. The calculation of the Galois group √G of equation (1.2) is easy. The Picard–Vessiot extension is given by K/L = K( x, ln x)/K. Then, denoting α1 = √ x and α2 = ln x, we have α12 ∈ K and, as in the classical Galois theory of polynomials, σ(α1 ) = λα1 , with λ2 = 1, for σ ∈ G. On the other hand, from α2 ∈ K we obtain (σ(α2 ) − α2 ) = 0, and hence σ(α2 ) = α2 + ν, for ν ∈ C. The linear action of the Galois group G on the fundamental system of solutions (y1 y2 ) = (α1 α1 α2 ) is           λ μ σ y1 y2 = σ(y1 ) σ(y2 ) = λy1 λy2 + λνy1 = y1 y2 , 0 λ−1 with λ2 = 1 and μ ∈ C. Thus, the Galois group of equation (1.2) is    λ μ  2 λ G= = 1, μ ∈ C ,  0 λ−1 i.e., the group G2 in Proposition 1.1.6. Of course, as the equation is integrable, the identity component of the Galois group is solvable, isomorphic in this case to the additive group (C, +). It is well known that any second-order differential equation can be transformed into a general Riccati equation through a classical logarithmic change of variable. The following proposition recalls this and summarizes some other related transformations. Proposition 1.1.10. Let K be a differential field and consider functions a0 (x), a1 (x), a2 (x), r(x), q(x), p(x) belonging to K (that, for simplicity, will be denoted without showing their explicit dependence on x). Consider now the following forms associated to any second-order differential equation (ode) and Riccati equation: (i) Second-order ode (in general form): y  + py  + qy = 0.

(1.3)

(ii) Second-order ode (in reduced form): ξ  = rξ.

(1.4)

(iii) Riccati equation (in general form): v  = a0 + a1 v + a2 v 2 ,

a2 = 0.

(1.5)

(iv) Riccati equation (in reduced form): w = r − w2 .

(1.6)

8

Chapter 1. Integrable Systems and Differential Galois Theory

Then, there exist transformations T , B, S and R mapping these equations into one another, as shown in the following diagram: v  = a0 + a1 v + a2 v 2

T

/ w = r − w2

S

 / ξ  = rξ.

B

 y  + py  + qy = 0

R

The new independent variables are defined by means of    a2 a1 1 y 1 , T : v=− + w, B : v = − − 2a22 2a2 a2 a2 y 1

pdx , S : y = ξ exp − 2

R: w=

ξ , ξ

and the functions r, q and p are given by  1 r= a0 + a1 α + a2 α 2 − α  , β    a2 a1 1 α=− + , β=− , 2 2a 2a2 a2  2  a2 p = − a1 + , q = a0 a2 , a2 p p2 + − q. r= 4 2 Proof. The proof is quite standard. [T ]: Applying the change v = α + βw we get the equation α + β  w + βw = a0 + a1 α + a1 βw + a2 α2 + 2a2 αβw + a2 β 2 w2 , which upon regrouping terms leads to

   1 β 2  a0 + a1 α + a2 α − α + a1 + 2a2 α − w = w + a2 βw2 . β β 

Since a2 = 0, we can take β = −1/a2 and, therefore, a2 β = −1. Taking this into account, the value of α such that the coefficient in w vanishes is given by    β 1 − a1 . α= 2a2 β The expressions for α, β, and r follow straightforwardly:     a2 a1 1 1 a0 + a1 α + a2 α 2 − α  , α = − + , β=− . r= β 2a22 2a2 a2 Moreover, it is clear that α, β, and r belong to K.

1.1. Differential Galois Theory

9

[B]: Imposing α = 0 and taking β = −1/a2 in the transformation T , we have v = −w/a2 and we obtain the Riccati equation   a w = −a0 a2 + a1 + 2 w − w2 . a2 Performing now the change of variables w = (log y) (or, equivalently, v = −a2 y  /y) we obtain the differential equation y  + py  + qy = 0, with 

a p = − a1 + 2 a2

 ,

q = a0 a2 .

Obviously, q and p belong to K. [S]: The change of variable y = μξ, with μ = μ(x) and ξ = ξ(x), leads us to     μ μ μ  + q + p ξ = 0. ξ + 2 +q ξ + μ μ μ 



In order to obtain the equation ξ  = rξ we need to impose 2

μ + q = 0, μ

μ μ + q + p = −r, μ μ

which gives rise to 1 μ = exp − q , 2

r=

q2 b + 0 − p. 4 2

Moreover, it is straightforward to check that r ∈ K. [R]: This is a particular case of transformation [B] with the particular choice a0 = r, a1 = 0 and a2 = −1. Finally, composing the transformations provided by [B], [R] and [S], y −a2 v = , y



1 y = ξ exp − 2



q ,



a q = − a1 + 2 a2

 ,

ξ = w, ξ

we recover the result given by [T ],  v=−

a1 a + 22 2a2 2a2

 −

1 w = α + βw, a2

which implies that, in some sense and taking ρ = r, the diagram commutes: S ◦B =R◦T. 

10

Chapter 1. Integrable Systems and Differential Galois Theory

From this proposition, it follows that the function v is algebraic over K if and only if the function w is also algebraic over K. Furthermore, in such case, the degrees over K of the two functions v and w coincide. We recall now the d’Alembert reduction: given a particular solution ξ1 (x) of ξ  = rξ, we can obtain a second independent solution by

dx ξ2 = ξ1 . (1.7) ξ12 Exercise. Prove that if the equation w = r − w2 has an algebraic solution over the field K, then the equation ξ  = rξ is integrable. Solution. Let w1 be the algebraic solution. Then ξ1 := exp( w1 dx) is a solution of the linear equation. The other solution, ξ2 , is given by d’Alembert reduction, ξ1 and ξ2 define Liouvillian extensions from K, and the Picard–Vessiot extension K ⊂ K(ξ1 , ξ2 , ξ1 , ξ2 ) is also Liouvillian. In fact, this sufficient condition for integrability is also necessary. Theorem 1.1.11 (Liouville [15]). The equation (1.1) is integrable if and only if the associated Riccati equation has an algebraic solution over the field K.

1.1.3 Kovacic Algorithm The Kovacic algorithm gives us a procedure to compute the Picard–Vessiot extension (i.e., a fundamental system of solutions) of a second-order differential equation, provided the equation is integrable. Conversely, if the differential equation is non-integrable, the algorithm does not work (see [13]). Given a second-order linear differential equation with coefficients in C(x), it is a classical fact that it can be transformed into the so-called reduced invariant form (1.8) ξ  − rξ = 0, with r = r(x) ∈ C(x). We remark that in this change we introduce the exponentiation of an integral and the integrability of the original equation is equivalent to the integrability of the equation (1.8) (check this!), although, in general, the Galois groups are not the same. The algorithm uses the following two general ingredients: (i) the classification of the algebraic subgroups of SL(2, C) given in Proposition 1.1.5 (the Galois group of the equation (1.8) is contained in SL(2, C)); and (ii) the transformation to a Riccati equation, by the change w = ξ  /ξ: w = r − w2 .

(1.9)

1.1. Differential Galois Theory

11

Then the differential equation (1.8) is integrable if and only if the equation (1.9) has an algebraic solution. The key point now is that the degree n of the associated (minimal) polynomial, which defines the algebraic curve Q(v, x) = v n + a1 (x)v n−1 + · · · = 0, with coefficients in C(x), belongs to the set

 Lmax = 1, 2, 4, 6, 12 ; n = 1 corresponds to case (i) in Proposition 1.1.5, n = 2 to case (ii), and the rest of values of n to case (iii). Only for cases (i), (ii), (iii) the equation is integrable, case (iv) corresponds to non-integrability. The Kovacic algorithm may possibly provide one solution, ξ1 , so the second one, ξ2 , can be obtained via

dx . (1.10) ξ2 = ξ1 ξ12 We will use the following notation. Given ξ  = ρξ = rξ, r = s/t, with s, t ∈ C[x], (1) we denote by Γ the set of (finite) poles of r, Γ = {c ∈ C : t(c) = 0}; (2) we write Γ = Γ ∪ {∞}; (3) by the order of r at a finite singular point c ∈ Γ , denoted ◦(rc ), we mean the multiplicity of c as a pole of r; (4) by the order of r at ∞, denoted ◦ (r∞ ), we mean the order of ∞ as a zero of r, i.e., ◦ (r∞ ) = deg(t) − deg(s). Let us consider the following four cases. √ √ √ Case 1. In this case [√r]c and [ r]∞ denote the Laurent series of r at c and the Laurent series of r at ∞, respectively. Furthermore, for p ∈ Γ, we put + − , α∞ will be defined in ε (p) ∈ {+, −}. Finally, the complex numbers αc+ , αc− , α∞ the first step. The present case includes, in particular, the differential equations without poles in the finite complex plane. Step 1. Examine for each c ∈ Γ and for ∞ the corresponding situation as follows: √ (c0 ) if ◦ (rc ) = 0, then [ r]c = 0 and αc± = 0; √ (c1 ) if ◦ (rc ) = 1, then [ r]c = 0 and αc± = 1; √ (c2 ) if ◦ (r√c ) = 2 and r = · · · + b(x − c)−2 + · · · , then [ r]c = 0 and αc± = (1 ± 1 + 4b)/2; −v

−2

(c3 ) if ◦ (rc ) = 2v ≥ 4 and r = (a (x − c) + · · · + d (x − c) )2 +b(x − c)−(v+1) +  √ −v −2 · · · , then [ r]c = a (x − c) + · · · + d (x − c) and αc± = ± ab + v /2; √ + − (∞1 ) if ◦ (r∞ ) > 2, then [ r]∞ = 0, α∞ = 0, and α∞ = 1;

12

Chapter 1. Integrable Systems and Differential Galois Theory

√ ± (∞2 ) √ if ◦ (r∞ ) = 2 and r = · · · + bx2 + · · · , then [ r]∞ = 0 and α∞ = (1 ± 1 + 4b)/2; √ 2 (∞3 ) if ◦ (r∞ ) = −2v ≤ 0 andr = (axv + · · · + d) + bxv−1 + · · · , then [ r]∞ = ± = ± ab − v /2. axv + · · · + d and α∞ Step 2. Find

    ε(∞) D = m ∈ Z+  m = α∞ − αcε(c) ,

 ∀ (ε (p))p∈Γ .

c∈Γ

If D = ∅, then we should start with the Case 2. Now, if #D > 0, then for each m ∈ D we search for ω ∈ C(x) such that  √  √  ω = ε (∞) r ∞ + ε (c) r c + αcε(c) (x − c)−1 . c∈Γ

Step 3. For each m ∈ D, search for a monic polynomial Pm of degree m with   + 2ωPm + (ω  + ω 2 − r)Pm = 0. If one succeeds, then ξ1 = Pm exp( ω) is a Pm solution to the differential equation. Else, Case 1 cannot hold. Case 2. Examine for each c ∈ Γ and for ∞ the corresponding situation as follows. Step 1. Examine for each c ∈ Γ and ∞ the sets Ec = ∅ and E∞ = ∅. For each of them, we define Ec ⊂ Z and E∞ ⊂ Z as follows: (c1 ) if ◦ (rc ) = 1, then Ec = {4}; (c2 ) if ◦ (rc ) = 2 and r = · · · + b(x − c)−2 + · · · , then Ec = k = 0, ±2};

√ 2 + k 1 + 4b |

(c3 ) if ◦ (rc ) = v > 2, then Ec = {v}; (∞1 ) if ◦ (r∞ ) > 2, then E∞ = {0, 2, 4};



 (∞2 ) if ◦ (r∞ ) = 2 and r = · · · + bx2 + · · · , then E∞ = 2 + k 1 + 4b | k = 0, ±2 ;

(∞3 ) if ◦ (r∞ ) = v < 2, then E∞ = {v}. Step 2. Find    1  ec , D = m ∈ Z+  m = e∞ − 2 

 ∀ep ∈ Ep , p ∈ Γ .

c∈Γ

If D = ∅, then we should start the Case 3. Now, if #D > 0, then for each m ∈ D we search a rational function θ defined by 1  ec . θ= 2 x−c  c∈Γ

Step 3. For each m ∈ D, search for a monic polynomial Pm of degree  m, such     + 3θPm + (3θ + 3θ2 − 4r)Pm + θ + 3θθ + θ3 − 4rθ − 2r Pm = 0. If that Pm Pm does not exist, then Case 2 cannot hold. If such a polynomial  is found, set φ = θ + P  /P and let ω be a solution of ω 2 + φω + φ + φ2 − 2r /2 = 0. Then, ξ1 = exp( ω) is a solution.

1.1. Differential Galois Theory

13

Case 3. Examine for each c ∈ Γ and for ∞ the corresponding situation as follows. Step 1. Examine for each c ∈ Γ and ∞ the sets Ec = ∅ and E∞ = ∅. For each of them we define Ec ⊂ Z and E∞ ⊂ Z as follows: (c1 ) if ◦ (rc ) = 1, then Ec = {12};



(c2 ) if ◦ (rc ) = 2 and r = · · · + b(x − c)−2 + · · · , then Ec = 6 + k 1 + 4b | k = 0, ±1, ±2, ±3, ±4, ±5, ±6}; √

(∞) if ◦ (r∞ ) = v ≥ 2 and r = · · · + bx2 + · · · , then E∞ = 6 + 12k 1 + 4b/n | k = 0, ±1, ±2, ±3, ±4, ±5, ±6}, n ∈ {4, 6, 12}.   12k √ 1 + 4b : k = 0, ±1, ±2, ±3, ±4, ±5, ±6 , n ∈ {4, 6, 12}. E∞ = 6 + n Step 2. Find    n  ec , D = m ∈ Z+  m = e∞ − 12 

 ∀ep ∈ Ep , p ∈ Γ .

c∈Γ

In this case we start with n = 4 to obtain the solution, afterwards n = 6, and finally n = 12. If D = ∅, then the differential equation has no Liouvillian solution because it belongs to the Case 4. Now, if #D > 0, then for each m ∈ D with its respective n, search for a rational function n  ec , θ= 12 x−c  c∈Γ

and a polynomial S defined as S =

 c∈Γ

(x − c).

Step 3. Search for each m ∈ D, with its respective n, for a monic polynomial Pm = P of degree m, such that its coefficients can be determined recursively by P−1 = 0, Pn = −P , and Pi−1 = −SPi − ((n − i) S  − Sθ) Pi − (n − i) (i + 1) S 2 rPi+1 , where i ∈ {0, 1 . . . , n − 1, n}. If P does not exist, then the differential equation has no Liouvillian solution (i.e., it is not integrable) and it belongs to the Case 4. Now, if such P exists, search for ω such that n  SiP ω i = 0, (n − i)! i=0

and a solution of the differential equation is given by ξ = exp( ω), where ω is a solution of the previous polynomial of degree n. Corollary 1.1.12 (Necessary conditions). The following conditions are necessary for the respective cases to hold:

14

Chapter 1. Integrable Systems and Differential Galois Theory

Case 1: For any pole of r, c ∈ Γ , ◦(rc ) must be even or else equal to 1; the order of r at infinity, ◦ (r∞ ) = deg(t) − deg(s), must be even or else greater than 2. Case 2: There exist at least one pole of r, c ∈ Γ , such that ◦(rc ) is either odd and greater than 2, or else equal to 2. Case 3: For any pole of r, c ∈ Γ , ◦(rc ) cannot be greater than 2 and ◦ (r∞ ) must be greater than 1. The condition for Case 3 means that every singular point, including ∞, must be a regular singular point (Fuchs criteria), and the existence of irregular singular points is an obstruction for the general solution to be algebraic, because for irregular singular points we have exponential grow of some solutions. In Kovacic’s paper [13] the necessary condition for Case 3 was supplemented by conditions related with the rationality of the solutions of the initial equation about the singular points but, in what follows, we shall not use these conditions. As a general strategy, before formally applying Kovacic’s algorithm, it is convenient to use Corollary 1.1.12 to possibly discard some cases. Exercise. Prove that if r(x) in (1.8) is a polynomial of odd degree, P2n+1 (x), then the equation (1.8) is not integrable. Solution. The necessary conditions in Corollary 1.1.12 are not satisfied. In fact, in Case 1, ◦ (r∞ ) = −2n − 1 is odd and not greater than 2; in Case 2, P2n+1 has no poles; and in Case 3, ◦ (r∞ ) = −2n − 1 is not greater than 1 (i.e., the point at ∞ is an irregular singular point). Exercise (Weber equation). The goal is to study the integrability of equation (1.8) if r(x) is a polynomial of degree two: (i) prove that we can write r(x) = (ax + d)2 + b; (ii) prove that, by an affine change of variable in x, we obtain the Weber equation   1 2 x − λ ξ = 0, λ ∈ C; (1.11) ξ  − 4 (iii) prove that (1.11) is integrable if and only if λ = 1/2 + n, with n ∈ Z.

1.1.4 Algebrization Here, we only consider a particular case of the algebrization mechanism following [3], without discussing the complex analytical geometrical meaning of the general case and without proofs (see [19, Appendix B] for the general theoretical situation and [1] where the Hamiltonian algebrization was introduced). Given a second-order differential equation in reduced form (without term in the first derivative) d2 y − s(x)y = 0, (1.12) dx2

1.1. Differential Galois Theory

15

where s(x) is not a rational function, we can not apply directly to it the Kovacic algorithm. However, in some cases, by means of a change of variable z = z(x), it is possible to obtain a new equation with rational coefficients dy d2 y + q(z)y = 0. + p(z) 2 dz dz Then, passing to the reduced form d2 ξ − r(z)ξ = 0, dz 2 we can apply the Kovacic algorithm. Definition 1.1.13 (Hamiltonian change of variable). A change of variable z = z(x) is called Hamiltonian if (z(x), z  (x)) is a solution curve of the autonomous classical one-degree-of-freedom Hamiltonian system ∂x z = ∂w H ∂x w = −∂z H, with H = H(z, w) = w2 /2 + V (z) for some suitable V . Assume that we algebrize equation (1.12) through a Hamiltonian change of variables z = z(x), i.e., V ∈ C(z). Then, we consider the coefficient field of equation (1.12) as K0 = C(z, z  (x), z  (x), . . .), but we have the algebraic relation (x (z))2 = 2h − 2V (z), with h = H(z, x (z)) ∈ C so that K0 = C(z, z  (x)) is an algebraic extension of C(z). Theorem 1.1.14 (Hamiltonian algebrization). The differential equation (1.12) is algebrizable through a Hamiltonian change of variable z = z(x) if and only if there exist f = f (z) and α = α(z) such that α /α, f /α ∈ C(z), f (z(x)) = s(x), and α(z) = 2(H − V (z)) = z  (x)2 . Furthermore, the algebraic form of equation (1.12) becomes d2 y 1 α dy f − y = 0, (1.13) + dz 2 2 α dz α and the identity components of the Galois groups of equations (1.12) and (1.13) are the same. Exercise. Algebrize the Mathieu equation d2 y − (a + b cos x)y = 0. dx2

(1.14)

16

Chapter 1. Integrable Systems and Differential Galois Theory

Solution. With the change z = cos x, we get α(z) = 1 − z 2 and (1 − z 2 )

d2 y dy − (a + bz)y = 0, −z 2 dz dz

which is an algebraic form of (1.14). We remark that the algebrization is, in general, not unique. For instance, the change z = eix gives another algebrization of the Mathieu equation (check this out!).

1.1.5 Bessel, Whittaker, and Hypergeometric Equations Exercise. Determine for which values of the parameter ν the Bessel equation x2

d2 y dy + (x2 − ν 2 )y = 0, +x 2 dx dx

(1.15)

with ν ∈ C, is integrable. Solution (Idea). Passing to the reduced form by means of the change

y = ξ exp − 1/2 pdx = x−1/2 ξ, and applying Kovacic’s algorithm, we get ν ∈ 1/2 + Z for integrability. This characterization of the integrability of the Bessel equation was initially obtained by Kolchin [12]. For an analytical proof, see [19]. We remark that the Galois group of equation (1.15) is contained in SL(2, C) (why?). Furthermore, it is a classical fact that the Bessel functions become elementary functions for ν ∈ 1/2 + Z. Now, if in the Bessel equation (1.15) we make the change to the reduced form y = x−1/2 ξ and the change of the independent variable z = 2ix, then we get   1 4ν 2 − 1 d2 ξ + − η = 0. dz 2 4 4z 2 This equation is a particular member (with parameters κ = 0 and μ = ν) of the family of Whittaker equations   1 κ 4μ2 − 1 d2 ξ − + − ξ = 0, (1.16) dz 2 4 z 4z 2 with complex parameters κ and μ. Using Kovacic’s algorithm, after some lengthly computations, one can obtain a complete characterization of the integrability of (1.16), which generalizes the above exercise.

1.1. Differential Galois Theory

17

Theorem 1.1.15 (Martinet and Ramis [18]). The Whittaker differential equation (1.16) is integrable if and only if either κ + μ ∈ 1/2 + N, or κ − μ ∈ 1/2 + N, or −κ + μ ∈ 1/2 + N, or −κ − μ ∈ 1/2 + N. The original proof of Martinet and Ramis was given using analytical tools, such as Stokes multipliers, etc. The hypergeometric (or Riemann) equation is the most general second-order linear differential equation over the Riemann sphere with three regular singular singularities. If we place the singularities at x = 0, 1, ∞, the equation is given by      1 − γ − γ  dy αα 1 − α − α γγ  ββ  − αα γγ  d2 y + + + + + y = 0, dx2 x x−1 dx x2 (x − 1)2 x(x − 1) (1.17) where (α, α ), (γ, γ  ), (β, β  ) are the exponents at the singular points and must satisfy the Fuchs relation α + α + γ + γ  + β + β  = 1. Now, Kimura’s Theorem below provides necessary and sufficient conditions for the integrability of the hypergeometric equation. Let λ = α − α , μ = β − β  and ν = γ − γ  . Theorem 1.1.16 (Kimura [11]). The hypergeometric equation (1.17) is integrable if and only if either (i) at least one of the four numbers λ + μ + ν, −λ + μ + ν, λ − μ + ν, λ + μ − ν is an odd integer; or (ii) the numbers λ or −λ, μ or −μ, and ν or −ν belong (in an arbitrary order) to some of the following fifteen families: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1/2 + l 1/2 + l 2/3 + l 1/2 + l 2/3 + l 1/2 + l 2/5 + l 2/3 + l 1/2 + l 3/5 + l 2/5 + l 2/3 + l 4/5 + l 1/2 + l 3/5 + l

1/2 + m 1/3 + m 1/3 + m 1/3 + m 1/4 + m 1/3 + m 1/3 + m 1/5 + m 2/5 + m 1/3 + m 2/5 + m 1/3 + m 1/5 + m 2/5 + m 2/5 + m

arbitrary complex number 1/3 + q 1/3 + q 1/4 + q 1/4 + q 1/5 + q 1/3 + q 1/5 + q 1/5 + q 1/5 + q 2/5 + q 1/5 + q 1/5 + q 1/3 + q 1/3 + q

l + m + q even l + m + q even l+m+q l+m+q l+m+q l+m+q l+m+q l+m+q l+m+q l+m+q l+m+q

even even even even even even even even even

Here l, m, and q are integers. We remark that it is very difficult to obtain this theorem by a direct application of Kovacic’s algorithm, because the computations are quite involved for Case

18

Chapter 1. Integrable Systems and Differential Galois Theory

3 of the algorithm. This case corresponds to purely algebraic solutions of the hypergeometric equation, and in his proof Kimura used the previous characterization due to Schwarz for the algebraic solutions to the mentioned equation (“Schwarz’s table”). The Whittaker equation (1.16) is one of the general forms of the confluent hypergeometric equation, obtained by confluence of two of the regular singular points (1 and ∞) to obtain an irregular singular point (at ∞) in a suitable way. The success in the application of Kovacic’s algorithm to equation (1.16) comes from the fact that this equation cannot fall into Case 3 of the algorithm, because the necessary condition of Corollary 1.1.12 is not satisfied: there are always exponential terms in some solutions, coming from the behaviour in a neighborhood of the irregular singular point, i.e., the general solution must be transcendent and, hence, the Galois group is not a finite group.

1.2

Non-Integrability of Hamiltonian Systems

Here, we only consider the application of the Picard–Vessiot theory to the nonintegrability of homogeneous potentials. For other non-integrability applications see the book [19], the survey [22] (and references therein), as well as the nice introductory course [4].

1.2.1 Introduction Given a dynamical system, z˙ = X(z),

(1.18)

with a particular integral curve z = φ(t), at the end of the nineteenth century Poincar´e introduced the variational equation (VE ) along z = φ(t), ξ˙ = X  (φ(t))ξ,

(1.19)

as the fundamental tool to study the behavior of (1.18) in a neighborhood of φ(t), see [28]. Equation (1.19) describes the linear part of the flow of (1.18) along z = φ(t). We have the following General Principle. If we assume that the dynamical system (1.18) is “integrable” in any reasonable sense, then it is natural to conjecture that the linearized differential equation (1.19) must also be “integrable”. It seems clear that, in order to convert this principle into a genuine conjecture, it is necessary to clarify what kind of “integrability ” one considers for equations (1.18) and (1.19). As (1.19) is a linear differential equation, it is natural to consider its integrability in the context of the Galois theory of linear differential equations. In order to do that, we need to assume that the field of constants is the complex field.

1.2. Non-Integrability of Hamiltonian Systems

19

Therefore, we have go over to the complex analytical category, i.e., all equations are complex analytic and defined over complex analytic spaces. For complex analytic Hamiltonian systems the General Principle works well and we obtained the following result, which in some sense may be considered as a generalization of a 1982 result by Ziglin (see [32]). The essential idea is to consider in the General Principle not only integrability of the variational equations (characterized by the solvability of the identity component of its Galois group), but also commutativity of the identity component of the Galois group of the variational equations. This is natural because, for integrable Hamiltonian systems, we have an abelian Poisson Lie algebra of first integrals of maximal dimension. Let H be a complex analytical Hamiltonian function defined on a symplectic manifold M of (complex) dimension 2n and let XH be the Hamiltonian system defined by H. In canonical coordinates z = (x1 , . . . , xn , y1 , . . . , yn ), XH it is classically given by ∂H ∂H x˙ i = , y˙ i = − , i = 1, . . . , n. ∂yi ∂xi We recall here the definition of integrability for Hamiltonian systems. One says that XH = (∂H/∂yi , ∂H/∂xi ), i = 1, . . . , n, is completely integrable or Liouville integrable if there are n functions f1 = H, f2 , . . . , fn , such that (i) they are functionally independent, i.e., the 1-forms dfi , i = 1, 2, . . . , n, are ¯ = M; linearly independent on a dense open set U ⊂ M , U (ii) they form an involutive set, i.e., {fi , fj } = 0, for i, j = 1, 2, . . . , n. We recall that, in canonical coordinates, the Poisson bracket has the classical expression n  ∂f ∂g ∂f ∂g − . {f, g} = ∂y ∂x ∂x i i i ∂yi i=1 We remark that, by the item (ii) above, the functions fi , i = 1, . . . , n, are first integrals of XH . It is very important to be precise regarding the degree of regularity of these first integrals. In our text we will assume, unless otherwise stated, that the first integrals are meromorphic. Sometimes, to recall this fact we shall talk about meromorphic (complete) integrability. Now we can write the variational equations along a particular integral curve z = φ(t) of the vector field XH ,  (φ(t))ξ. ξ˙ = XH

(1.20)

Using the linear first integral dH(z(t)) of the variational equation, it is possible to reduce it to the so-called normal variational equation (NVE ) which, in suitable coordinates, can be written as a linear Hamiltonian system η˙ = JS(t)η where, as usual,   0 I J= −I 0

20

Chapter 1. Integrable Systems and Differential Galois Theory

is the standard matrix of the symplectic form of dimension 2(n − 1). More generally, if, including the Hamiltonian, there are m meromorphic first integrals independent over Γ and in involution, we can reduce the number of degrees of freedom of the variational equation (1.20) by m, and obtain the NVE which, in suitable coordinates, can be written as a 2(n − m)-dimensional linear system η˙ = JS(t)η, (1.21) where now J is the matrix of the symplectic form of dimension 2(n − m). For more details about the reduction to the NVE, see [19, 20]). Theorem 1.2.1 ([19, 20]). Assume a complex analytic Hamiltonian system is meromorphically completely integrable in a neighborhood of the integral curve z = φ(t). Then, the identity components of the Galois groups of the variational equations (1.20) and of the normal variational equations (1.21) are commutative. This theorem is today extended to higher-order variational equations and even to some non-Hamiltonian systems using the General Principle (see [5, 22, 23]).

1.2.2 Homogeneous Potentials Consider an n-degrees-of-freedom Hamiltonian system with Hamiltonian H(x, y) = T + V =

   1 2 y1 + · · · + yn2 + V x1 , . . . , xn , 2

(1.22)

where V is a complex homogeneous function of integer degree k, and n ≥ 2. From the homogeneity of V , it follows that there is an invariant plane x = z(t)c, y = z(t)c, ˙ where z = z(t) is a solution of the (scalar) hyperelliptic differential equation z˙ 2 =

 2 1 − zk k

(where we assume k = 0), and c = (c1 , . . . , cn ) is a solution of the equation c = V  (c).

(1.23)

This is our particular solution Γ along which we compute the variational equation VE and the normal variational equation NVE. We shall call these the homothetic solutions of the Hamiltonian system (1.22) and refer to solutions of (1.23) as homothetic points. In most of the works devoted to the integrability of the homogeneous potentials, the solutions of (1.23) are called Darboux points (see [17], for instance); we use the terminology that is standard in Celestial Mechanics.

1.2. Non-Integrability of Hamiltonian Systems

21

The VE along Γ is given by η¨ = −z(t)k−2 V  (c)η in the temporal parametrization. Assume V  (c) is diagonalizable. Due to the symmetry of the Hessian matrix V  (c), it is possible to express the VE as a direct sum of second-order equations η¨i = −z(t)k−2 λi ηi , i = 1, . . . , n, where we keep η for the new variable, λi being the eigenvalues of the matrix V  (c). We call these eigenvalues Yoshida coefficients. One of the above second-order equations is the tangential variational equation, say, the equation corresponding to λn = k − 1. This equation is trivially solvable, whereas the NVE is an equation in the variables ξ := (η1 , . . . , ηn−1 ) := (ξ1 , . . . , ξn−1 ), i.e., ξ¨ = −z(t)k−2 diag(λ1 , . . . , λn−1 )ξ. Now, following Yoshida [31], we consider the change of variable (which happens to be a finite branched covering map) Γ → P1 given by t → x, where x =: z(t)k (here, Γ is the compact hyperelliptic Riemann surface of the hyperelliptic curve w2 = 2(1 − z k )/k; see [19, 21] for the notation and technical details). Thanks to the symmetries of this problem, we obtain as NVE a system of independent hypergeometric differential equations in the new independent variable x:   dξ k − 1 3k − 2 λi d2 ξ − x + ξ = 0, i = 1, 2, . . . , n − 1. (ANVEi ) x(1 − x) 2 + dx k 2k dx 2k Each of these equations (ANVEi ), corresponding to the Yoshida coefficient λi , is part of the system called the algebraic normal variational equation ANVE. In fact, the ANVE splits into a system of n − 1 independent equations (ANVEi ), i = 1, . . . , n − 1. Then, it is clear that the ANVE is integrable if and only if each of the (ANVEi ) is also integrable. That this, the identity component of the Galois group of the ANVE is solvable if and only if each one of the identity components of the Galois group of the (ANVEi ) i = 1, . . . , n − 1, is solvable. As was observed by Yoshida, each one of the above (ANVEi ) is an hypergeometric equation with three regular singular points at x = 0, x = 1 and x = ∞. By (a slight generalization of) Theorem 1.1.14, the identity component of the Galois group of the NVE is the same as the identity component of the Galois group of the ANVE. By adapting Kimura’s table (Theorem 1.1.16) of integrable hypergeometric equations to the new hypothesis, namely, assuming that the Galois differential group of each of the variational equations must have a commutative identity component, we obtain the following result (in fact, for this particular family of hypergeometric equations all the cases of integrability have a commutative identity component). Theorem 1.2.2 ([19, 21]). Let XH be a Hamiltonian system defined by (1.22) and c a homothetic point such that V  (c) is diagonalizable. If XH is meromorphically

22

Chapter 1. Integrable Systems and Differential Galois Theory

completely integrable, then each pair (k, λi ) matches one of the following items (p being an arbitrary integer): k

λ

k

λ

1

k

p + p (p − 1) k2

10

−3

2

2

arbitrary z ∈ C

11

3

3

−2

12

3

13

3

14

3

15

4

16

5

17

5

18

k

4

arbitrary z ∈ C  10 2 1 −5 49 40 − 40 3 + 10p

5

−5

49 40

6

−4

9 8

7

−3

8

−3

25 1 24 − 24 (2 3 25 1 24 − 24 2

9

−3

25 24



1 40

(4 + 10p)  2 − 18 43 + 4p

2

2

+ 6p) 2 + 6p 6 2 1 − 24 5 + 6p

25 24



1 24

1 − 24 +

 12 5

+ 6p

2 2

1 24

(2 + 6p) 3 2 1 1 − 24 + 24 2 + 6p 6 2 1 1 − 24 + 24 5 + 6p  12 2 1 1 − 24 + 24 5 + 6p  2 − 18 + 18 43 + 4p  10 2 9 1 − 40 + 40 3 + 10p 9 − 40 +  1 k−1 2

k

1 40

(4 + 10p)

(1.24)

2

+ p (p + 1) k



Hence, in order to prove the non-integrability of a given Hamiltonian system with a homogeneous potential, we (i) find the homothetic points, solutions ci of the equation c = V  (c); (ii) prove that, for some of the ci ’s in (i), at least one of the eigenvalues of V  (ci ) is not in the table of Theorem 1.2.2. This theorem is a generalization of a necessary condition of integrability obtained by Yoshida using Ziglin’s approach [31], and it is the starting point of several important results, such as the non-integrability of some celestial mechanics problems (three-body-problem, etc.), a complete classification of two-degrees-offreedom polynomial homogeneous potentials of degree 3 and 4 (works by Maciejewski, Przybylska, etc.), . . . ; see [22]. Here, we only illustrate it with some examples proposed by Yoshida, following [19]. Exercise. Consider the collinear three-body problem with a homogeneous potential V (q1 , q2 , q3 ) = |q1 − q2 |k + |q1 − q3 |k + |q2 − q3 |k ,

(1.25)

k ∈ Z. By a reduction to the center of mass this problem is transformed to a two-degrees-of-freedom problem with the potential k  √ k √ V (x1 , x2 ) = 3x1 + x2 + − 3x1 + x2 + (2x2 )k . Find a necessary condition for integrability in terms of the parameter k.

1.3. Integrability of the Schr¨ odinger Equation

23

Solution. For an arbitrary integer k , we obtain a hyperelliptic integral curve with ANVE having Yoshida parameter λ = 3(k − 1)/(1 + 2k−1 ). By applying the theorem we conclude that for k = −2, 1, 2, 4 the system is not integrable. It is well-known that the three-body problem with potential (1.25) is integrable for the four cases k = −2, 1, 2, 4. Thus the problem of the integrability of this family is completed solved. Exercise. The homogeneous H´enon–Heiles Hamiltonian is given by H=

 e 1 2 y1 + y22 + x31 + x1 x22 , 2 3

e ∈ C. A rotation in the configuration space converts (1.26) into √ 2−e e+1 3 1 2 ξ13 + ξ1 ξ22 + ξ , H = (η12 + η22 ) + 2 3(e − 1) 3 e−1 2

(1.26)

(1.27)

if e = 1. Obtain necessary conditions for integrability of the Hamiltonian system defined by (1.26) in terms of the parameter e. ˆ = eˆ − 1, corSolution. The values of the Yoshida’s parameters are λ = 2/e, λ responding to (1.26) and (1.27), respectively. From Theorem 1.2.2, the Yoshida ˆ can take the following six possible values: f1 (p) := p+3p(p− parameters (λ and λ) 1)/2, f2 (p) := 1/3 + 3p(p − 1)/2, f3 (p) := −1/24 + (2 + 6p)2 /24, f4 (p) := −1/24 + (3/2+6p)2 /24, f5 (p) := −1/24+(6/5+6p)2 /24, f6 (p) := −1/24+(12/5+12p)2 /24, p ∈ Z. The integrability of the Hamiltonian (1.26) is only compatible with the idenˆ + 1, where λ and λ ˆ take values in the above six families fi (p), tity 2/λ = e = λ i = 1, . . . , 6. ˆ It is clear that λ+1 ≥ 1. On the other hand, we have 2/f1 (p) ≤ 2, 2/f2 (p) ≤ 6, 2/f3 (p) ≤ 16, 2/f4 (p) ≤ 192/5, 2/f5 (p) ≤ 1200/11, and 2/f6 (p) ≤ 1200/119. Then, we only need to check a finite number of cases and the only ones of these that verify the above identity are e = 1, 2, 6, 16. Thus, for e = 1, 2, 6, 16 the system is not integrable. It is well known that for e = 1, 6, 16 the system (1.26) is integrable. One can also show that it is not integrable for e = 2, but we have to work harder using the higher-order variational equations (see [23]).

1.3

Integrability of the Schr¨ odinger Equation

This section is devoted to the Differential Galois Theory point of view on the non-relativistic stationary Schr¨ odinger equation. The main algorithmic tools used here are the Kovacic algorithm for solving second-order linear differential equations and the algebrization method for obtaining linear differential equations with rational coefficients. In fact, here we only give an introduction to this topic: for

24

Chapter 1. Integrable Systems and Differential Galois Theory

space limitations we do not consider some interesting aspects, like the Darboux transformations, shape invariant potentials, the connections with spectral theory, etc. We partially follow [3], to which we refer for more information.

1.3.1 The Schr¨odinger Equation In classical mechanics the Hamiltonian corresponding to the energy (kinetic plus potential) is given by H=

− → p 2 − + U (→ x ), 2m

→ − p = (p1 , . . . , pn ),

→ − x = (x1 , . . . , xn ),

− − while in quantum mechanics the momentum → p is given by → p = −ı∇, the Hamiltonian operator is the (non-relativistic, stationary) Schr¨odinger operator given by H=−

2 2 − ∇ + U (→ x ). 2m

− The corresponding Schr¨odinger equation is HΨ = EΨ, where → x is the coordinate, the eigenfunction Ψ is the wave function, the eigenvalue E is the energy level, − U (→ x ) is the potential or potential energy, and the solutions of the Schr¨ odinger equation are the states of the particle. We only consider the one-dimensional Schr¨ odinger equation HΨ = EΨ,

H=−

d2 + V (z), dz 2

(1.28)

where z = x (cartesian coordinate) or z = r (radial coordinate), and we normalize the units such that  = 1 and 2m = 1. In this exposition we are using the (in some sense naive) approach of simply thinking of the Schr¨ odinger equation (1.28) as a differential equation with a parameter E and we will focus on the following problem: Problem 1.3.1. For which values of the parameter E ∈ C is the equation (1.28) integrable? We will denote by Λ ⊂ C the set of values of E for which (1.28) is integrable and call Λ the algebraic spectrum of the Schr¨odinger equation (1.28) (see [3] for the motivation of this terminology).

1.3.2 Application of the Picard–Vessiot Theory We study three families of equations: the harmonic oscillator, the Liouville potential, and the Coulomb potential.

1.3. Integrability of the Schr¨ odinger Equation

25

The harmonic oscillator The potential is 1 2 1 1 kx = mω 2 x2 = ω 2 x2 , 2 2 4  k/m, and we normalize to m = 1/2. Then, the where m is the mass, ω = Schr¨odinger equation (1.28) becomes V (x) =

d2 Ψ = dx2



 1 2 2 ω x − E Ψ. 4

(1.29)

We are now in a particular case of an exercise proposed in Subsection 1.1.3, related to the Weber equation (1.11), that we specialized to this situation. Exercise. Transform the equation (1.29) to the Weber equation d2 Ψ = dx2



 1 2 x − λ Ψ. 4

(1.30)

Solution. We use a linear change of variable x → αx, transforming the equation to   1 2 2 d2 Ψ E = ω x − Ψ dx2 4α4 α2 (we do not change the notation for√the dependent and independent variables of the transformed equation). For α = ω we obtain equation (1.30), with λ = E/ω. Now it is very easy to obtain the algebraic spectrum of equation (1.30) (set of values of λ for integrability) and hence the algebraic spectrum for (1.29). The Weber equation (1.30) is integrable if and only if λ = 1/2 + n, for some n ∈ Z. Then, as E = ωλ, the algebraic spectrum of (1.29) is  En =

 1 + n ω, 2

n ∈ Z.

We remark that, for natural n, this gives us the classical discrete spectrum of the harmonic oscillator. The Liouville potential The potential is V (x) = e−2x , so we have the Schr¨odinger equation  d2 Ψ  −2x = e − E Ψ. 2 dx

(1.31)

It is possible to algebrize this equation by converting it into a Bessel equation.

26

Chapter 1. Integrable Systems and Differential Galois Theory

Exercise. Transform equation (1.31) into a Bessel type equation (1.15): z2

d2 Ψ dΨ + (z 2 − ν 2 )Ψ = 0. +z 2 dz dz

Obtain the algebraic spectrum of (1.31). Solution. By means of the change z = e−x , we obtain z2

d2 Ψ dΨ + (E − z 2 )Ψ = 0, +z dz 2 dz

which is “almost” a Bessel equation. In fact, by the change z → iz, we obtain the Bessel equation: d2 Ψ dΨ + (z 2 + E)Ψ = 0. z2 2 + z dz dz Hence, here E = −ν 2 and the algebraic spectrum is En = −(1/2 + n)2 . The Coulomb potential The Schr¨odinger equation can be written as   ( + 1) e2 d2 Ψ e4 + = − − E Ψ, dr2 r2 r 4( + 1)2

 ∈ Z.

(1.32)

This equation is the radial equation (reduced to standard form), obtained when we separate coordinates in the three-dimensional Schr¨odinger equation of the motion of the electron in the hydrogen atom, e is the charge of the electron, and l is a quantum number that comes from the angular part in the process of separation of variables (see any quantum mechanics book, for instance [14]). Exercise. Transform equation (1.32) into a Whittaker type equation (1.16). Solution (Hint). By means of the change  2 −4 ( + 1) E + e4 r, r → +1 one obtains a Whittaker differential equation in which the parameters are given by e2 ( + 1) 1 κ=  , μ=+ . 2 2 −4 ( + 1) E + e4 Applying the Martinet–Ramis Theorem 1.1.15, we can impose ±κ ± μ half integer. Thus the algebraic spectrum is given by En =

e4 λn , 4( + 1)2

1.4. Integrability of Fields on the Plane where

 λn ∈

 1−

+1 +1+n

27

    2  2  +1    n ∈ Z+ ∪ 1 −  n ∈ Z+ . −n

In [3] the algebraic spectrum is obtained by a direct application of Kovacic’s algorithm, which is equivalent in some sense to the proof of the Martinet–Ramis Theorem.

1.4

Integrability of Fields on the Plane

Here, we consider applications of the Picard–Vessiot theory to the integrability of some differential fields on the plane. We partially follow [2].

1.4.1 Riccati Fields Given a differential system in C2 , dx = x˙ = P (x, y), dt

dy = y˙ = Q(x, y), dt

(1.33)

we consider its associated differential vector field X = P (x, y)

∂ ∂ + Q(x, y) . ∂x ∂y

(1.34)

The integral curves of the field X, x = x(t), y = y(t), are solutions of (1.33). By eliminating the time, we obtain the equation of the phase curves (orbits) or, in a more geometrical language, the foliation associated to the field X. This foliation is given by the solutions of the first-order differential equation y =

Q(x, y) dy = . dx P (x, y)

(1.35)

This expression (1.35) is often written as a Pfaff equation Ω = 0,

(1.36)

where Ω = Q(x, y)dx − P (x, y)dy is the corresponding differential 1-form. The connection between integral curves of the vector field X and solutions of Ω = 0 is clear: geometrically, it is given by Ω·X = 0, which means that the vector field X is tangent to the leaves of the foliation (the phase curves) defined by (1.36). On the other hand, dynamically, the general solution to equation (1.36), say H(x, y) = constant, is given by a first integral H of the original vector field X, that is, a non-constant scalar function which remains constant along any of its solutions (x(t), y(t)). Since Ω · X = 0, this is equivalent to X(H) = 0 and to the existence

28

Chapter 1. Integrable Systems and Differential Galois Theory

of a suitable scalar function f such that Ω = f dH. In fact, 1/f is an integrating factor of the field X, or of the 1-form Ω. In these notes we identify the field X with the associated foliation, although from a rigorous point of view they are not exactly the same, but are strongly related as it is illustrated by the following elementary exercise. Exercise. (i) Describe a method to obtain the integral curves of the field X from the phase curves defined by the foliation H(x, y) = constant. Apply it to the integration of the harmonic oscillator mechanical system with field X=y

∂ ∂ − ω2 x , ∂x ∂y

where ω is a parameter. (ii) Prove that the differential form Ω is closed if and only if the field X is Hamiltonian if and only if the divergence of the field X is zero. (iii) Prove that one integrating factor μ can be physically and geometrically interpreted as the density of the stationary flow defined by the field X (hint: the divergence of the field μX must be zero and this gives exactly the continuity equation of a stationary velocity flow of a fluid). We say that a field X is Riccati if the associated foliation is of Riccati type, i.e., Ω = 0 can be reduced to a Riccati equation v  = a0 (x) + a1 (x)v + a3 (x)v 2 ,

(1.37)

where the ai ’s belong to some differential field. Now we know that Riccati equations are related to the Picard–Vessiot theory through the connection with second-order linear differential equations (Proposition 1.1.10), and we say that the equation Ω = 0 (or the field X) is integrable if the equation (1.37) is integrable, i.e., if it has an algebraic solution (equivalent to the Picard–Vessiot integrability of the associated linear second-order equation). In particular, if the coefficients in (1.37) are rational, then, passing to the reduced form, Kovacic’s algorithm gives precisely one algebraic solution (if it exists).

1.4.2 Quadratic Polynomial Fields The study of the integrability of the quadratic polynomial vector field x˙ = a20 x2 + a11 xy + a02 y 2 + a10 x + a01 y + a00 , y˙ = b20 x2 + b11 xy + b02 y 2 + b10 x + b01 y + b00 , with aij , bi,j ∈ C is, in its general form, a difficult task. One of the possible approaches is to deal with the so-called linear-quadratic case, when one of the two components is a polynomial of degree one. In [16] it is proved that its study in the

1.4. Integrability of Fields on the Plane

29

vicinity of a finite equilibrium point (the origin) can be reduced to two families of systems. Using the notation introduced therein, we refer to these families as (S1)-type, x˙ = x, y˙ = εx + λy + b20 x2 + b11 xy + b02 y 2 ,

(S1)

and (S2)-type, x˙ = y, y˙ = εx + λy + b20 x2 + b11 xy + b02 y 2 .

(S2)

In [16], the authors prove that the linear-quadratic systems having a global analytic first integral are those satisfying: (a1 ) b02 = λ = 0, (b1 ) b02 = 0 and λ = −p/q ∈ Q− , in the case of (S1)-type systems, and (a2 ) b20 = b02 = λ = 0 and εb11 = 0, (b2 ) b20 = b11 = λ = 0 and εb02 = 0, (c2 ) b11 = λ = 0 and b20 = 0, for (S2)-type systems. Furthermore, they also provide the explicit form of the corresponding first integrals. It is important to notice that all of them are Liouvillian. Our aim in this example is to show that these results can be recovered using arguments coming from the Galois theory of linear differential equations. We shall only address here the (S1)-type, whose associated foliation is given by the Riccati equation:   λ + b11 x dy b02 2 = (ε + b20 x) + y . (1.38) y+ dx x x By Lemma 1.1.10, this equation can be transformed into the reduced form w = r(x) − w2 , with r(x) =

1 κ 4μ2 − 1 − + , 4 x 4x2

1 κ=  2 b11 − 4b20 b02

 b02 ε +

 b11 (1 − λ) , 2

μ=

λ , 2

provided b211 − 4b20 b02 = 0, and into the form ξ  = r(x)ξ. This equation is a Whittaker equation (1.16) to which one can apply the Martinet–Ramis Theorem 1.1.15. Exercise. Prove that (a1 ) and (b1 ) above are necessary and sufficient conditions for integrability.

30

Chapter 1. Integrable Systems and Differential Galois Theory

Solution. Theorem 1.1.15 asserts that a Whittaker equation is integrable if and only if at least one of the conditions ±κ ± μ ∈ 1/2 + N is verified; or, equivalently, 2 (κ ± μ) ∈ 2Z + 1. In our case one has 2 (κ ± μ) =

2b20 ε + b11 (1 − λ)  ± λ, b211 − 4b20 b02

and conditions (a1 ) and (b1 ) read 2(κ ± μ) = 1 ∈ 2Z + 1 and 2(κ + μ) = (1 + (p/q)) + (−p/q) = 1 ∈ 2Z + 1, respectively.

Bibliography [1] P.B. Acosta-Hum´ anez and D. Bl´ azquez-Sanz, Non-integrability of some Hamiltonians with rational potentials, Discrete Contin. Dyn. Syst. 10 (2008), 265–293. [2] P.B. Acosta-Hum´ anez, J.J. Morales-Ruiz, J.T. L´ azaro, and C. Pantazi, On the integrability of polynomial fields in the plane by means of Picard–Vessiot theory, arXiv:1012.4796. [3] P.B. Acosta-Hum´anez, J.J. Morales-Ruiz, and J.A. Weil, Galoisian approach to integrability of Schr¨ odinger equation, Reports on Mathematical Physics 67 (2011), 305–374. [4] M. Audin, Les Syst`emes Hamiltoniens et leur Int´egrabilit´e, Cours Sp´ecialis´es, Collection SMF 8 Soci´et´e Mathematique de France, Marseille (2001). English translation: Hamiltonian Systems and their Integrability, AMS, Providence, Rhode Island (2008). [5] M. Ayoul and N.T. Zung, Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris 348(23–24) (2010), 1323–1326. [6] F. Beukers, Differential Galois theory, In: “From Number Theory to Physics”, W. Waldschmidt, P. Moussa, J.M. Luck, C. Itzykson Ed., Springer-Verlag, Berlin (1995), 413–439. [7] A. Borel, Linear Algebraic Groups, Springer-Verlag, New York (1991). [8] J.E. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York (1981). [9] I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris (1976). [10] N.M. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), 203–239. [11] T. Kimura, On Riemann’s equations which are solvable by quadratures, Funkcial. Ekvac. 12 (1969), 269–281. [12] E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York (1973). 31

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[13] J.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Computation 2 (1986), 3–43. [14] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Pergamon Press, Oxford (1977). [15] J. Liouville, M´emoire sur l’int´egration d’une classe d’´equations diff´erentielles du second ordre en quantit´es finies explicites, J. de Math´ematiques Pures et Appliqu´ees 1re s´erie, tome 4 (1839), 423–456. [16] J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems, Ergod. Th. and Dynam. Sys. 31 (2011), 245–258. [17] A.J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys. 46 (2005), 062901.1–062901.33. [18] J. Martinet and J.P. Ramis, Th´eorie de Galois diff´erentielle et resommation, In: “Computer Algebra and Differential Equations”, E. Tournier ed. Academic Press, London (1989), 117–214. [19] J.J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkh¨auser, Basel (1999). [20] J.J. Morales-Ruiz and J.P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems, Methods and Applications of Analysis 8 (2001), 33–96. [21] J.J. Morales-Ruiz and J.P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods and Applications of Analysis 8 (2001), 113–120. [22] J.J. Morales-Ruiz and J.P. Ramis, Integrability of dynamical systems through differential Galois theory: a practical guide, In: “Differential algebra, complex analysis and orthogonal polynomials”, (P.B. Acosta-Hum´anez and F. Marcell´ an eds.), Contemp. Math. 509, Amer. Math. Soc., Providence, RI (2010), 143–220. [23] J.J. Morales-Ruiz, J.P. Ramis, and C. Sim´o, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sc. ´ Ecole Norm. Sup. 40 (2007), 845–884. [24] J.J. Morales-Ruiz and C. Sim´ o, Picard–Vessiot theory and Ziglin’s theorem. J. Diff. Equations 107 (1994), 140–162. [25] E. Picard, Sur les groupes de transformation des ´equations diff´erentielles lin´eaires, C. R. Acad. Sci. Paris 96 (1883), 1131–1134. [26] E. Picard, Sur ´equations diff´erentielles et les groupes alg´ebriques des transformation, Ann. Fac. Sci. Univ. de Toulouse (1) 1 (1887), A1–A15. [27] E. Picard, Trait´e d’Analyse, Tome III, Gauthiers-Villars, Paris (1928).

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[28] H. Poincar´e, Les M´ethodes Nouvelles de la M´ecanique C´eleste, Vol. I. Gauthiers-Villars, Paris (1892). [29] M. van der Put and M. Singer, Galois Theory of Linear Differential Equations, Springer, Berlin (2003). [30] M.E. Vessiot, Sur l’int´egration des ´equations diff´erentielles lin´eaires, Ann. ´ Sci. de l’Ecole Norm. Sup. 9(3) (1892), 197–280. [31] H. Yoshida, A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica D 29 (1987), 128–142. [32] S.L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl. 16 (1982), 181–189.

Chapter 2

Singularities of bi-Hamiltonian Systems and Stability Analysis Alexey Bolsinov 1 2.1

Integrable Systems: Singularities and Bifurcations

The main topics in this section will be: Integrable Systems, the Lagrangian fibration and its singularities, the importance of singularities, and the non-degeneracy of singularities and their basic properties.

2.1.1 Integrable Systems, Lagrangian Fibrations and their Singularities We consider a symplectic manifold (M, ω), i.e., a smooth manifold M endowed with a closed non-degenerate differential 2-form ω, dω = 0

and

det(ωij ) = 0.

A Hamiltonian system on M is understood as a system of ordinary differential equations (a dynamical system) of the form x˙ = XH (x) = ω −1 (dH(x))

(2.1)

1 These lecture notes are based on the results of a joint project we have been working on with Anton Izosimov and Andrey Oshemkov for several years. I thank Eva Miranda, Vladimir Matveev, Francisco Presas, and Iskander Taimanov, coordinators of the Research Programme “Geometry and Dynamics of Integrable Systems” for organising this minicourse at Centre de Recerca Matem` atica in September 2013. I thank CRM for hospitality, support and inspiring atmosphere. I am also very grateful to David Dowell, Tam´ as F. G¨ orbe, Sonja Hohloch, Anton Izosimov, and Andriy Panasyuk for very valuable comments and remarks and, especially, to Alexander Motorin for drawing all the figures for these lecture notes.

© Springer International Publishing Switzerland 2016 A. Bolsinov et al., Geometry and Dynamics of Integrable Systems, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-319-33503-2_2

35

36

Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

or, in coordinates, ∂H , ∂xj where H : M → R is a smooth function called the Hamiltonian of the system, and ω ij are the components of the inverse matrix to ω = (ωjk ), i.e., ω ij ωjk = δki . In what follows, we shall mostly discuss a more general situation, namely, Hamiltonian systems on Poisson manifolds, but for the purposes of this section it is more convenient to consider the symplectic case. The passage from “symplectic” to “Poisson” is more or less straightforward. Among all Hamiltonian systems we distinguish a very special and important subclass, the so-called Liouville integrable systems. This is the main subject of the course. Recall that a Hamiltonian system (2.1) is said to be Liouville integrable if there exist n smooth functions f1 , . . . , fn : M → R satisfying three properties: x˙ i = ω ij (x)

(i) f1 , . . . , fn are first integrals of XH (x); (ii) f1 , . . . , fn pairwise commute with respect to the Poisson bracket on M ; (iii) f1 , . . . , fn are independent almost everywhere. These properties immediately imply thatM is fibered by common levels of

integrals, La = f1 (x) = a1 , . . . , fn (x) = an , a = (a1 , . . . , an ) ∈ Rn , each of which is invariant under the flow of XH . The dynamics of the integrable Hamiltonian system on a regular fiber La is described by the classical Liouville–Arnold theorem. Theorem 2.1.1 (Dynamics). Let La be regular, compact, and connected. Then, La is an n-dimensional torus and the dynamics of XH on La is quasi-periodic. In a sufficiently small neighbourhood of La , the fibration by common levels of the integrals is trivial, in particular, all neighboring fibers are tori with quasiperiodic dynamics. Moreover, the structure of this fibration is standard, in the sense that, locally in a neighbourhood of La , the fibration is symplectomorphic to the following canonical model. Let Mreg = T n × Dn be the direct product n of a torus T n and a disc Dn , endowed with the symplectic structure ω = i=1 dsi ∧ dφi , where φ1 , . . . , φn are angle coordinates on T n and s1 , . . . , sn are coordinates on Dn . Then the functions s1 , . . . , sn commute and define the (trivial) fibration of Mreg into tori. Theorem 2.1.2 (Fibration). Let La be regular, compact and connected. Then there exist a neighbourhood U (La ) and a fiberwise symplectomorphism from U (La ) to this canonical model F : U (La ) → Mreg . If La consists of several regular components, then all of them are tori and the conclusion of the Liouville–Arnold theorem (Theorems 2.1.1 and 2.1.2) holds true for each of them separately.

2.1. Integrable Systems: Singularities and Bifurcations

37

By regularity we mean that the differentials df1 (x), . . . , dfn (x) are linearly independent at all points x ∈ La . However, for some fibers Lb the regularity condition may fail; thus, in fact, we deal with a singular Lagrangian fibration whose fibers are, by definition, connected components of common levels of the commuting integrals f1 , . . . , fn . Remark 2.1.3. This fibration is Lagrangian in the sense that all regular fibers La are Lagrangian submanifolds, i.e., the restriction of ω on La vanishes. Notice that every Lagrangian fibration is locally defined by a collection of commuting functions. The main goal of the topology of integrable systems is to describe and classify such fibrations, and their singularities and invariants. From the viewpoint of Singularity Theory, the structure on the phase space M we are interested in, is defined by the smooth map   Φ = f1 , . . . , fn : M −→ Rn , the so-called momentum map. In what follows, we always assume that Φ is proper. The fibers are connected components of preimages Φ−1 (a), a ∈ Rn . Basically, we want to understand the structure of singularities of this map, bifurcations of regular fibers, global invariants, and so on. However, in our setting, there is one quite essential difference compared with the usual Singularity Theory. The point is that the functions f1 , . . . , fn satisfy a very strong additional condition, namely, they Poisson commute. This property dramatically affects the structure of the singularities and of the fibration as a whole. To clarify the structure of the fibration, we introduce the set of critical points 

S = x ∈ M | rank(df1 (x), . . . , dfn (x)) < n and the bifurcation diagram Σ as the image of S under the momentum map, Σ = Φ(S) ⊂ Rn . Example. Consider the Neumann system on S 2 , describing the motion of a point x ∈ S 2 in a quadratic potential. The phase space for this system is the cotangent bundle M 4 = T ∗ S 2 considered as a symplectic manifold with respect to the canonical symplectic form ω = dx1 ∧ dp1 + dx2 ∧ dp2 . In sphero-conical coordinates (x1 , x2 ), the Hamiltonian and first integral of the Neumann system are, respectively, H=

−P (x1 )p21 + P (x2 )p22 + x1 + x2 , x1 − x2

F =

x2 P (x1 )p21 − x1 P (x2 )p22 − x1 x2 , x1 − x2

where P (x) = (x + a)(x + b)(x + c). So, following the general scheme, we can define the momentum map   Φ : T ∗ S 2 −→ R2 , Φ(x, p) = H(x, p), F (x, p) ,

38

Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

and study its bifurcation diagram

 Σ = Φ(x, p) | rank dΦ(x, p) < 2 ⊂ R2 (H, F ), i.e., the image of the set of critical points S. This critical set S can be described by analysing the rank of the Jacobian matrix, namely, ⎛ ∂H ∂H ∂H ∂H ⎞ (x, p) ∈ S

⇐⇒

rank⎝

∂p1

∂p2

∂x1

∂x2

∂F ∂p1

∂F ∂p2

∂F ∂x1

∂F ∂x2

⎠ 0, and nilpotent (ξ, ξ) = 0. Equivalently, one may use the sign of det ξ in the standard representation.

Figure 2.10: sl(2) case: two other possibilities. This example leads us to the following conclusion: Non-degeneracy ⇐⇒ ξ is semisimple.

Example (Non-semisimple case). Consider the Lie ⎧⎛ ⎨ 0 e(2) = so(2) +φ R2 = ⎝−x ⎩ 0 The corresponding Lie–Poisson bracket is ⎛ 0 ξ3 A = ⎝−ξ3 0 0 ξ2 where ξ1 , ξ2 , ξ3 are the dual coordinates ⎛ 0 B = ⎝−b3 b2

algebra ⎞⎫ x y ⎬ 0 z⎠ . ⎭ 0 0

⎞ −ξ2 0 ⎠, 0

to x, y, z. The constant bracket is ⎞ b3 −b2 0 b1 ⎠ . −b1 0

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Take the corresponding linear pencil Πe(2),B . Casimir functions on e(2)∗ are FA = ξ22 + ξ32 and FB = a1 ξ1 + a2 ξ2 + a3 ξ3 . These functions give a non-degenerate singularity if and only if b1 = 0. The algebraic reformulation is: Non-degeneracy ⇐⇒ Ker B is a Cartan subalgebra.

Example (Semisimple Lie algebras of higher dimension). Let g be a semisimple Lie algebra different from so(3) and sl(2); for instance, so(n) or sl(n). As we know, the “argument shift” pencil gives a polynomial integrable system on g. Is the origin 0 ∈ g = g∗ a non-degenerate critical point? For the sake of simplicity, let g = sl(4). The family of polynomial shifts in this case is “produced” from the polynomial invariants of the (co)adjoint representation, which are the functions of the form Tr X 2 , Tr X 3 , Tr X 4 , where X ∈ sl(4). If we replace X by X + λA and expand in powers of λ, we obtain nine homogeneous commuting polynomials whose degrees are 4, 3, 3, 2, 2, 2, 1, 1, 1. These are generators of the algebra FA of polynomial shifts. In other words, each polynomial Casimir of degree m generates m commuting polynomials of degree 1, 2, . . . , m − 1, m. To verify the non-degeneracy condition at the origin, we need to restrict these functions to the symplectic leaf of the constant bracket { , }A (which is defined by three linear equations h1 = 0, h2 = 0, and h3 = 0, where h1 , h2 , h3 are exactly the three linear polynomials from the set of shifts). Obviously, the origin is a critical point for all of them and, therefore, is indeed a common equilibrium. Is this singularity non-degenerate? The answer is no, and the reason is very simple. A necessary (but not sufficient!) condition for non-degeneracy is that the Hessians of the commuting functions must be linearly independent. However, in our case, the set of commuting functions contains two cubic polynomials and one of degree 4. Their Hessians vanish at the origin. Thus, this example leads us to the conjecture that the Casimir functions of the Lie algebras we are interested in (i.e., with non-degenerate singularities) must be at most quadratic. This conjecture is indeed true.

2.3.5 Classification of Non-Degenerate Linear Pencils The following theorem describes all “good” Lie algebras g (equivalently, Lie– Poisson brackets A) which may “produce” non-degenerate linear pencils, and then states a necessary and sufficient condition for a constant bracket B on g∗ to give indeed a non-degenerate pencil Π = Πg,B . Such Lie algebras and linear pencils are called non-degenerate. Theorem 2.3.6 (A. Izosimov [12, 40]). A linear pencil Π = Πg,B is non-degenerate (in the complex case) if and only if the Lie algebra g is isomorphic to + + + C , so(3, C) ⊕ D /h0 ⊕

2.3. Linearisation of Poisson Pencils and a Criterion of Non-Degeneracy

65

where D is the,diamond Lie algebra, h0 is a commutative ideal which belongs to the centre of ( D), and Ker B is a Cartan subalgebra of g. This theorem basically states that all non-degenerate complex Lie algebras can be obtained from very “simple blocks”, namely so(3), D, and Abelian Lie algebras7 , by taking direct sums and quotienting with respect to a sub-centre. What is the diamond Lie algebra D? D is a four dimensional Lie algebra generated by e, f, t, h with the following relations [t, e] = f,

[t, f ] = −e,

[e, f ] = h,

[h, D] = 0.

(2.2)

In other words, D (as a complex Lie algebra) is the non-trivial central extension of e(2, C). The matrix representation is ⎛ ⎞ 0 α β 2γ ⎜0 0 −θ β ⎟ ⎟ αe + βf + θt + γh −→ ⎜ ⎝0 θ 0 −α⎠ 0 0 0 0 and the Casimir functions are F1 = f 2 + e2 + 2 and F2 = h. Before stating the answer in the real case, we make a couple of remarks. First of all, it is the real case that is important in applications. From the dynamical viewpoint, the real case is also much richer. In integrable systems, we can observe elliptic, hyperbolic, and focus singularities in different combinations and the dynamics of a system will essentially depend on the types of its singularities. If g is a real Lie algebra and gC is its complexification, then g and gC are simultaneously degenerate or non-degenerate. This easily follows from the simple observation that the family of shifts for gC can be obtained from that of g by replacing real variables xi by complex variables zi . Since the non-degeneracy condition is in essence algebraic, this “operation” does not change anything. In other words, all real forms of a non-degenerate complex Lie algebra are non-degenerate too. Finally, k complex commuting functions on a complex symplectic manifold of dimension n can be considered as 2k commuting functions on a real symplectic manifold of dimension 2n. It is easy to see that the “non-degeneracy condition” is preserved under this passage from complex to real. Moreover, the real singularity will be of focus type (elliptic and hyperbolic components cannot appear in this way). This observation immediately implies that every non-degenerate n-dimensional complex Lie algebra g treated as a 2n-dimensional real Lie algebra will be non-degenerate in the real sense too. Thus, in the real case, the elementary blocks from which non-degenerate Lie algebras can be built include the real forms of so(3, C) and D, as well as the 7 Recall that the complex Lie algebras so(3, C) and sl(2, C) are isomorphic, so the difference between them will be essential only in the real case.

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algebras so(3, C) and D themselves, treated as real Lie algebras (of dimension 6 and 8, respectively). The orthogonal Lie algebra so(3, C) has two different real forms, namely, so(3, R), and sl(3, R). And the complex diamond Lie algebra D has two different real forms too: gell defined by (2.2), and ghyp defined by [t, e] = e, [t, f ] = −f , and [e, f ] = h. The above remarks clarify the nature of the classification theorem below, but should not be considered as its formal justification. Theorem 2.3.7 (A. Izosimov [12, 40]). A real Lie algebra g is non-degenerate if and only if + + + g so(3, R) ⊕ sl(2, R) ⊕ so(3, C) + + + + ⊕ gell ⊕ ghyp ⊕ gfoc /h0 ⊕ R , where gell and ghyp are the non-trivial central extensions of e(2) and e(1, 1) (equivalently, they are real forms of D), gfoc = D is treated as real Lie algebra, and h0 is a commutative ideal which belongs to the centre. A linear pencil Πg,B is non-degenerate if g is non-degenerate and Ker B is a Cartan subalgebra of g. The type of the singularity is naturally defined by the “number” of elliptic, hyperbolic, and focus components in the above decomposition. The only exception is the Lie algebra sl(2, R) for which, as we have seen in Example 2.3.4, two cases are possible, hyperbolic and elliptic, depending on the sign of the Killing form.

2.3.6 General Non-Degeneracy Criterion Finally, we explain how to verify the non-degeneracy of singularities for bi-Hamiltonian systems in the general case. Let Π = {A + λB} be an arbitrary pencil of compatible Poisson brackets (of Kronecker type). We consider the commutative family of functions FΠ and a singular point x ∈ SΠ . Our goal is to verify the non-degeneracy condition for x. Recall that we assume that almost all Poisson structures Aλ ∈ Π have maximal rank at x. We know already (see Section 2.2) that x is singular for the family of commuting functions FΠ if and only if there is at least one value of the parameter λ such that the rank of the corresponding Poisson structure Aλ = A + λB drops at the point x. In algebraic terms, this means that the pencil Π(x) = {A(x) + λB(x)} of skew-symmetric forms possesses a non-trivial spectrum Λ(x) = {λ1 , . . . , λk } (see Jordan–Kronecker decomposition theorem in Section 2.2). On the other hand, from the geometric viewpoint, we can reformulate this condition by saying that x belongs to the singular sets Sλ1 , . . . , Sλk , where

 Sλi = y ∈ M | rank Aλi (x) < rank Π .

2.3. Linearisation of Poisson Pencils and a Criterion of Non-Degeneracy

67

For each λi ∈ Λ, we can consider the λi -linearisation of the pencil Π. It turns out that these λi -linearisations contain enough information to verify the non-degeneracy condition for x. Theorem 2.3.8 (A. Izosimov [12, 40]). Let Π = {A + λB} be a pencil of compatible Poisson brackets, FΠ be the associated commutative family of functions, and x ∈ M a singular point for FΠ . This point is non-degenerate if and only if the following two conditions hold for every λi ∈ Λ(x): (i) the λi -linearisation of the pencil Π at x is non-degenerate, (ii) the corank of the λi -linearisation equals corank Π. Let us make several comments about Theorem 2.3.8. The second condition can be interpreted in both algebraic and geometric terms. Algebraically, it means that the pencil Π(x) = {A(x) + λB(x)} is diagonalisable in the sense that the Jordan–Kronecker decomposition for Π(x) contains no non-trivial  Jordan blocks,  i.e., all Jordan blocks are 2 × 2 and of the form 0 λi − λ Ai + λBi = (there could be several blocks with the same λi ; in 0 λ − λi other words, elements of the spectrum may have non-trivial multiplicities). Equivalently, one can say that the recursion operator, which can be naturally defined for the Jordan part of this decomposition, is diagonalisable. And from the geometric viewpoint, condition (ii) is equivalent to a very natural property of the (transversal) linearisation of Aλi at the point x. It is easy to see that the rank of the linearised Poisson structure Alinear can, in general, be smaller than that of Atransv . The second condition simply means that after the linearisation, the rank of Aλi does not drop, but remains the same. In other words, the rank is preserved under linearisation. The type of the singularity x ∈ SΠ is just the “sum” of types of the λi linearisations. Recall that each λi -linearisation is represented as a finite-dimensional Lie algebra g with a cocycle B. The type of the singularity for Πg,B is basically defined by the types of components in the decomposition of g into elementary blocks (see Theorem 2.3.7). The only exception is the sl(2)-block, which requires some additional analysis involving B. The spectrum Λ(x) may contain pairs of complex conjugate numbers αj ±iβj . ¯ j -linearisations will be represented by complex Lie algebras and they The λj - and λ will contribute as focus components. As we know, stable singularities are of purely elliptic type. Thus, stability of bi-Hamiltonian systems is, in essence, determined by the algebraic type of the λi linearisations (see Theorem 2.3.7). Again, the only exceptions are sl(2)-components which may “produce” both hyperbolic and elliptic singularities. Theorem 2.3.8 provides a very useful tool which essentially simplifies the analysis of singularities of bi-Hamiltonian systems. Indeed, following the standard scheme we need to (i) choose some basis f1 , . . . , fs in the family of first integrals;

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Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

∂fi (ii) compute the Jacobian matrix J = ∂x in appropriate local coordinates k and find those points where the rank of J drops; and, finally, 2 fi at such points. (iii) analyse the Hessians J = ∂x∂k ∂x j Already for systems with two or three degrees of freedom, this straightforward approach requires very non-trivial (but yet reasonable) analytic computations. For many degrees of freedom, technical difficulties becomes unsurmountable and we need to use some additional hidden structure. That is exactly the main idea of this approach. All the information we need is hidden in the algebraic properties of the underlying bi-Hamiltonian structure and Theorem 2.3.8 explains how to uncover it. Instead of analysing the differentials of integrals, we only need to study the singular sets of the involved Poisson brackets. Since these brackets usually have an algebraic nature, the questions we are interested in can often be treated by purely algebraic means. The next section is devoted to examples showing how to use the above techniques in practice.

To read about • Poisson structure, splitting and linearisation: Weinstein [69] and Dufour and Zung [24]; • Non-degeneracy: Bolsinov and Izosimov [12].

2.4

How Does It Work? Examples and Applications

In the present section, we will develop four examples: the Rubanovskii case in rigid body dynamics, the Mischenko–Fomenko systems on semisimple Lie algebras, the Euler–Manakov top on so(n), and the periodic Toda lattice. Some of the results presented below are well known to experts in the field and were obtained quite a while ago. We use them as an illustration of the biHamiltonian approach, which essentially simplifies the original proofs. We also would like to point out an important feature of the method we are using. Our main goal is to study qualitative properties of a dynamical system (e.g., stability of solutions) as well as the properties of the singular Lagrangian fibration defined by the commuting integrals of the system. It might look strange but, for our analysis, we need neither the equation of motion, nor the explicit formulas for the integrals. The only important information is the Poisson pencil associated with the given system.

2.4.1 Rubanovskii Case This integrable system is a generalisation of the famous Steklov–Lyapunov case of the Kirchhoff equations discovered by V. N. Rubanovskii. The most convenient

2.4. How Does It Work? Examples and Applications

69

and elegant way to describe this system is the Lax pair found by Yu. Fedorov:  dL(λ)  = L(λ), A(λ) , dt where

λ ∈ C,

- λ − bγ (zγ + λpγ ) + gγ λ − bγ ,  1 = εαβγ (λ − bα )(λ − bβ ) (bγ zγ − gγ ), λ

Lαβ (λ) = εαβγ A(λ)αβ

L(λ), A(λ) ∈ so(3),



and εαβγ denotes the Levi–Civita symbol. In this representation, z and p are dynamical variables, and b and g are geometric parameters. The analysis of singularities, based on the bi-Hamiltonian approach, was carried out by I. Basak [6]. The initial point for this analysis is a description of the corresponding pencil of compatible Poisson brackets. Proposition 2.4.1. The Rubanovskii system is Hamiltonian with respect to the pencil defined on R6 (z, p) by the compatible Poisson brackets ⎛ ⎞ 0 b3 z3 − g3 −b2 z2 + g2 0 0 0 ⎜−b3 z3 + g3 0 b 1 z1 − g 1 0 0 0 ⎟ ⎜ ⎟ ⎜ b2 z2 − g2 −b1 z1 + g1 0 0 0 0 ⎟ ⎟ P0 = ⎜ ⎜ 0 0 0 0 p3 −p2 ⎟ ⎜ ⎟ ⎝ 0 0 0 −p3 0 p1 ⎠ 0 0 0 p2 −p1 0 and



P∞

0 ⎜−z3 + b3 p3 ⎜ ⎜ z2 − b 2 p 2 =⎜ ⎜ 0 ⎜ ⎝ −p3 p2

z3 − b 3 p3 0 −z1 + b1 p1 p3 0 −p1

−z2 + b2 p2 z1 − b 1 p 1 0 −p2 p1 0

0 −p3 p2 0 0 0

p3 0 −p1 0 0 0

⎞ −p2 p1 ⎟ ⎟ 0 ⎟ ⎟, 0 ⎟ ⎟ 0 ⎠ 0

where z, p are coordinates in the phase space R6 , and b and g are geometric parameters. The algebraic structure of P0 −λP∞ becomes clear when we apply the change of variables z˜i = zi + λpi + gi /(λ − bi ), with the pi ’s remaining unchanged. After this change of variables, the pencil reads ⎛ ⎞ 0 (b3 − λ)˜ z3 −(b2 − λ)˜ z2 ⎜−(b3 − λ)˜ ⎟ z3 0 (b1 − λ)˜ z1 ⎜ ⎟ ⎜ (b2 − λ)˜ ⎟ z2 −(b1 − λ)˜ z1 0 ⎟. P0 − λP∞ = ⎜ ⎜ 0 p3 −p2 ⎟ ⎜ ⎟ ⎝ −p3 0 p1 ⎠ p2 −p1 0

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Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

Thus, P1 − λP0 splits into the direct sum of two brackets, one of which is the standard so(3)-bracket and the other is isomorphic to either so(3), or sl(2), depending on the signs of bi − λ, i = 1, 2, 3. What are the critical points for the integrals? The answer is universal: those points where the rank of P0 − λP∞ drops. So we obtain Theorem 2.4.2 (Basak I. [6]). A point (z, p) is critical if and only if there is λ ∈ C \ {b1 , b2 , b3 } such that zi + λpi + gi /(λ − bi ) = 0, i = 1, 2, 3. Common equilibrium points are easy to describe too. Theorem 2.4.3 (Basak I. [6]). A point (z, p) is a common equilibrium if and only if ⎛ ⎞ p1 z1 − b1 p1 g1 − b1 z1 rank ⎝p2 z2 − b2 p2 g2 − b2 z2 ⎠ = 1. p 3 z 3 − b 3 p 3 g 3 − b 3 z3 The non-degeneracy condition for corank one singularities is easy to obtain by using the λ-linearisation techniques. Theorem 2.4.4 (Basak I. [6]). Let γ be a critical closed trajectory passing through (z, p) with parameter λ. Then γ is non-degenerate if and only if C = (λ − b1 )(λ − b2 )(λ − b3 )

3   i=1

gi (λ − bi )pi − λ − bi

2

1 = 0. λ − bi

Moreover, if C > 0 then γ is stable, and if C < 0 then γ is unstable. For corank zero singularities (i.e., common equilibria) the result is similar. Notice that all these results are obtained by straightforward application of general theorems discussed in Sections 2.2 and 2.3 to the pencil P0 − λP∞ .

2.4.2 Mischenko–Fomenko Systems on Semisimple Lie Algebras Let g be a finite-dimensional (real) Lie algebra, and g ∗ its dual space, endowed with two Lie–Poisson brackets     {f, g}(x) = x [df (x), dg(x)] and {f, g}a (x) = a [df (x), dg(x)] , where f, g : g∗ → R are arbitrary smooth functions, x, a ∈ g∗ , and a is fixed and regular. This is a typical example of a linear pencil (see Section 2.3). The pencil { , } + λ{ , }a leads to the family of commuting Casimirs

 Fa = f (x + λa) | f is a Casimir of { , }, λ ∈ R . Below, we discuss the properties of Fa for semisimple (and even compact) Lie algebras. In this case, g  g∗ and the family Fa possesses a natural basis

2.4. How Does It Work? Examples and Applications

71

consisting of s = (dim g + ind g)/2 homogeneous polynomials f1 , . . . , fs . In other words, Fa is freely generated by them. Mischenko–Fomenko systems on g can be understood as Hamiltonian systems with quadratic Hamiltonians H ∈ Fa . It can be shown that they are automatically bi-Hamiltonian with respect to the pencil { , } + λ{ , }a . We want to study the properties of the corresponding momentum map Φa : g → Rs , Φa (x) = (f1 (x), . . . , fs (x)), by applying the bi-Hamiltonian approach presented in Sections 2.2 and 2.3. Theorem 2.4.5 (Mischenko and Fomenko [52]). If g is semisimple and a ∈ g∗ is regular, then the collection of commuting polynomials Fa is complete on g  g∗ . In other words, f1 , . . . , fs are functionally independent on g. This theorem is a particular case of the following general result. Theorem 2.4.6 (Bolsinov [7]). In the case of an arbitrary finite-dimensional g, the family Fa is complete, i.e., it contains s = (dim g + ind g)/2 functionally ∗ independent functions, if and only if codim Sing ≥ 2, where Sing ⊂ gC is the set of singular elements. The proof is obvious. We just need to apply the codimension two principle and use, in the semisimple case, the well-known algebraic fact that codim Sing = 3 for all semisimple Lie algebras. Theorem 2.4.7 (Bolsinov [7]). An element x ∈ g is a critical point of the momentum map Φa if and only if there exists λ ∈ C such that x + λa is a singular element in gC . In other words, the set of critical points Sa of Φa is (the real part of ) the cylinder over the set of singular elements Sing, with the generating line parallel to a; that is, Sa = (Sing + C · a) ∩ g. Recall that x ∈ g is said to be a common equilibrium point for Fa if, for all f ∈ Fa , we have Xf (x) = [df (x), x] = 0. Theorem 2.4.8 (Bolsinov and Oshemkov [17]). A point x ∈ g is a common equilibrium point for Fa if and only if x ∈ ha , where ha is the Cartan subalgebra generated by a ∈ g. The number of equilibrium points on each regular orbit is the order of the Weil group. Let x ∈ ha be a common equilibrium point. Is x non-degenerate? Theorem 2.4.9 (Sufficient condition [17]). Let α1 , . . . , αs be the positive roots asC sociated with the complexification hC a ⊂ g , s = (dim g + ind g)/2. Consider the collection of numbers λi = αi (x)/αi (a). If all these numbers are distinct, then x ∈ g is a non-degenerate equilibrium point. Moreover, if g is compact, then x is of purely elliptic type and, therefore, is stable. Proof. The proof easily follows from the general non-degeneracy criterion. The only thing we need to do is to analyse the λi -linearisation of the pencil at the point x ∈ ha , for each λi from the spectrum Λ(x). Recall that λ ∈ Λ(x) if and only

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Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

if x+λa is a singular point. Since x, a both belong to the Cartan subalgebra ha , the element x + λa becomes singular if for some root αi we have αi (x + λa) = 0, i.e., λ = −λi = −αi (x)/αi (a). The −λi -linearisation in this case is easy to describe. The Lie algebra g−λi is the centraliser of x − λi a,

 g−λi = ξ ∈ g | [ξ, x − λi a] = 0 , and the 2-form on this centraliser is defined by the same element a ∈ g−λi . If all λi are distinct, then αi is the only root that vanishes at x + λi a and, therefore, g−λi = span {ha , eαi , e−αi }, i.e., is isomorphic to either so(3) + Rr−1 , or sl(2) + Rr−1 , or so(3, C) + Cr−1 (if λi ∈ C) and the kernel of the 2-form is still the same Cartan subalgebra ha . Thus, this linearisation is obviously non-degenerate  for each λi and, hence, x is a non-degenerate singular point. What about corank 1 singularities? Let x ∈ g be a critical point of corank 1 of the momentum map Φa . Then there is a unique value of the parameter λ ∈ R such that x + λa is a singular element of g, and the centraliser gλ of x + λa has dimension ind g + 2. If we assume that x + λa is a semisimple element (this situation is generic), then the centraliser gλ of x + λa is isomorphic to the direct sum u ⊕ Rr−1 , where u is either so(3), or sl(2). According to the general scheme, we now should take the restriction of the a-bracket to gλ and verify if the kernel of this restriction is a Cartan subalgebra. As we already discussed, this restriction is defined by the natural projection of a onto g∗λ . Since the centre Rr−1 of gλ does not play any role in this construction, we come to the following conclusion. Theorem 2.4.10. Let x ∈ g be a critical point of corank 1 of the momentum map Φa , and let λ ∈ R be the unique value of the parameter such that x + λa is a singular element of g. Assume that x + λa is semisimple and u is the semisimple part of the centralizer of x + λa, and consider the natural orthogonal projection b = pru a of a onto u. Then, x is non-degenerate if and only if b ∈ u is semisimple and non-zero. Moreover, if (b, b) > 0 then the singularity is hyperbolic, and if (b, b) < 0 then the singularity is elliptic, where ( , ) is the Killing form on u. In particular, in the case of a compact Lie algebra g, all corank 1 singularities are non-degenerate and of elliptic type. In this case, there are no hyperbolic singularities. It follows from this that the set of regular values of Φa in Rs is connected s and each non-trivial regular level Φ−1 a (y), y ∈ R , consists of a Liouville torus. Remark 2.4.11. The assumption that x + λa is semisimple is not necessary for the non-degeneracy of x. For some non-semisimple elements x + λa, the centraliser gx+λa may have the form D ⊕ centre, which is still a non-degenerate Lie algebra.

2.4. How Does It Work? Examples and Applications

73

2.4.3 Euler–Manakov Tops on so(n) The Euler–Manakov top on so(n) is an n-dimensional generalisation of the classical Euler equations in rigid body dynamics. The Euler–Manakov equations are   d X = R(X), X , X ∈ so(n), dt where R : so(n) → so(n) is a linear operator which, in terms of matrix coefficients, takes the form R(X)ij = (bi − bj )/(ai − aj )Xij . As was shown by S. V. Manakov, this system admits a family of commuting integrals of the form Tr(X + λA)k , where A = diag(a1 , . . . , an ) is the diagonal matrix with diagonal elements ai . The fact that this family is complete (i.e., sufficient for Liouville integrability) was proved by Mischenko and Fomenko. Our aim is to study the singularities of this integrable system. First of all, we need to describe the bi-Hamiltonian structure for the Euler– Manakov top. Along with the standard commutator [X, Y ] = XY − Y X on the space of skew-symmetric matrices, we introduce a new operation [X, Y ]A = XAY − Y AX, where A is a symmetric matrix. It is easy to see that these two brackets are compatible in the sense that their linear combination [ , ]A + λ[ , ] = [ , ]A+λE satisfies the Jacobi identity. Hence, on the dual space so(n)∗ , we obtain a pencil of compatible linear Poisson brackets. It turns out that the Euler–Manakov top is Hamiltonian with respect to this k  pencil. The Casimirs of the bracket { , }A+λE are of the form Tr X(A + λE)−1 and it is not difficult  these integrals are equivalent to those found by

to verify that k . This family admits a basis consisting of exactly = Tr(X + λA) Manakov, F A   s = dim so(n) + ind so(n) /2 commuting polynomials. Theorem 2.4.12 (Mischenko and Fomenko [52]). If the eigenvalues of A are all distinct, then the family of Manakov’s integrals FA is complete on so(n). Here is a bi-Hamiltonian version of the proof. Proof. Almost all brackets in the family are semisimple, so the singular set of each of them has codimension 3. The corank of those brackets which are not semisimple coincides with the corank of the pencil. Thus, the statement immediately follows from the codimension two principle (see Section 2.2).  Theorem 2.4.13. X ∈ so(n) is a critical point of FA if and only if there exists λ ∈ C such that X is singular for the bracket { , }A+λE . Equivalently, "    n "  1/2 1/2 SA = S−ai ∪ (A + λE) Sing (A + λE) ∩ so(n, R) , i=1

¯ λ=−ai λ∈C,

where Sing ⊂ so(n, C) is the set of singular points, and S−ai is the singular set of the bracket { , }A−ai E .

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Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

Proof. The first part is just a reformulation of Theorem 2.2.12 for this particular pencil. The second part follows from the observation that, for λ = −ai , the bracket [ , ]A+λE is isomorphic to the standard so(n) bracket, and X → (A+λE)1/2 X(A+ λE)1/2 is the corresponding (generally speaking, complex !) isomorphism for the dual spaces.  Notice that, for λ = −ai , the bracket { , }A−ai E becomes isomorphic to the e(n − 1) bracket. This passage from so(n) to e(n − 1) as λ → −ai is known as the contraction between these two Lie algebras, and one might expect that their singular sets are closely related. By continuity, Sλ should “converge” to some part of S−ai as λ → −ai and therefore, (A − ai E)1/2 Sing (A − ai E)1/2 ⊂ S−ai . It would be interesting to understand the relationship between these two sets in detail. In low dimensions, this can be done by a straightforward computation. In dimension 3, for example, Sλ = {0} for λ = ai , whereas S−ai is a line. However, in dimension 4, the set S−ai is the limit of Sλ as λ → −ai , in particular, (A − ai E)1/2 Sing (A − ai E)1/2 = S−ai . Indeed, so(4) = so(3) ⊕ so(3) and the set of singular points Sing in the complex Lie algebra so(4, C) is the union of two 3-dimensional subspaces ⎛

0 ⎜ −z3 V1 = ⎜ ⎝ z2 −z1

z3 0 −z1 −z2

−z2 z1 0 −z3

⎞ z1 z2 ⎟ ⎟, z3 ⎠ 0



0 ⎜ z3 V2 = ⎜ ⎝ −z2 −z1

−z3 0 z1 −z2

z2 −z1 0 −z3

⎞ z1 z2 ⎟ ⎟. z3 ⎠ 0

Under the transformation X → (A + λE)1/2 X(A + λE)1/2 , these two subspaces “move” and for λ = −ai coincide (at this very moment the transformation ceases to be invertible). This single subspace is exactly the singular set S−ai . As a result, the description of the set of critical points for FA becomes simpler: " SA = Viλ , ¯ i=1,2, λ∈C

where Viλ = (A + λE)1/2 Vi (A + λE)1/2 . This gives a natural parametrisation for SA by means of four parameters z1 , z2 , z3 , λ. The next theorem describes the (common) equilibria for the Euler–Manakov system. Theorem 2.4.14 (Feh´er and Marshall [28]). The set of common equilibrium points of FA (with A diagonal) is the union of those Cartan subalgebras h ⊂ so(n) which are common Cartan subalgebras for all commutators [ , ]A+λE . One of these Cartan

2.4. How Does It Work? Examples and Applications subalgebras is standard: ⎧⎛ 0 ⎪ ⎪ ⎪⎜ ⎪ −x ⎪ ⎨⎜ 12 ⎜ h0 = ⎜ ⎜ ⎪ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

75



x12 0 0 −x34

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

x34 0 ..

.

xi,i+1 ∈ R

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

All the others are obtained from h0 by conjugation h0 → P h0 P −1 , where P is a permutation matrix. We give a bi-Hamiltonian version of the proof. Proof. According to Theorem 2.2.13, common equilibria for FA are those points where the kernels of all the brackets coincide. For the so(n) bracket, this kernel at a regular point X is the Cartan subalgebra containing X. Thus, we need to find common Cartan subalgebras for all the brackets [ , ]A+λE . An explicit description of such subalgebras is an easy algebraic exercise.  We finally describe non-degenerate equilibria. Let X be a 2×2 block-diagonal skew-symmetric matrix (as above). For each pair of blocks we set         0 ω ai 0 0 xi,i+1 λ1 0 = , = −xi,i+1 0 −ω 0 0 ai+1 0 λ2 and consider the function f (x) = (x−λ21 )(x−λ22 )/(ω 2 (λ1 +λ2 )2 ), formally assuming that f (∞) = 1/(ω 2 (λ1 + λ2 )2 ). By drawing the graphs of all of these functions on the same plane R2 , we obtain a collection of parabolas called the parabolic diagram P. For simplicity, we assume that n is even. We say that this diagram is generic if any two parabolas intersect exactly at two points (including complex intersections and intersections at infinity). Theorem 2.4.15 (Izosimov [38, 39]). (i) The equilibrium point X ∈ so(n) is non-degenerate if and only if the parabolic diagram P is generic, i.e., (i1 ) each intersection point in the upper half-plane corresponds to an elliptic component; (i2 ) each intersection point in the lower half-plane corresponds to a hyperbolic component; (i3 ) each complex intersection corresponds to a focus component. (ii) If P is generic, and all intersections are real and located in the upper halfplane, then the equilibrium is stable. (iii) If there is either a complex intersection or an intersection point in the lower half-plane, then the equilibrium point is unstable.

76

Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

2.4.4 Periodic Toda Lattice The phase space MT of the periodic Toda lattice with n sites is Rn+ × Rn , endowed with the Flaschka variables a = (a1 , . . . , an ) ∈ Rn+ and b = (b1 , . . . , bn ) ∈ Rn . It is ∞ convenient to treat a and b as infinite n-periodic sequences a ∈ R∞ + and b ∈ R , with ai+n = ai and bi+n = bi . The equations of motion are  a˙ i = ai (bi+1 − bi ), b˙ i = 2(a2i − a2i−1 ). It is well known that these equations are bi-Hamiltonian. The corresponding pencil ΠT = {P0 + λP∞ } is given by −ai ai+1 , {bi , bi+1 }0 = −2a2i 2 = −ai .

{ai , bi }0 = ai bi , {ai , bi+1 }0 = −ai bi+1 , {ai , ai+1 }0 = {ai , bi }∞ = ai , {ai , bi+1 }∞

The corresponding Hamiltonians are ⎧ n  ⎪ ⎪ ⎨ bi for λ = ∞, Hλ = i=1 n n   ⎪ 1 2 ⎪ b2i for λ = ∞. ⎩ ai + 2 i=1

i=1

The integrals of the Toda lattice are the Casimir functions of the pencil ΠT . Taking all of these Casimirs, we obtain a complete family of polynomials FT in bi-involution. Our goal is to study the singularities of FT . According to the general scheme, to this end we need to proceed as follows: (i) For each point x ∈ MT determine the spectrum of the pencil at x. The point x is singular if and only if the spectrum is non-empty. (ii) If x is singularthen, for each λin the spectrum, check the following conditions: dim Ker P∞ (x) |Ker Pλ (x) = corank ΠT (x), and the linearized pencil dλ ΠT (x) is non-degenerate. The point x is non-degenerate if and only if these conditions are satisfied for each λ in the spectrum. (iii) If x is non-degenerate, determine its type by adding up the types of dλ Π(x). First of all, notice that each Poisson structure Pλ ∈ ΠT (x) is of corank two. Moreover, P∞ is of constant corank two (recall that we are assuming ai > 0). Next, the map (a, b) → (a, b − λ) transforms the bracket Pλ into P0 . Thus, all these brackets are isomorphic. This observation reduces the study of singularities of the pencil to the singularities of P0 . To that end, consider the infinite Lax matrix ⎞ ⎛ .. .. .. . . ⎟ ⎜ . ⎟ ⎜ a b ai i−1 i ⎟. L(a, b) = ⎜ ⎟ ⎜ ai bi+1 ai+1 ⎠ ⎝ .. .. .. . . .

2.4. How Does It Work? Examples and Applications

77

Proposition 2.4.16. The singular set S0 of the Poisson structure P0 consists of those infinite periodic sequences a, b ∈ R∞ for which the equation L(a, b) ξ = 0 has either two periodic or two anti-periodic solutions, i.e., zero is a multiplicity-two periodic or anti-periodic eigenvalue of L(a, b). If (a, b) ∈ S0 then corank P0 (a, b) = 4 and the transversal linearisation of P0 at the point (a, b) ∈ S0 is isomorphic to the Lie algebra sl(2, R) ⊕ R (in particular, this Lie algebra is non-degenerate!). Thus, the structure of the singular set S0 is quite simple (and so is that of Sλ , since P0 and Pλ are isomorphic). The next step is to restrict the Poisson structure P∞ to Ker P0  (sl(2, R) ⊕ R)∗ . It can be shown by a straightforward computation that the kernel of this restriction is a Cartan subalgebra of sl(2, R) ⊕ R and, moreover, this subalgebra is of elliptic type. If we replace P0 by Pλ , all the conclusions remain unchanged. The only difference is that the equation L(a, b) ξ = 0 should be replaced by L(a, b) ξ = λξ so that λ must be a multiplicity-two periodic or anti-periodic eigenvalue of L(a, b). Thus, the structure of singularities of the family of commuting Casimirs FT can be described as follows. If L(a, b) has no multiple periodic or anti-periodic eigenvalues, then the integrals of the Toda lattice are independent at the point (a, b) ∈ MT . If L(a, b) has k multiplicity-two periodic or anti-periodic eigenvalues λ1 , λ2 , . . . , λk , then these λi ’s are exactly the elements of the spectrum Λ of the pencil of compatible Poisson structures Pλ at the point (a, b) ∈ MT . Since each λi -linearisation is non-degenerate and elliptic, we come to the following Theorem 2.4.17. All singularities of the periodic Toda lattice are non-degenerate and of elliptic type. In particular, all of them are (orbitally) stable. By using different methods, this result was obtained by Foxman and Robbins.

To read about • Rubanovskii case: Rubanovskii [60], Fedorov [27], and Basak [6]; • Mischenko–Fomenko systems on semisimple Lie algebras: Mishchenko and Fomenko [52], Bolsinov [7], and Bolsinov and Oshemkov [17]; • Euler–Manakov top: Manakov [47], Mischenko and Fomenko [52], Feh´er and Marshall [28], and Izosimov [38, 39]; • Periodic Toda lattice: Flaschka [29], Foxman and Robbins [33], and Bolsinov and Izosimov [12].

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Chapter 3

Geometry of Integrable non-Hamiltonian Systems Nguyen Tien Zung 3.1

Introduction

This text is an expanded version of the lecture notes for a minicourse taught by the author at the summer school “Advanced Course on Geometry and Dynamics of Integrable Systems” organized by Vladimir Matveev, Eva Miranda and Francisco Presas at Centre de Recerca Matem` atica (CRM) Barcelona, from September 9th to 14th, 2013. The aim of this minicourse was to present some geometrical aspects of integrable non-Hamiltonian systems. Here, the adjective non-Hamiltonian does not mean that the systems in question cannot be Hamiltonian, it simply means that we consider general dynamical systems which may or may not admit a Hamiltonian structure, and even when they are Hamiltonian we can sometimes forget about their Hamiltonian nature. The notion of integrability for Hamiltonian systems can be traced back to the paper [55] by Joseph Liouville, written in 1855, some 160 years ago. Compared to that, the similar notion of integrability for more general dynamical (nonHamiltonian) systems is very young. Though integrable non-Hamiltonian systems, especially non-holonomic systems, have been studied since more than a century ago by people like Chaplygin, Suslov, Routh, etc., it is only in the 1990s that people started writing explicitly about the notion of integrability in the non-Hamiltonian context. Some of the earlier papers on this subject are by Fedorov and Kozlov [36] from 1995, Bogoyavlenskij [9] and Dragovic, Gajic and Jovanovic [29] from 1998, Bates and Cushman [6] from 1999, Stolovitch [71] from 2000, Cushman and Duistermaat [27] from 2001, etc. Many more articles on integrable non-Hamiltonian © Springer International Publishing Switzerland 2016 A. Bolsinov et al., Geometry and Dynamics of Integrable Systems, Advanced Courses in Mathematics - CRM Barcelona, DOI 10.1007/978-3-319-33503-2_3

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systems have appeared recently, including a whole issue of the journal Regular and Chaotic Dynamics [13], with papers by Borisov, Bolsinov, Mamaev, Kilin, Kozlov, and other people. These papers provide a lot of interesting examples of integrable non-Hamiltonian systems, from the more simple (e.g., rolling disks, rolling balls, Chinese tops, Chaplygin’s skates) to the more complicated ones. Though different people arrive at integrable non-Hamiltonian systems from different points of view, they converge at the following definition, which was probably first written down explicitly by Bogoyavlenskij [9], who called it broad integrability: a dynamical system is called integrable if it admits a sufficiently large family of commuting vector fields and common first integrals. More precisely, we have: Definition 3.1.1. A vector field X on a manifold M is said to be integrable of type (p, q), where p ≥ 1, q ≥ 0, p + q = dim M , if there exist p vector fields X1 = X, X2 , . . . , Xp and q functions F1 , . . . , Fq on M which satisfy the following conditions: (i) the vector fields X1 , . . . , Xp commute pairwise, [Xi , Xj ] = 0, ∀i, j; (ii) the functions F1 , . . . , Fq are common first integrals of X1 , . . . , Xp , that is, Xi (Fj ) = 0, ∀i, j; (iii) independence condition: X1 ∧ X2 ∧ · · · ∧ Xp = 0 and dF1 ∧ · · · ∧ dFq = 0 almost everywhere. Under the above conditions, we will also say that (X1 , . . . , Xp , F1 , . . . , Fq ) is an integrable system of type (p, q). There are many reasons why the above definition of integrability for general dynamical systems is the right one. Let us list some of them here: (i) Hamiltonian systems on a symplectic 2n-dimensional manifold which are integrable in the sense of Liouville are also integrable of type (n, n) in the above sense (i.e., n commuting first integrals together with their n commuting Hamiltonian vector fields). Hamiltonian systems which are integrable of various generalized types, e.g., non-commutatively integrable Hamiltonian systems in the sense of Fomenko and Mischenko [39], are also integrable in the sense of the above definition. Conversely, the cotangent lifting of an integrable non-Hamiltonian system is an integrable Hamiltonian system in the sense of Liouville [3]. (ii) Liouville’s theorem [55] is still valid for integrable non-Hamiltonian systems. In particular, under additional regularity and compactness conditions, the motion of an integrable dynamical system is quasi-periodic, just like in the Hamiltonian case. (iii) The theories of normal forms for Hamiltonian and non-Hamiltonian systems are essentially identical. In particular, an analytic integrable dynamical system always admits a local analytic normalization near a singular point, be it

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Hamiltonian or not [80, 84]. An analytic Hamiltonian vector field will admit a local analytic Birkhoff normalization if and only if it admits a local analytic Poincar´e–Dulac normalization when forgetting about the Hamiltonian structure [80, 84]. (iv) Many other results in the theory of integrable systems do not need the Hamiltonian structure either. In particular, Morales–Ramis–Sim´ o’s theorem on Galoisian obstructions to meromorphic integrability [61] turns out to be valid also for non-Hamiltonian systems [3]. Various topological invariants for integrable Hamiltonian systems, e.g., the monodromy and the Chern class (see [31, 83]) can be naturally extended to the case of integrable nonHamiltonian systems as well (see, e.g., [12, 27]). (v) Integrability in both classical and quantum mechanics means some kind of commutativity, and Definition 3.1.1 fits well into this philosophy. In fact, the equation Xi (Fj ) = 0 can be rewritten as [Xi , fj ] = 0 using the Schouten bracket. If one considers the functions Fi as zeroth-order linear differential operators, and the vector fields Xi as first-order linear differential operators on the space of functions on the manifold M , then the conditions [Xi , Xj ] = 0 and Xi (Fj ) = 0 mean that these p + q differential operators commute, like in the case of a quantum integrable system. (vi) Sometimes it is useful to forget about the symplectic or Poisson structure and consider integrable Hamiltonian systems on the same footing as more general integrable dynamical systems. In particular, we will show in Subsection 3.2.1 a simple, short, and conceptual proof of the existence of action-angle variables using this approach. When applying this approach to other underlying structures, i.e., Dirac and contact structures, one obtains new results about action-angle variables [87, 89]. The above points show, mainly, the similarities between Hamiltonian and nonHamiltonian systems. Now, let us indicate some differences between them: (i) The class of integrable non-Hamiltonian systems is much larger than the class of integrable Hamiltonian systems. There are many problems coming from control theory, economics, biology, etc. which are a priori non-Hamiltonian, but which can still be integrable. Not every integrable system admits a Hamiltonian structure, and the problem of Hamiltonization is very interesting and non-trivial, even in the case of dimension 2, see, e.g., [10, 90]. (ii) The geometry and topology of integrable non-Hamiltonian systems are much richer than those of integrable Hamiltonian systems. In particular, there are manifolds which do not admit any symplectic structure, but which admit nondegenerate integrable non-Hamiltonian systems. Integrable non-Hamiltonian systems also display many kinds of interesting singularities which are not available for Hamiltonian systems.

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(iii) Being Hamiltonian has certain advantages. For example, Noether’s theorem about the relationship between symmetries and first integrals needs the symplectic form, and there is no analogue of this theorem in the non-Hamiltonian case. There are also mechanisms of integrability which are specific to Hamiltonian systems, e.g., the fact that bi-Hamiltonian systems are automatically integrable under some mild additional assumptions. (iv) For Hamiltonian systems, reduction (with respect to symmetry groups) commutes with integrability, i.e., a proper symmetric Hamiltonian system is integrable if and only if the reduced system is integrable. On the other hand, a non-Hamiltonian system which becomes integrable after a reduction is not necessarily integrable before reduction, see [48, 85]. In this text, we will concentrate on two aspects of integrable dynamical systems that I am most familiar with. Namely, in Section 3.2, we will study local and semi-local normal forms and associated torus actions for integrable systems, and in Section 3.3 we will study the geometry of integrable systems of type (n, 0). This class of systems of type (n, 0) is a particular but very important class in the geometric study of integrable systems, because every integrable system becomes a system of type (n, 0) when restricted to a common level set of the first integrals.

3.2 Normal forms, action-angle variables, and associated torus actions 3.2.1 Liouville torus actions and action-angle variables Liouville’s theorem. In 1855, Liouville [55] showed that if a Hamiltonian system XH on a symplectic manifold (M 2n , ω) is integrable with a momentum map F = (F1 , . . . , Fn ) : M 2n → Rn , F1 = H, then each connected component of a compact regular level set of the momentum map F is diffeomorphic to an m-dimensional torus on which the vector fields XH , XF2 , . . . , XFn are constant, i.e., are invariant under the action of the torus on itself by translations. Each such connected level set N is called a Liouville torus. The torus Tn acts not only on N , but also on the nearby Liouville tori by the same arguments, and so we have a torus Tn -action in a tubular neigborhood of N , which preserves XH , XF2 , . . . , XFn , and whose orbits are regular connected compact level sets of the momentum map. This torus action is called the Liouville torus action near N . Liouville’s theorem can be naturally extended to the case of integrable nonHamiltonian systems. Theorem 3.2.1 (Non-Hamiltonian version of Liouville’s theorem). Assume that (X1 , . . . , Xp , F1 , . . . , Fq ) is an integrable system of type (p, q) on a manifold M which is regular at a compact level set N . Then, in a tubular neighborhood U (N ) there is, up to automorphisms of Tp , a unique free torus action ρ : Tp × U(N ) →

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U (N ) which preserves the system (i.e., the action preserves each Xi and each Fj ) and whose orbits are regular level sets of the system. In particular, N is diffeomorphic to Tp and U (N ) ∼ = Tp × B q , with periodic coordinates θ1 (mod 1), . . . , p θp (mod 1) on T , and coordinates (z1 , . . . , zq ) on a q-dimensional ball B q , such that F1 , . . . , Fq depend only on the variables z1 , . . . , zq , and the vector fields Xi are of the type p  ∂ Xi = aij (z1 , . . . , zq ) . (3.1) ∂θ j j=1 The proof of the above theorem is absolutely similar to the case of integrable Hamiltonian systems on symplectic manifolds, see, e.g., [9, 85]. It consists of two main points: (i) the map (F1 , . . . , Fq ) : U (N ) → Rq from a tubular neighborhood of N to Rq is a topologically trivial fibration by the level sets, due to the compactness of N and the regularity of (F1 , . . . , Fq ); and (ii) the vector fields X1 , . . . , Xp generate a transitive action of Rp on the level sets near N , and the level sets are compact and of dimension p, which imply that each of them is a p-dimensional compact quotient of Rp , i.e., a torus. Similarly to the Hamiltonian case, the regular level sets in the above theorem are called Liouville tori, and the torus action is also called the Liouville torus action. Theorem 3.2.1 shows that the flow of the vector field X = X1 of an integrable system is quasi-periodic under some natural compactness and regularity conditions. This quasi-periodicity is the most fundamental geometrical property of proper integrable dynamical systems. Remark 3.2.2. There is also a version of Theorem 3.2.1 for integrable stochastic dynamical systems, see [92]. One may suspect that a version of this Liouville torus action exists for quantum systems as well. Structure-preserving property of Liouville torus actions. The Liouville torus actions, and other associated torus actions that we will discuss later in this section, preserve not only the system, but has the following much stronger structure preserving property: roughly speaking, anything which is preserved by the system must also be preserved by these torus actions. In other words, one may view these torus actions as a kind of double commutant. In order to formulate this result more precisely, we first need to define the notion of tensor fields which are invariant with respect to an integrable system. Definition 3.2.3. Let (X = X1 , . . . , Xp , F1 , . . . , Fq ) be a smooth integrable system of type (p, q) on a manifold M . A tensor field G ∈ Γ(⊗k T M ⊗h T ∗ M ) is said to be preserved by the system (i.e., is invariant under the system) (X1 , . . . , Xp , F1 , . . . , Fq ) if at least one of the following two conditions is satisfied: (i) G is invariant under the vector fields X1 , . . . , Xp ; (ii) G is invariant under the vector field X = X1 , and moreover the orbits of X1 are dense in a dense family of Liouville tori in a tubular neighborhood of N .

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Chapter 3. Geometry of Integrable non-Hamiltonian Systems p In other words, if we write X1 = i=1 ai (z1 , . . . , zq )∂/∂θi in a canonical coordinate system as in (3.1), then for a dense family of the values of (z1 , . . . , zq ), the numbers a1 (z1 , . . . , zq ), . . . , ap (z1 , . . . , zq ) are incommensurable.

We will say that a tensor field G is conformally invariant under a vector field X if there exists a (smooth) function g such that LX G = gG. Similarly to Definition 3.2.3, we also have the notion of tensor fields which are conformally invariant under an integrable system. Instead of tensor fields, one can also consider subbundles of the vector bundles on M which can be constructed from the trivial bundle and the tangent and cotangent bundles on M using the natural operations (sums and tensor products). Any tensor field may be viewed as such a subbundle, and so these subbundles may be viewed as general underlying geometric structures on M . We will call them natural vector subbundles on M . One can naturally extend Definition 3.2.3 to the case of natural vector subbundles on M . Now we can formulate precisely the structure-preserving property of Liouville torus action. Theorem 3.2.4 ([87, 89]). (i) Let (X1 , . . . , Xp , F1 , . . . , Fq ) be a smooth integrable system of type (p, q) on a manifold M , which preserves a smooth tensor field G ∈ Γ(⊗k T M ⊗h T ∗ M ) (of any type), and let N = {F1 = c1 , . . . , Fq = cq } be a Liouville torus of the system. Then G is also invariant under the Liouville torus Tp -action in a tubular neighborhood of N . (ii) If G is not invariant but only conformally invariant under the system, then it is also conformally invariant under the Liouville torus action near N . (iii) If G is not a tensor field, but rather a natural vector subbundle which is invariant under the system, then it is also invariant under the Liouville torus action near N . The tensor field (or vector bundle) G in the above theorem can be of any nature. For example, when G is an infinitesimal generator of a Lie group action, we obtain that if a connected Lie group action preserves the system then it commutes with the Liouville torus action. When the system is isochoric, i.e., preserves a volume form, then that volume form is also preserved by the Liouville torus action, etc. G can also be a contact distribution, or a Dirac structure, for example. Proof. We will give here the proof of assertion (i) (the proofs of the other two assertions are similar, though a bit more complicated). We will assume that G satisfies condition (ii) of Definition 3.2.3 (the case when condition (i) is satisfied is completely similar). Fix a canonical coordinate system (θ1 (mod 1), . . . , θp (mod 1), z1 , . . . , zq ) in a tubular neighborhood U (N ) of N as given by Theorem 3.2.1. We will make a filtration of the space Γ(⊗k T M ⊗h T ∗ M ) of tensor fields of contravariant order k and contravariant order h as follows. The subspace Tsh,k consists of sections of ⊗k T M ⊗h T ∗ M whose expression in the coordinates (θ1 (mod 1), . . . , θp (mod 1), z1 , . . . , zq , w1 , . . . , wr ) contains only

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terms of the type ∂ ∂ ∂ ∂ ⊗ ··· ⊗ ⊗ ⊗ ··· ⊗ ⊗ dθi1 ⊗ · · · ⊗ dθic ⊗ dzj1 ⊗ · · · ⊗ dzjd ∂θi1 ∂θia ∂zj1 ∂zib up to permutations of the factors in the tensor product, with b + c ≤ s. For example, ⎧ ⎫ ⎨ ⎬ ∂ ∂ fi,j  ⊗ ··· ⊗ ⊗ dzj1 ⊗ · · · ⊗ dzjk . T0h,k = ⎩  ⎭ ∂θi1 ∂θih i,j

h,k = {0}. It is clear that Put T−1 h,k h,k ⊂ T0h,k ⊂ T1h,k ⊂ · · · ⊂ Th+k = Γ(⊗k T M ⊗h T ∗ M ). {0} = T−1

It is also clear that the above filtration is stable under the Lie derivative with respect to the vector field X1 , i.e., we have LX1 Λ ∈ Tsh,k ,

s = 0, . . . , k + h,

∀Λ ∈ Tsh,k .

Since LX1 G = 0 by our hypothesis, and the Liouville torus action commutes with the vector field X1 , we also have that LX1 G = 0, where the overline means the average of a tensor with respect to the Liouville torus action. Thus, we also have LX1 Gˆ = 0, where Gˆ = G − G has average equal to 0. The equality LX1 Gˆ = 0 implies that in Gˆ the coefficients of the terms which h,k are not in Th+k−1 , i.e., the terms of the type ∂ ∂ ⊗ ··· ⊗ ⊗ dθi1 ⊗ · · · ⊗ dθik , ∂zj1 ∂zih are invariant under X1 . This means that these coefficient functions are constant on the orbits of X1 . By continuity, it follows that they are constant on Liouville tori for which the orbits of X1 are dense. But since the family of such Liouville tori is dense in the space of all Liouville tori near N , we see that these functions are constant on every Liouville torus near N , i.e., they are constant with respect to the Liouville Tp -action in a neighborhood of N . But any Tp -invariant function with average 0 is a trivial function, so in fact Gˆ does not contain any term other h,k h,k , i.e., we have Gˆ ∈ Th+k−1 . By the same arguments, one can verify than Th+k−1 h,k h,k ˆ that if G ∈ Ts with s ≥ 0 then, in fact, Gˆ ∈ Ts−1 . So, by induction, we have Gˆ = 0, i.e., G = G is invariant under the Liouville torus action.  In the case of linear differential operators, we also have the following result, obtained recently in a joint work with Nguyen Thanh Thien, whose proof is absolutely similar to the proof of Theorem 3.2.4.

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Theorem 3.2.5 ([92]). Under the assumptions of Theorem 3.2.1, let Λ be a linear differential operator on M which satisfies at least one of the following two conditions: (i) Λ is invariant under X1 , . . . , Xp . (ii) Λ in invariant under X1 , and moreover, the orbits of X1 are dense in a dense family of orbits of the Liouville Tp -action near N . Then Λ is invariant under the Liouville Tp -action in a neighborhood of N . Action-angle variables. In the case of integrable Hamiltonian systems on symplectic manifolds, one can deduce easily from Theorem 3.2.4 the following wellknown theorem about the existence of action-angle variables. Theorem 3.2.6. If N is a Liouville torus of an integrable Hamiltonian system on a symplectic manifold (M 2n , ω) given by a momentum map F = (F1 , . . . , Fn ) : M 2n → Rn , then in a neigborhood U (N ) of N there is a system of canonical coordinates (θ1 (mod 1), . . . , θn (mod 1), z1 , . . . , zn ), called action-angle variables in which the F1 , . . . , Fn are functions of the action variables n z1 , . . . , zn only, and the symplectic structure ω has the canonical form ω = i=1 dzi ∧ dθi . We remark that the above action-angle variables theorem was first proved by Henri Mineur in 1935 [57, 58], though it is often called Arnold–Liouville theorem. It would be more appropriate to call it Liouville–Mineur–Arnold theorem. Proof. According to Theorem 3.2.4, the symplectic structure is preserved by the Liouville torus Tn -action, because it is preserved by the Hamiltonian vector fields XF1 , . . . , XFn of the system. But, since N is a Lagrangian torus, i.e., the pullback of ω to N is trivial, the cohomology class of ω in a tubular neighborhood U (N ) of N is trivial, and so this action is a Hamiltonian action in U (N ), i.e., it is given by a momentum map (z1 , . . . , zn ) : U (N ) → Rn . Define periodic coordinates (θ1 , . . . , θn ) on U (N ) in such a way that the zero section S = {θ1 = 0, . . . , θn = 0} is a Lagrangian submanifold and ∂/∂θi = Xzi for all i = 1, . . . , n. Then, one verifies n easily that ω = i=1 dzi ∧ dθi on S. But, since both forms are Tn -invariant, it follows that they are equal everywhere in a neighborhood of N .  Many generalizations of the Liouville–Mineur–Arnold theorem, including Nekhoroshev’s theorem about partial action-angle variables for noncommutatively integrable Hamiltonian systems (when the Liouville tori are isotropic instead of Lagrangian) [62], Fass`o–Sansonetto’s generalization of action-angle variables for integrable Hamiltonian systems on almost-symplectic manifolds (with a nondegenerate but non-closed 2-form) [35], Laurent–Miranda–Vanhaecke’s generalization of action-angle variables theorem to the case of Poisson manifolds [54] and, more recently, our results about action-angle variables on manifolds equipped with Dirac structures, contact structures, or other geometric structures [87, 89], can also be deduced from Theorem 3.2.4. For example, for integrable systems on contact

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manifolds, one recovers easily the action-angle variables results due to Banyaga and Molino [4] and Jovanovic [49] in the transversal case (when the contact distribution is transverse to the invariant tori), and also obtains a new action-angle variables result for the case when the contact distribution is tangent to the invariant tori at a point [89].

3.2.2 Local normal forms of singular points Poincar´e–Birkhoff normal forms. It is well-known that every smooth or analytic vector field near an equilibrium point admits a formal Poincar´e–Birkhoff normal form (Birkhoff in the Hamiltonian case, and Poincar´e–Dulac in the nonHamiltonian case), see, e.g., [14, 68]. It turns out that Poincar´e–Birkhoff normalization is closely related to (intrinsic formal or smooth) torus actions around singular points [80, 84]. These torus actions play the same role for singular points of dynamical systems as Liouville actions do for regular orbits of integrable systems. For singular orbits of integrable systems, one can combine these two torus actions together to get the “big torus action”, which leads to normal forms near such orbits. We will briefly recall here the theory of Poincar´e–Birkhoff normalization, and show the role of torus actions in this theory. Let X be a given formal or analytic vector field in a neighborhood of 0 in Km , where K = R or C, and let X(0) = 0. When K = R, we may also view X as a complex vector field by complexifying it. Denote by X = X (1) + X (2) + X (3) + · · · the Taylor expansion of X in some local system of coordinates, where X (k) is a homogeneous vector field of degree k for each k ≥ 1. In the Hamiltonian case on a symplectic manifold, X = XH , m = 2n, K2n has a standard symplectic structure in a canonical system of coordinates, and X (j) = XH (j+1) , where H (j+1) is the term of degree j + 1 in the Taylor expansion of H. The algebra of linear vector fields on Km , under the standard Lie bracket, is nothing but the reductive algebra gl(m, K) = sl(m, K) ⊕ K. In particular, we have X (1) = X s + X nil , where X s (resp., X nil ) denotes the semi-simple (resp., nilpotent) part of X (1) . There is a complex linear system of coordinates (xj ) in Cm in which X s has diagonal form, m  s X = γj xj ∂/∂xj , j=1

where γj are complex coefficients, called the eigenvalues of X (or X (1) ) at 0.

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In the Hamiltonian case, X (1) ∈ sp(2n, K), which is a simple Lie algebra, and we also have the decomposition X (1) = X s + X nil , which corresponds to the decomposition H (2) = H s + H nil . There is a complex canonical linear system of coordinates (xj , yj ) in C2n in which H s has diagonal form: Hs =

n 

λj x j yj ,

j=1

where λj are complex coefficients, called frequencies of H (or H (2) ) at 0. For each natural number k ≥ 1, the vector field X s acts linearly on the space of homogeneous vector fields of degree k by the Lie bracket, and the monomial vector fields are the eigenvectors of this action, m /

n 0  γj xj ∂/∂xj , xb11 xb22 · · · xbnn ∂/∂xl = bj γj − γl xb11 xb22 · · · xbnn ∂/∂xl .

j=1

j=1

When an equality of the type m 

bj γ j − γ l = 0

j=1

 holds for some nonnegative integer m-tuple (bj ) with bj ≥ 2, we will say that the monomial vector field xb11 xb22 · · · xbmm ∂/∂xl is a resonant term, and that the mtuple (b1 , . . . , bl − 1, . . . , bl ) is a resonance relation for the eigenvalues (γi ). More precisely, a resonance relation for the n-tuple of eigenvalues  (γj ) of a vector field ) of integers satisfying the relation cj γj = 0, such that X is an m-tuple (c j  cj ≥ 1, and at most one of the cj may be negative. cj ≥ −1, In the Hamiltonian case, H s acts linearly on the space of functions by the Poisson bracket. The resonant terms (i.e., generators of the kernel of this action)  a j bj are the monomials xj yj that satisfy the following resonance relation, with c j = a j − bj : n  cj λj = 0. j=1

Denote by R the subset of Zm (or sublattice of Zn in the Hamiltonian case) consisting of all resonance relations (cj ) for a given vector field X. The number r = dimZ (R ⊗ Z) is called the degree of resonance of X. Of course, the degree of resonance depends only on the eigenvalues of the linear part of X, and does not depend on the choice of local coordinates. If r = 0, then we say that the system is nonresonant at 0. The vector field X is said to be in Poincar´e–Birkhoff normal form if it commutes with the semisimple part of its linear part, [X, X s ] = 0. In the Hamiltonian case, this equation can also be written as {H, H s } = 0.

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The above equations mean that if X is in normal form, then its nonlinear terms are resonant. A transformation of coordinates (which is symplectic in the Hamiltonian case) which puts X in Poincar´e–Birkhoff normal form is called a Poincar´e–Birkhoff normalization. Theorem 3.2.7 (Poincar´e–Dulac–Birkhoff). Any analytic formal or vector field which vanishes at 0 admits a formal Poincar´e–Birkhoff normalization. The proof of this result is based on the classical method of step-by-step normalization: at each step one eliminates a non-zero nonresonant monomial term Cb,l xb11 xb22 · · · xbnn ∂/∂xl of lowest degree by a local coordinate transformation (diffeomorphism) of the type m  bj γ j − γ l , exp Cb,l xb11 xb22 · · · xbnn / j=1

where exp denotes the time-1 flow of the vector field. The total number of steps is infinite in general, and the composition of all these consecutive normalizing maps converges in the formal category, but does not necessarily converge in the analytic category in general. Toric characterization of local normal forms. Denote by Q ⊂ Zm the sublattice of Zm consisting of m-dimensional vectors (ρj ) ∈ Zm which satisfy the properties m 

ρj cj = 0 ∀ (cj ) ∈ R,

and quadρj = ρk if γj = γk

(3.2)

j=1

(where R is the set of resonance relations as before). In the Hamiltonian case, Q is defined by n  ρj cj = 0 ∀ (cj ) ∈ R. (3.3) j=1

We will call the number d = dimZ Q

(3.4)

the (complex) toric degree of X at 0. Of course, this number depends only on the eigenvalues of the linear part of X, and we have the following (in)equality: r+d = n in the Hamiltonian case (where r is the degree of resonance), and r + d ≤ m in the non-Hamiltonian case. Let (ρ1j ), . . . , (ρdj ) be a basis of Q. For each k = 1, . . . , d, define the following diagonal linear vector field Zk : Zk =

m  j=1

ρkj xj ∂/∂xj ,

(3.5)

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in the non-Hamiltonian case, and Zk = XF k , where k

F =

n 

ρkj xj yj

(3.6)

j=1

in the Hamiltonian case. The vector fields Z1 , . . . , Zr have the following remarkable properties: (i) They commute pairwise and commute with X s and X nil , and they are linearly independent almost everywhere. √ (ii) iZj is a periodic vector field of period 2π for each j ≤ r (here i = −1). This means that if we write iZj = (iZj ) + i(iZj ), then (iZj ) is a periodic real vector field in Cn = R2n that preserves the complex structure. (iii) Together, iZ1 , . . . , iZr generate an effective linear Tr -action in Cn which preserves X s and X nil (and the symplectic structure in the Hamiltonian case). Another equivalent way to define the toricdegree is as follows: it is the smalld est number d such that one can write X s = j=k αk Zk , where αk are complex m k coefficients and each Zk has the form Zk = j=1 ρj xj ∂/∂xj with ρkj ∈ Z. The minimality of d is equivalent to the fact that the numbers α1 , . . . , αd are incommensurable. A simple calculation shows that X is in Poincar´e–Birkhoff normal form, i.e., [X, X s ] = 0 if and only if [X, Zk ] = 0,

k = 1, . . . , r.

These commutation relations say that if X is in normal form, then it is preserved by the effective r-dimensional torus action generated by iZ1 , . . . , iZr . Conversely, if there is a torus action which preserves X, then, since the torus is a compact group, we can linearize this torus action (using Bochner’s linearization theorem [8] in the non-Hamiltonian case, and the equivariant Darboux theorem in the Hamiltonian case, see, e.g., [25, 41]), leading to a normalization of X. In other words, we have the following result. Theorem 3.2.8 ([80, 84]). A holomorphic (Hamiltonian) vector field X in a neighborhood of 0 in Cm (or C2n with a standard symplectic form) admits a locally holomorphic Poincar´e–Birkhoff normalization if and only if it is preserved by an effective holomorphic (Hamiltonian) action of a real torus of dimension t, where t is the toric degree of X (1) as defined in (3.4), in a neighborhood of 0 in Cm (or C2n ), which has 0 as a fixed point and whose linear part at 0 has appropriate weights (given by the lattice Q defined in (3.2), (3.3), which depends only on the linear part X (1) of X).

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97

The above theorem is true in the formal category as well. But, of course, any vector field admits a formal Poincar´e–Birkhoff normalization, and a formal torus action. This (formal) torus action is intrinsically associated to the singular point, and we will call it the associated torus action of the system (i.e., vector field) at the singular point. Remark 3.2.9. The above toric point of view of normalization for vector fields was first developed in [80, 84]. It was later extended to the case of diffeomorphisms (i.e., discrete-time dynamical systems) by Raissy in [67], and also used by Chiba [22] in singular perturbation and renormalization methods. Theorem 3.2.8 has many important implications. One of them is the following. Proposition 3.2.10. A real analytic vector field X (Hamiltonian or non-Hamiltonian) in the neighborhood of an equilibrium point admits a local real analytic Poincar´e–Birkhoff normalization if and only if it admits a local holomorphic Poincar´e–Birkhoff normalization when considered as a holomorphic vector field. The proof of this proposition (see [84]) is based on the fact that the complex conjugation induces an involution on the torus action which governs the Poincar´e–Birkhoff normalization (which preserves the symplectic structure in the Hamiltonian case). Recall that, even when the vector field is real, the torus acts not in the real space, but in the complexified space in general. Only a subtorus of this associated torus acts in the real space. The dimension of this real subtorus action can be called the real toric degree of the system. Structure-preserving property of associated torus actions. The associated torus action at a singular point of a vector field has the same structure-preserving property as the Liouville torus actions discussed above. Theorem 3.2.11. If a formal or analytic tensor field G is preserved by a formal or analytic vector field X which vanishes at a point O, then the associated torus action of X at O also preserves G. Proof (Sketch). We can assume that X is already in Poincar´e–Birkhoff normal d form, i.e., [X, X s ] =0, where X s = k=1 αk Zk is the semisimple linear part m k ρ x ∂/∂x with ρkj ∈ Z, d is the toric degree, and of X at O, Zk = j j j=1 j √ √ −1Z1 , . . . , −1Zd are the generators of the associated torus action. One verifies easily that the Lie derivative LX is a linear operator on the space of formal tensor fields of the type of G, whose semisimple part (in the Jordan decomposition) is LX s , so LX G = 0 implies that LX s G = 0, which implies that LX s K = 0 for every monomial term K of G, because each monomial d tensor term is an eigenvector of the linear operator LX s . Thus, we have k=1 αk LZk K = 0 for every monomial term K of G. But, since the numbers α1 , . . . , αd are incommensurable and each LZk K is an integer multiple of K, we must have that LZk K = 0 for every k = 1, . . . , d. 

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In particular, in the Hamiltonian case, the symplectic structure is automatically preserved by the associated torus action. As a consequence, the existence of an analytic Birkhoff normalisation is equivalent to the existence of a Poincar´e– Dulac normalization (forgetting about the symplectic structure). Local normalization of analytic integrable systems. When the vector field is analytically integrable, then Theorem 3.2.8 leads to the following strong result about the existence of an analytic Poincar´e–Birkhoff normalization. Theorem 3.2.12 ([80, 84]). Let X be a local analytic vector field in (Km , 0), where K = R or C, such that X(0) = 0. If X is analytically integrable, then it admits a local analytic Poincar´e–Birkhoff normalization in a neighborhood of 0 (which is compatible with the volume form or the symplectic structure if X is an isochoric or a Hamiltonian vector field). Partial cases of the above theorem were obtained earlier by many authors, including R¨ ussmann [69] (the non-degenerate Hamiltonian case with two degrees of freedom), Vey [73, 74] (the non-degenerate Hamiltonian and isochoric cases), Ito [44] (the nonresonant Hamiltonian case), Ito [46] and Kappeler, Kodama, and N´emethi [51] (the Hamiltonian case with a simple resonance), and Bruno and Walcher [15] (the non-Hamiltonian case with m = 2). These authors, except Vey whose approach was more geometric, relied on long and heavy analytical estimates to show the convergence of an infinite normalizing coordinate transformation process. On the other hand, the proof of Theorem 3.2.12 in [80, 84] is a geometrical proof which uses resolution of singularities and L ojasiewicz inequalities in order to show the existence of an analytic torus action (i.e., to show that the associated torus action is not just formal, but analytic), and is relatively short. Remark 3.2.13. Also in the case of infinite-dimensional systems, one can talk about Poincar´e–Birkhoff normalization, and there should be an analytic associated torus action, see, e.g., [52, 53]. Toric action for a commuting family of vector fields. It is well-known that if X1 , . . . , Xp is a family of formal or analytic pairwise commuting vector fields which vanish at a point O, then they admit a simultaneous formal Poincar´e– Birkhoff normalization at O, see; e.g., [71, 72] and references therein. This fact corresponds to the existence of an intrinsic formal associated torus action for the family X1 , . . . , Xp at O, whose dimension will be called the toric degree of the d family at O: it is the smallest number d such that we can write Xis = k=1 αik Zk s for every i = 1, . . . , p, where Xi is the semisimple linear part of Xi , and the vector m fields Zk = j=1 ρkj xj ∂/∂xj are like in the case of a single vector field. Another equivalent definition is that this torus degree is the toric degree of a generic linear p c combination i=1 i Xi of the family (X1 , . . . , Xp ). Using this associated torus action for a family of commuting vector fields, we get the following simultaneous version of Theorem 3.2.12, whose proof remains the same.

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Theorem 3.2.14. Let (X1 , . . . , Xp , F1 , . . . , Fq ) be a local analytic integrable system of type (p, q) in (Km , 0), where K = R or C, such that X1 (0) = · · · = Xp (0) = 0. Then, the vector fields X1 , . . . , Xp admit a simultaneous local analytic Poincar´e– Birkhoff normalization in a neighborhood of 0 (which is compatible with the volume form or the symplectic structure if the system is isochoric or Hamiltonian).

3.2.3 Geometric linearization of non-degenerate singular points Non-degenerate singular points and linear systems. Consider an integrable system (X1 , . . . , Xp , F1 , . . . , Fq ) of type (p, q) on a manifold M , and let O ∈ M be a singular point of the system. The number k = dim Span(X1 (O), . . . , Xp (O)) is called the rank of O. If k = 0, then we say that O is a fixed point. If k > 0, then we make a local reduction in order to obtain a system of type (p − k, q) with a fixed point as follows: without loss of generality, we can assume that X1 (O) ∧ · · · ∧ Xk (O) = 0, i.e., X1 , . . . , Xk generate a local free Rk -action in a neighborhood U (O) of O. Since the (p+q −k)-tuple (Xk+1 , . . . , Xp , F1 , . . . , Fp ) is invariant under this local Rk -action, it can be projected to an integrable system of type (p − k, q) with a fixed point on the quotient of U (O) by this local Rk -action. The definition of nondegeneracy below will not depend on the choice of local reduction. Assume now that O is a fixed point. Denote by Yi the linear part of Xi at O, and by Gj the homogeneous part (i.e., the non-constant terms of lowest degree in the Taylor expansion) of Fj in some coordinate system. The first terms of the Taylor expansion of the identities [Xi , Xk ] = 0 and Xi (Fj ) = 0 show that the vector fields Y1 , . . . , Yp commute with each other and have G1 , . . . , Gq as common first integrals. Hence, (Y1 , . . . , Yp , G1 , . . . , Gq ) is again an integrable system of type (p, q), which shall be called the linear part of the system (X1 , . . . , Xp , F1 , . . . , Fq ), provided that the independence conditions Y1 ∧· · ·∧Yp = 0 and dG1 ∧· · ·∧dGq = 0 (almost everywhere) still hold. Definition 3.2.15. An integrable system (Y1 , . . . , Yp , G1 , . . . , Gq ) of type (p, q) is called linear if the vector fields Y1 , . . . , Yp are linear and the functions G1 , . . . , Gq are homogeneous. If, moreover, the linear vector fields Y1 , . . . , Yp are semisimple, then (Y1 , . . . , Yp , G1 , . . . , Gq ) is called a non-degenerate linear integrable system. A singular point O of rank k of an integrable system of type (p, q) is called non-degenerate singular point if after a local reduction it becomes a fixed point of an integrable system of type (p − k, q) whose linear part is a non-degenerate linear integrable system. Remark 3.2.16. If z is an isolated singular point of X1 in an integrable system (X1 , . . . , Xp , F1 , . . . , Fq ), then it is automatically a fixed point of the system. Indeed, if Xi (z) = 0 for some i, then due to the commutativity of X1 with Xi , X1 will vanish not only at z, but on the whole local trajectory of Xi which goes

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through z, and so z will be a non-isolated singular point of X1 . In the definition of nondegeneracy of linear systems, we don’t require the origin to be an isolated singular point. For example, (x1 ∂/∂x1 , x2 ) is a non-degenerate linear system of type (1, 1), for which the origin is a non-isolated singular point. Remark 3.2.17. In the Hamiltonian case on a symplectic manifold, p = q = n, when Y1 , . . . , Yn are linear Hamiltonian vector fields in a canonical coordinate system, we can take G1 , . . . , Gn to be their respective quadratic Hamiltonian functions: Yi = XGi . Definition 3.2.15 generalizes in a natural way the well-known notion of non-degenerate singular points of integrable Hamiltonian systems (see, e.g., [33, 73, 78]): in the Hamiltonian case on a symplectic manifold with p = q = n, the fact that Y1 , . . . , Yn are linear semisimple means that they generate a Cartan subalgebra of the simple Lie algebra sp(2n, K) of linear symplectic vector fields. It is well-known that, already in the Hamiltonian case, not every integrable linear system is non-degenerate. For example, in R4 , take G1 = x 1 y1 − x 2 y 2 , Y1 = x1

G2 = y1 y2 ,

∂ ∂ ∂ ∂ − y1 − x2 + y2 , ∂x1 ∂y1 ∂x2 ∂y2

Y2 = y 2

∂ ∂ + y1 . ∂x1 ∂x2

Then this is a degenerate (non-semisimple) integrable linear Hamiltonian system. Non-degenerate linear systems as linear torus actions. Consider a non-degenerate linear integrable system (Y1 , . . . , Yp , G1 , . . . , Gq ). Recall that the Lie algebra of linear vector fields on Km , where K = R or C, is naturally isomorphic to the reductive Lie algebra gl(m, K). Since Y1 , . . . , Yp are commuting semisimple elements in gl(m, K), then they can be diagonalized simultaneously over C. In other words, there is a complex coordinate system in which Y1 , . . . , Yp are diagonal: Yi =

m  i=j

cij xj

∂ . ∂xj

The linear independence of Y1 , . . . , Yp means that the matrix (cij )i=1,...,p j=1,...,m is of rank p. The set of polynomial common firstintegrals of Y1 , . . . , Yp is the vector α m space spanned by the monomial functions j=1 xj j that satisfy the resonance equation m  αj cij = 0, i = 1, . . . , p. (3.7) j=1

The set of nonnegative integer solutions of equation (3.7) is the intersection

 m m S ∩ Zm , where S = (α ) ∈ R | i + j=1 αj cij = 0 for all i = 1, . . . , p is the qdimensional space of all real solutions of (3.7), and Zm + is the set of nonnegative mtuples of integers. The functional independence of G1 , . . . , Gq implies that this set m S ∩ Zm + must have dimension q over Z. In particular, the set S ∩ R+ has dimension

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q over R, and the resonance equation (3.7) is equivalent to a linear system of equations with integer coefficients. In other  words, using a linear transformation to replace Yi by new vector fields Zi = j aij Yj with an appropriate invertible matrix (aij ) with constant coefficients, we may assume that Zi =

m  i=j

c˜ij xj

∂ , ∂xj

√ √ where c˜ij = k aik ckj ∈ Z for all i, j. The vector fields −1Z1 , . . . , −1Zp are the generators of a linear effective Tp -action, which is exactly the associated torus action of the family (Y1 , . . . , Yp ) in the sense of local normal form theory. In particular, in this case (and for non-degenerate fixed points of integrable systems of type (p, q)), the toric degree is equal to p. Thus, a non-degenerate linear integrable system of type (p, q) is essentially the same as an effective linear torus Tp -action in Cm . Hence, the classification of non-degenerate linear integrable systems is essentially the same as the classification of linear torus actions. Keep in mind that, when the system is real, the torus will act in the complexified space. 

Geometric equivalence. Geometrically, an integrable system of type (p, q) may be viewed as a singular fibration given by the level sets of the map (F1 , . . . , Fq ) : M → Kq , such that on each fiber there is an infinitesimal Kp -action generated by the commuting vector fields (X1 , . . . , Xp ). Denote by F the algebra of common first integrals of X1 , . . . , Xp . Instead of taking F1 , . . . , Fq , we can choose from F any other family of q functionally independent functions, and they will still form with X1 , . . . , Xp an integrable system. Moreover, in general, there is no natural preferred choice of q functions in F. For example, consider a linear integrable 4-dimensional system of type (1, 3), i.e., with one vector field and three functions. The vector ∂ ∂ ∂ ∂ +x2 ∂x −x3 ∂x −x4 ∂x . The corresponding resonance equation field is Y = x1 ∂x 1 2 3 4 is α1 + α2 − α3 − α4 = 0. The algebra of algebraic first integrals is generated by the functions x1 x3 , x1 x4 , x2 , x3 , x2 x4 ; it has functional dimension three but cannot be generated by just three functions. Thus, instead of specifying q first integrals, from the geometrical point of view it is better to look at the whole algebra F of first integrals of an integrable system of type (p, q). Notice also that, if fij ∈ F, i, j = 1, . . . , p, are such that the matrix (fij ) is invertible then, by putting  fij Xj , i = 1, . . . , p, Xˆi = ij

we get another integrable system (Xˆ1 , . . . , Xˆp , F1 , . . . , Fq ) which, from the geometric point of view, is essentially the same as the original system. So we have the following definition. Definition 3.2.18. Two integrable dynamical systems (X1 , . . . , Xp , F1 , . . . , Fq ) and (X1 , . . . , Xp , F1 , . . . , Fq ) of type (p, q) on a manifold M are said to be geometrically

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equal, if they have the same algebra of first integrals (i.e., F1 , . . . , Fp are functionally dependent of F1 , . . . , Fp and vice versa), and there exists an invertible matrix (fij )j=1,...,p i=1,...,p (i.e., whose determinant does not vanish anywhere), whose entries fij are first integrals of the system, such that one can write  Xi = fij Xj , i = 1, . . . , p. j

Two integrable systems are said to be geometrically equivalent if they become geometrically equal after applying a diffeomorphism. Remark 3.2.19. Though the choice of first integrals is not important in Definition 3.2.18 of geometric equivalence, the q-tuple F1 , . . . , Fq of first integrals in Definition 3.2.15 of non-degeneracy must be chosen so that not only they are functionally independent, but their homogeneous parts are also functionally independent. (According to a simple analogue of Ziglin’s lemma [77], in the analytic case, such a choice is always possible.) It’s clear that, near a regular point, i.e., a point z such that X1 ∧· · ·∧Xp (z) = 0, any integrable system of type (p, q) will be locally geometrically equivalent to the rectified system (X1 = ∂/∂x1 , . . . , Xp = ∂/∂xp , F1 = xp+1 , . . . , Fq = xm ). The question about the local structure of integrable systems becomes interesting only at singular points. Linearization and rigidity of non-degenerate singular points. The theorems presented in this subsubsection are from the paper [88]. Due to resonances, it is impossible to linearize integrable vector fields in general (if there were no resonance relations, there would be no formal first integral either). So, the following result about geometric linearization, i.e., linearization up to geometric equivalence, is the best that one can hope for in the case of analytic integrable systems. Theorem 3.2.20. An analytic (real or complex) integrable system near a nondegenerate fixed point is locally geometrically equivalent to a non-degenerate linear integrable system, namely its linear part. Proof. The above theorem is a simple consequence of Theorem 3.2.14 about the existence of an effective analytic torus Tp -action in the neighborhood of a nondegenerate fixed point of an integrable system of type (p, q) (because the toric degree is equal to p in this case), and the fact that non-degenerate linear integrable systems of type (p, the same as linear Tp -actions. √ √ q) are essentially Indeed, let −1Z1 , . . . , −1Zp be the generators of the associated analytic torus action near a non-degenerate fixed point O of an analytic integrable system (X1 , . . . , Xp , F1 , . . . , Fq ). We can assume that the torus action is already linearized, i.e., Z1 , . . . , Zp are linear vector fields. Since, for every i, Zi is tangent to the level sets of (F1 , . . . , Fq ), and the tangent space to such a level set at a generic point is

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spanned by X1 , . . . , Xp , we have that Zi ∧ X1 ∧ · · · ∧ Xp = 0, for all i = 1, . . . , p. Since Z1 , . . . , Zp are independent, by dimensional consideration, the inverse is also ∧· · ·∧Zp = 0 for all i = 1, . . . , p. Lemma 3.2.21 below says that we can true: Xi ∧Z1 write Xi = j fij Zj in a unique way, where fij are local analytic functions, which are also first integrals of the system. The fact that the matrix (fij ) is invertible, i.e., it has non-zero determinant at O, is also clear because (Z1 , . . . , Zp ) is nothing but a linear transformation of the linear part of (X1 , . . . , Xp ). What we have proved is that, near a non-degenerate fixed point, an integrable system is geometrically equivalent to its linear part, at least in the complex analytic case. In the real analytic case, the vector fields (Z1 , . . . , Zp ) are not real in general, but the proof will remain the same after a complexification, because the Poincar´e– Dulac normalization in the real case can be chosen to be real.  Lemma 3.2.21 (Division lemma). If (Y1 , . . . , Yp , G1 , . . . , Gq ) is a non-degenerate linear integrable system, and X is a local analytic vector field which commutes  f i Yi with Y1 , . . . , Yp and such that X ∧ Y1 ∧ · · · ∧ Yp = 0, then we can write X = in a unique way, where fi are local analytic functions which are common first integrals of Y1 , . . . , Yp .  Proof. Without loss of generality, we may assume that Yi = j cij Zj , where cij are integers,  and Zi = xi ∂/∂xi in some coordinate system (x1 , . . . , xm ). We will write gi X = i gi Zi , where xi gi are analytic functions. The main point is to prove  that αi are analytic functions, and the rest of the lemma will follow easily. Let x be  αi i i a polynomial first integral of the linear system. Then we also mhave X( i xi ) = 0,  which implies that i αi gi = 0. If α1 = 1, then x1 g1 = (− i=2 x1 gi )/α1 vanishes g1 is analytic. Thus, when x1 = 0, and so x1 g1 is divisible by x1 , which means  that i such that αi = 0, then for each i, if we can choose a monomial first integral i xα i  i gi is analytic. Assume now that all monomial first integrals i xα i must have α1 = 0. This means that all the first integrals are also invariant under the vector field Z1 = x1 ∂/∂x1 . Then Z1 must be a linear combination of Y1 , . . . , Yp (because the system is already “complete” and one cannot add another independent commuting vector field to it), and we have [Z1 , X] = 0. From this relation it follows easily  that g1 is also analytic in this case. Thus, all functions gi are analytic. Theorem 3.2.20 can be extended to the case of non-fixed non-degenerate singular points in an obvious way, with the same proof, using the toric characterization of local normalizations of vector fields. Theorem 3.2.22. Any analytic integrable dynamical system near a non-degenerate singular point is locally geometrically equivalent to a direct product of a linear non-degenerate integrable system and a constant (regular) integrable system. Another related result is the following deformation rigidity theorem for nondegenerate singular points.

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Theorem 3.2.23. Let (X1,θ , . . . , Xp,θ , F1,θ , . . . , Fq,θ ) be an analytic family of integrable systems of type (p, q) depending on a parameter θ which can be multidimensional: θ = (θ1 , . . . , θs ), and assume that z0 is a non-degenerate fixed point when θ = 0. Then there exists a local analytic family of fixed points zθ , such that zθ is a fixed point of (X1,θ , . . . , Xp,θ , F1,θ , . . . , Fq,θ ) for each θ, and moreover, up to geometric equivalence, the local structure of (X1,θ , . . . , Xp,θ , F1,θ , . . . , Fq,θ ) at zθ does not depend on θ. Proof. We can put the integrable systems in this family together to get one “big” integrable system of type (p, q + s), with the last coordinates xm+1 , . . . , xm+s as additional first integrals. Then z0 is still a non-degenerate fixed point for this big integrable system, and we can apply Theorem (3.2.20) to get the desired result.  We also have an extension of Ito’s theorem [44] to the non-Hamiltonian case. Ito’s theorem says that an analytic integrable Hamiltonian system at a non-resonant singular point (without the requirement of non-degeneracy of the momentum map at that point) can also be locally geometrically linearized (i.e., locally, one can choose the momentum map so that the system becomes non-degenerate and geometrically linearizable). For Hamiltonian vector fields, there are many autoresonances due to their Hamiltonian nature, which are not counted as resonance in the Hamiltonian case. So, in the non-Hamiltonian case, we have to replace the adjective “non-resonant” by “minimally-resonant”. Definition 3.2.24. A vector field X in a integrable dynamical system (X1 = X, . . . , Xp , F1 , . . . , Fq ) of type (p, q) is called minimally resonant at a singular point z if its toric degree at z is equal to p (maximal possible). Theorem 3.2.25. Minimally-resonant singular points of analytic integrable systems are also locally geometrically linearizable in the sense that one can change the auxiliary commuting vector fields (keeping the first vector field and the functions intact) in order to obtain a new integrable system which is locally geometrically linearizable. The proof of this result is also similar to the proof of Theorem 3.2.20 and is a direct consequence of the toric characterization of the Poincar´e–Birkhoff normalization. Geometric linearization in the smooth case. In the smooth case, we still have the same definitions of linear part, geometric equivalence, non-degeneracy and geometric linearization, as in the analytic case. We have the following conjecture, which is the smooth version of Theorem 3.2.22. Conjecture 3.2.26. Any smooth integrable dynamical system near a non-degenerate singular point is locally geometrically smoothly equivalent to a direct product of a linear non-degenerate integrable system and a constant system.

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As of this writing, the above conjecture is still open in the general case. The smooth case is much more complicated than the analytic case, because when the real toric degree is smaller than the toric degree, one cannot complexify a smooth system to find the torus action (whose dimension is equal to the toric degree) in general. Nevertheless, we know that the conjecture is true in the following particular cases: (i) Hamiltonian systems. The smooth linearization theorem for non-degenerate singular points of smooth integrable Hamiltonian systems was proved by Eliasson [32, 33]. Strictly speaking, Eliasson [32, 33] wrote down only a sketch of the proof in the case of non-elliptic singularities, though all the main ingredients were there. See also [19, 30, 59, 75, 81] for details and other methods of proof. For elliptic singularities of integrable Hamiltonian systems, one can also use the toric characterization to prove the local geometric linearization theorem, like in the analytic case. (For hyperbolic singularities the situation is more complicated, because in general one cannot complexify a smooth system in order to find a torus action.) (ii) Systems of type (m, 0), i.e., a family of m independent commuting vector fields on a m-dimensional manifold. In this case, we have: Theorem 3.2.27. [88] Let O be a non-degenerate singular point of a rank k smooth integrable system (X1 , . . . , Xm ) of type (m, 0). Then, there exists a smooth local coordinate system (x1 , . . . , xm ) in a neighborhood of O, nonnegative integers h, e ≥ 0 (which do not depend on the choice of coordinates) such that h + 2e = m − k, and a real invertible n × n matrix (vij ) such that n the vector fields Yi = j=1 vij Xj have the following form: ⎧ ∂ ⎪ , i = 1, . . . , h, Yi = xi ∂x ⎪ i ⎪ ⎪ ∂ ∂ ⎨Y h+2j−1 = xh+2j−1 ∂xh+2j−1 + xh+2j ∂xh+2j , ∂ ∂ ⎪ Yh+2j = xh+2j−1 ∂xh+2j − xh+2j ∂xh+2j−1 , j = 1, . . . , e, ⎪ ⎪ ⎪ ⎩Y = ∂ , i = m − k + 1, . . . , m. i

∂xi

One can prove this theorem using the following arguments, which are due to a referee of the paper [88]. In the case of  a fixed point, the linear fields part of an appropriate linear combination E = ai Xi of the vector n X1 , . . . , Xm is a radial vector field, i.e., has the form E (1) = i=1 xi ∂/∂xi . By Sternberg’s theorem, E is smoothly linearizable, i.e., we can assume that m Since the E = i=1 xi ∂/∂xi after a smooth change of the coordinate system. m vector fields Xi commute with the radial vector field E = i=1 xi ∂/∂xi , they are automatically linear in the new coordinate system. The case of a singular point of positive rank k > 0 can be reduced to the case of a fixed point, by considering the (m − k)-dimensional isotropy algebra of the infinitesimal Rm -action generated by X1 , . . . , Xm at the singular point, and showing the

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Chapter 3. Geometry of Integrable non-Hamiltonian Systems existence of a (m − k)-dimensional invariant submanifold of the subaction of this isotropy algebra, which is transverse to the local orbit through the singular point of the Rm -action.

(iii) Systems of type (1, m − 1), i.e., a vector field with a complete set of first integrals. In this case we have Theorem 3.2.28. [86] Let (X, F1 , . . . , Fn−1 ) be a smooth integrable system of type (1, n−1) with a fixed point O which satisfies the following non-degeneracy conditions: (i) the semisimple part of the linear part of X at O is non-zero, and the ∞-jets of F1 , . . . , Fn−1 at O are functionally independent; (ii) if, moreover, 0 is an eigenvalue of X at O with multiplicity k ≥ 1, then the differentials of the functions F1 , . . . , Fk are linearly independent at O, i.e., dF1 (O) ∧ · · · ∧ dFk (O) = 0. Then there exists a local smooth coordinate system (x1 , . . . , xn ) in which X can be written as X = F X (1) , where X (1) is a semisimple linear vector field in (x1 , . . . , xn ), and F is a smooth first integral of X (1) such that F (O) = 0. The nondegeneracy condition (ii) in the above theorem is conjectured to be superfluous. (iv) Some low-dimensional cases. In particular, the case of non-Hamiltonian focus-focus singular points of smooth integrable systems of type (2, 2) was studied by Jiang Kai (unpublished, talk presented in Barcelona in 09/2013). Let us indicate now why do we believe that Conjecture 3.2.26 is true, and some methods which could be used to prove it: (1) By geometric arguments similar to the ones used in [78, 80, 84], we can show the existence of a smooth torus Td -action which preserves the system, where d is the real toric degree of the system. Up to geometric equivalence, we can also assume that the vector fields which generate this torus action are part of our system. The remaining vector fields of the system are hyperbolic and invariant under this smooth torus action. (2) Theorem 3.2.20 is also true in the formal case with the same proof, and so we can apply a formal linearization to our smooth system. Together with Borel’s theorem, this implies that there is a local smooth coordinate system in which our system is already geometrically linear up to a flat term. (3) After the above step (2), one can try to use results and techniques on finite determinacy of mappings a` la Mather [56] to find a matrix whose entries are smooth first integrals, such that when multiplying this matrix by our vector fields, we obtain a new geometrically equivalent system whose vectors are linear, plus flat terms.

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(4) One can now try to invoke an equivariant version of the Sternberg–Chen theorem [21, 70], due to Belitskii and Kopanskii [7], which says that smooth equivariant hyperbolic vector fields which are formally linearizable are also smoothly equivariantly linearizable. Of course, we will have to do it simultaneously for all commuting hyperbolic vector fields. So we need an extension of the result of Belitskii and Kopanskii to the situation of a smooth Rk -action with some hyperbolicity property which is formally linear. Maybe we would also need Belitskii–Kopanskii–Sternberg–Chen type result for vector fields having first integrals. (5) The results and techniques of Chaperon [18, 20] for the smooth linearization of (Rk × Zh )-actions may be very useful here. Of course, techniques from the integrable Hamiltonian case, e.g., [24, 30, 33], in particular division lemmas for non-degenerate smooth systems, can probably be extended to the nonHamiltonian case as well. (6) Geometrically, at least at the linear level, via a spectral decomposition, a complicated non-degenerate singular point can be decomposed into a direct product of its indecomposable components. For example, in the Hamiltonian case, there are only three kinds of indecomposable non-degenerate singular components, namely two-dimensional elliptic, two-dimensional hyperbolic, and four-dimensional focus-focus (see, e.g., [78]). One can try to reduce the linearization problem for a complicated singular point to the decomposition problem plus the linearization problem for each of its components.

3.2.4 Semi-local torus actions and normal forms Consider a level set of an integrable system (X1 , . . . , Xp , F1 , . . . , Fq ) of type (p, q) on a manifold M , say N = {F1 = c1 , . . . , Fq = cq }. We will assume that N is connected compact, but the Liouville’s theorem fails for N , because N contains a singular point of the system. The set N is partitioned into the orbits of the Rp -action generated by the commuting vector fields X1 , . . . , Xp . Under some mild conditions, N will contain a compact orbit O of this Rp -action. A natural question arises: does there exist a natural associated torus action near N or near O, which preserves the system and which is transitive on O? We know that the answer is yes, at least in the case of non-degenerate singularities. For example, in the case of integrable Hamiltonian systems, it has been shown in [78] that, if N is a connected compact non-degenerate singular level set of rank k (i.e., dim O = k, where O is a compact orbit of the system in N ), there exists a Hamiltonian torus Tk -action in a neigborhood of N which preserves the system and which is transitive on O. A semi-local normal form (linearization) theorem near a compact non-degenerate singular orbit O of a smooth integrable Hamiltonian system was obtained by Miranda and Zung [60], based on this torus action and on the virtual commutativity of the automorphism group. The Miranda–Zung linearization theorem [60] can probably be extended to the case of non-degenerate

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compact singular orbits of integrable non-Hamiltonian systems (modulo Conjecture 3.2.26). The non-degeneracy condition is a natural condition and most singularities are non-degenerate. However, there are also degenerate singularities, and one is also interested in the existence of associated torus actions for such singularities. It turns out that, in the real-analytic case, the so-called finite type condition, which is much weaker than the nondegeneracy condition, suffices. (In a “reasonable” system, all degenerate singularities will be of finite type.) To formulate this condition, denote by MC a small open complexification of our manifold M m on which the complexification XC , FC of X and F exists, where X = (X1 , . . . , Xp ) denotes the p-tuple of vector fields and F = (F1 , . . . , Fq ) : M → Rq denote the q-vector valued first integral of our integrable system. Denote by NC a connected component of F−1 C (F(O)) which contains N . Definition 3.2.29. With the above notations, the singular orbit O is called of finite type if there is only a finite number of orbits of the infinitesimal Cp -action generated by XC in NC , and NC contains a regular point of the map FC . Theorem 3.2.30 ([82]). With the above notations, if O is a compact finite-type singular orbit of dimension r, then there is a real analytic torus action of Tr in a neighborhood of O which preserves the integrable system (X, F), and which is transitive on O. If, moreover, N is compact, then this torus action exists in a neighborhood of N . The following theorem is a result very close to the above one, and its proof also uses the same ingredients. It is about the local automorphism group of an integrable system near a compact singular orbit. Denote by AO the local automorphism group of the integrable system (X, F) at O, i.e., the group of germs of local analytic diffeomorphisms in a neigborhood of O which preserve X and F. 1 , where Denote by A0O the subgroup of AO consisting of elements of the type gZ Z is an analytic vector field in a neighborhood of O which preserves the system, 1 is the time-1 map of the flow of Z. The torus in the previous theorem is, and gZ of course, an Abelian subgroup of A0O . Theorem 3.2.31 ([82]). If O is a compact finite-type singular orbit as above, then A0O is an Abelian normal subgroup of AO , and AO /A0O is a finite group. Theorem 3.2.30 reduces the study of the behavior of integrable systems near compact singular orbits to the study of fixed points with a finite Abelian group of symmetry (this group arises from the fact that the torus action is not free in general, only locally free). For example, as was shown in [79], the study of corank one singularities of Liouville-integrable systems is reduced to the study of families of functions on a two-dimensional symplectic disk which are invariant under the rotation action of a finite cyclic group Z/Zk , where one can apply the theory of singularities of functions with an Abelian symmetry developed by Wassermann [76] and other people. A (partial) classification up to diffeomorphisms of corank

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one degenerate singularities was obtained by Kalashnikov [50] (see also [40, 79]), and symplectic invariants were obtained by Colin de Verdi`ere [23]. Notice also that Theorem 3.2.30, together with Theorem 3.2.12 and the toric characterization of Poincar´e–Birkhoff normalization, provides an analytic Poincar´e–Birkhoff normal form in the neighborhood of a singular invariant torus of an integrable system. More precisely, with the above notations, we can formulate Theorem 3.2.32 below. First, make a reduction (i.e., quotient) of the system near O with respect to the torus action in Theorem 3.2.30. Then, the vector fields of the reduced system vanishes at the image of O under the reduction, and so we can talk about the toric  degree of this reduced system: it is equal to the toric deq gree of the reduction of i=1 ai Xi , where the numbers ai are in generic position. Denote this reduced toric number by t. Theorem 3.2.32. With the above notations, in a neighborhood of O in the complexified manifold MC , there is a natural effective analytic torus Tt+dim O -action which preserves the system (XC , FC ), and which leaves O invariant and is transitive on O. The torus Tdim O -action in Theorem 3.2.30 is a subaction of this torus action. Moreover, this torus action has the structure-preserving property: it preserves every tensor field which is preserved by the system. The proof of the above theorem has not been written down explicitly anywhere: it is left to the reader as an exercise.

3.3 Geometry of integrable systems of type (n, 0) Given an integrable system (X1 , . . . , Xp , F1 , . . . , Fq ) of any type (p, q), if we fix the values of the first integrals F1 = c1 , . . . , Fq = cq , then we get an invariant (possibly singular) p-dimensional manifold with an integrable system of type (p, 0) on it, i.e., only vector fields and no function. Thus, in order to understand the topology of integrable systems of type (p, q) in general, we need to understand systems of type (p, 0). Such systems will be studied in this section, where we will use the letter n (which denotes the dimension of the manifold) instead of p. Our main reference for this section is a recent paper with Nguyen Van Minh [91] (which won an outstanding paper award of Journal of Math. Soc. Japan in 2015). We refer to it for various details and proofs that will be omitted here. Recall that a smooth integrable system of type (n, 0) means an n-tuple of commuting smooth vector fields X1 , . . . , Xn on a n-dimensional manifold M n . We will always assume in this section that the system (X1 , . . . , Xn ) is non-degenerate, i.e., every singular point of it is non-degenerate. Moreover, we will assume that the commuting vector fields X1 , . . . , Xn are complete, i.e., they generate an action of Rn on M n , which we will denote by ρ : Rn × M n → M n . We will say that ρ is a non-degenerate Rn action on M n , and that X1 , . . . , Xn are the generators of ρ. Instead of talking about the system (X1 , . . . , Xn ), we will talk about the Rn -action ρ, which is the same thing. If a point z ∈ M n is of rank

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k with respect to the system (X1 , . . . , Xn ), then it is also of rank k with respect to ρ, in the sense that the orbit Oz of ρ through z is of dimension k. 1 n n n n If iv = (v , . . . , v ) ∈ R is a non-trivial element of R , then we put Xv = i=1 v Xi , and call Xv the generator of the action ρ associated to v. If v1 , . . . , vn ∈ Rn form a basis of Rn , then the vector fields Xv1 , . . . , Xvn also generate the same Rn -action as ρ, up to an automorphism of Rn . We will study the global geometry of non-degenerate Rn -actions on n-manifolds. But first, we need some local and semi-local normal form results.

3.3.1 Normal forms and automorphism groups Local normal forms and adapted bases. Recall from Theorem 3.2.27 that, if z is a singular point of rank k of a non-degenerate smooth action ρ : Rn × M n → M n generated by commuting vector fields X1 , . . . , Xn , then there is a smooth local coordinate system (x1 , . . . , xn ) in a neighborhood of z, and a basis (v1 , . . . , vn ) of Rn such that, locally, we have ⎧ ∂ ⎪ , i = 1, . . . , h, Xvi = xi ∂x ⎪ i ⎪ ⎪ ∂ ∂ ⎨X = x vh+2j−1 h+2j−1 ∂xh+2j−1 + xh+2j ∂xh+2j , (3.8) ∂ ∂ ⎪ ⎪Xvh+2j = xh+2j−1 ∂xh+2j − xh+2j ∂xh+2j−1 , j = 1, . . . , e, ⎪ ⎪ ⎩X = ∂ , i = n − k + 1, . . . , n, vi

where Xvi =

∂xi

n j=1

vij Xj is the generator of ρ associated to vi for i = 1, . . . , n.

Figure 3.1: Elbolic, hyperbolic, and regular components of Rn -actions on nmanifolds. The couple (h, e) in the above formula does not depend on the choice of coordinates and bases, and is called the HE-invariant of the action ρ at z. The

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number e is called the number of elbolic components, and h is called the number of hyperbolic components at z. The coordinate system (x1 , . . . , xn ) in the above formula is called a local canonical system of coordinates, and the basis (v1 , . . . , vn ) of Rn is called an adapted basis of the action ρ at p. Local canonical coordinate systems at a point p and associated adapted bases of Rn are not unique, but they are related to each other by the following theorem. Theorem 3.3.1. Let (x1 , . . . , xn ) be a canonical system of coordinates at a point z of a non-degenerate action ρ together with an associated adapted basis (v1 , . . . , vn ) of Rn . Let (y1 , . . . , yn ) be another canonical system of coordinates at z together with an associated adapted basis (w1 , . . . , wn ) of Rn . Then we have: (i) the vectors (v1 , . . . , vh ) are the same as the vectors (w1 , . . . , wh ) up to permutations, where h is number of hyperbolic components; (ii) the e-tuple of pairs of vectors ((vh+1 , vh+2 ), . . . , (vh+2e−1 , vh+2e )) is also the same as the e-tuple ((wh+1 , wh+2 ), . . . , (wh+2e−1 , wh+2e )) up to permutations and changes of sign of the type (vh+2i−1 , vh+2i ) −→ (vh+2i−1 , −vh+2i ) (only the second vector, the one whose corresponding generator of ρ is a vector field with 2π-periodic flow, changes sign); (iii) conversely, if (x1 , . . . , xn ) and (v1 , . . . , vn ) are as in formula (3.8), and (w1 , . . . , wn ) is another basis of Rn which satisfies the above conditions (i) and (ii), then (w1 , . . . , wn ) is the adapted basis of Rn for another canonical system of coordinates (y1 , . . . , yn ) at z. Remark 3.3.2. The fact that the last vectors (from wh+2e+1 to wn ) in an adapted basis can be arbitrary (provided that they form together with w1 , . . . , wh+2e a basis of Rn ) is very important in the global picture, because it allows us to glue different local canonical pieces together in a flexible way. It follows immediately from the local normal form formula (3.8) that the singular set S = {x ∈ M n | rank x < n} of a non-degenerate action ρ : Rn ×M n → M n is a stratified manifold, whose strata are Sh,e = {x ∈ M n | HE-invariant of x is (h, e)}, given by the HE-invariant, and dim Sh,e = n − h − 2e if Sh,e = ∅. If S = ∅, then dim S = n − 1 or dim S = n − 2. When there are singular points with hyperbolic components, then dim S = n − 1, and when there are only elbolic singularities (h = 0 for every point), then dim S = n − 2. Definition 3.3.3. If Oz is a singular orbit of corank 1 of a non-degenerate action ρ : Rn × M n → M n , i.e., the HE-invariant of Op is (1, 0), then the unique vector v ∈ Rn such that the corresponding generator Xv of ρ can be written as Xv = x∂/∂x near each point of Oz is called the associated vector of Oz .

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If Oz is a singular orbit of HE-invariant (0, 1) (i.e., corank 2 transversally elbolic), then the couple of vectors (v1 , ±v2 ) in Rn , where v2 is determined only up to a sign, such that Xv1 and Xv2 can be locally written as 

∂ ∂ + y ∂y , Xv1 = x ∂x ∂ ∂ Xv2 = x ∂y − y ∂x ,

is called the associated vector couple of Oz . Local automorphism groups and the reflection principle. Theorem 3.3.4. Let z be a non-degenerate singular point of HE-invariant (h, e) and rank r of an action ρ : Rn × M n → M n (n = h + 2e + r). Then, the group of germs of local isomorphisms (i.e., local diffeomorphisms which preserve the action) which fix the point z is isomorphic to Te × Re+h × (Z2 )h . The part Te × Re+h of this group comes from the action ρ itself (internal automorphisms given by the action of the isotropy group of ρ at z). The finite automorphism group (Z2 )k in the above theorem acts not only locally in the neighborhood of a singular point p of HE-invariant (h, e), but also in the neighborhood of a smooth closed manifold of dimension n − h − 2e which contains p. More precisely, we have the following reflection principle, which is somewhat similar to the Schwarz reflection principle in complex analysis. Theorem 3.3.5 (Reflection principle). (i) Let z be a point of HE-invariant (1, 0) of a non-degenerate Rn -action ρ on a manifold M n without boundary. Denote by v ∈ Rn the associated vector of z (i.e., of the orbit Oz ) as in Definition 3.3.3. Put Nv = {y ∈ M n | Xv (y) = 0 and Xv can be written as x1

∂ near y}. ∂x1

Then Nv is a smooth embedded hypersurface of dimension n−1 of M n (which is not necessarily connected), and there is a unique non-trivial involution σv : U (Nv ) → U(Nv ) from a neighborhood of Nv to itself which preserves the action ρ and which is the identity on Nv . (ii) If the HE-invariant of z is (h, 0) with h > 1, then we can write z ∈ Nv1 ,...,vh = Nv1 ∩ · · · ∩ Nvh , where Nvi are defined as in (i), (v1 , . . . , vh ) is a free family of vectors in Rn , the intersection Nv1 ∩ · · · ∩ Nvh is transversal and Nv1 ,...,vh is a closed smooth submanifold of codimension h in M . The involutions σv1 , . . . , σvh generate a group of automorphisms of (U (Nv1 ,...,vh ), ρ) isomorphic to (Z2 )h .

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Semi-local normal forms. Consider an orbit Oz = {ρ(t, z) | t ∈ Rn } though a point z ∈ M n of a given Rn -action ρ. Since Oz is a quotient of Rn , it is diffeomorphic to Rk × Tl for some nonnegative integers k, l ∈ Z+ . Definition 3.3.6. The HERT-invariant of an orbit Oz , or of a point z in it, is the quadruple (h, e, r, t), where h is the number of transversal hyperbolic components, e is the number of transversal elbolic components, and Rr × Tt is the diffeomorphism type of the orbit. An orbit is compact if and only if r = 0, in which case it is a torus of dimension t. We have the following linear model for a tubular neighborhood of a compact orbit of HERT-invariant (h, e, 0, t): • The orbit is {0} × {0} × Tt (Z2 )k , which lies in B h × B 2e × Tt (Z2 )k , where B h is a ball of dimension h, with coordinates (x1 , . . . , xh+2e ) on B h × B 2e and (z1 , . . . , zt ) mod 2π on Tt , and where k is some nonnegative integer such that k ≤ min(h, t). • The (infinitesimal) action of Rn is generated by the vector fields ⎧ ∂ ⎪ Yi = xi ∂x , i = 1, . . . , h, ⎪ i ⎪ ⎪ ∂ ∂ ⎨Y h+2j−1 = xh+2j−1 ∂xh+2j−1 + xh+2j ∂xh+2j , ∂ ∂ ⎪ − xh+2j ∂xh+2j−1 , j = 1, . . . , e, Yh+2j = xh+2j−1 ∂xh+2j ⎪ ⎪ ⎪ ⎩Y ∂ = i = 1, . . . , t, h+2e+i

∂zi

like in the local normal form theorem. • The Abelian group (Z2 )k acts on B h × B 2e × Tt freely, component-wise, and by isomorphisms of the action, so that the quotient is still a manifold with an induced action of Rn on it. The action of (Z2 )k on B h is, by an injection from (Z2 )k to the involution group (Z2 )h , generated by the reflections σi : (x1 , . . . , xi , . . . , xh ) → (x1 , . . . , −xi , . . . , xh ), its action on B 2e is trivial, and its action on Tt is via an injection of (Z2 )k into the group of translations on Tt . Theorem 3.3.7. Any compact orbit of a non-degenerate action ρ : Rn × M n → M n can be linearized, i.e., there is a tubular neighborhood of it which is, together with the action ρ, isomorphic to the linear model described above. More generally, for any point z lying in an orbit Oz of a HERT-invariant (h, e, r, t) which is not necessarily compact (i.e., the number r may be strictly positive), we still have the following linear model: • The intersection of the orbit with the manifold is {0} × {0} × Tt (Z2 )k × B r , which lies in (B h × B 2e × Tt (Z2 )k ) × B r , with coordinates (x1 , . . . , xh+2e ) on B h × B 2e , (z1 , . . . , zt ) mod 2π on Tt , and ζ1 , . . . , ζr on B r , and where k is some nonnegative integer such that k ≤ min(h, t).

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• The action of Rn is generated by the vector fields ⎧ ∂ , i = 1, . . . , h, Yi = xi ∂x ⎪ ⎪ i ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎨Yh+2j−1 = xh+2j−1 ∂xh+2j−1 + xh+2j ∂xh+2j , ∂ ∂ Yh+2j = xh+2j−1 ∂xh+2j − xh+2j ∂xh+2j−1 , ⎪ ⎪ ∂ ⎪ ⎪ Yh+2e+i = ∂zi , i = 1, . . . , t, ⎪ ⎪ ⎩ Yh+2e+t+i = ∂ζ∂ i , i = 1, . . . , r.

∀ j = 1, . . . , e,

• The Abelian group (Z2 )k acts on Rh × R2e × Tt freely in the same way as in the case of a compact orbit. Theorem 3.3.8. Any point q of any HERT-invariant (h, e, r, t) with respect to a non-degenerate action ρ : Rn × M n → M n admits a neighborhood which is isomorphic to the linear model described above. Theorem 3.3.8 it simply a parametrized version of Theorem 3.3.7, and can also be seen as a corollary of Theorem 3.3.7. Remark 3.3.9. The difference between the compact case and the non-compact case is that, when Oq is a compact orbit, we have a linear model for a whole tubular neighborhood of it, whereas when Oq is non-compact we have a linear model only for a neighborhood of a “stripe” in Oq . The twisting groups. The minimal required group (Z2 )k in Theorem 3.3.7 and Theorem 3.3.8 is naturally isomorphic to the group Gq = (Zρ (q) ∩ (Zρ ⊗ R))/Zρ .

(3.9)

Definition 3.3.10. The group Gq defined by the above formula is called the twisting group of the action ρ at q (or at the orbit Oq ). The orbit Oq is said to be nontwisted (and ρ is said to be non-twisted at q) if Gq is trivial, otherwise it is said to be twisted. Remark 3.3.11. The twisting phenomenon also appears in real-world physical integrable Hamiltonian systems, and it was observed, for example, by Fomenko and his collaborators in their study of integrable Hamiltonian systems with two degrees of freedom; see, e.g., [11].

3.3.2 Induced torus action and reduction The toric degree.

Given a non-degenerate action ρ : Rn × M n → M n , denote by Zρ = {g ∈ Rn | ρ(g, ·) = IdM n }

the isotropy group of ρ on M n . Since ρ is locally free almost everywhere due to its non-degeneracy, Zρ is a discrete subgroup of Rn , so we have Zρ ∼ = Zk for some

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115

integer k such that 0 ≤ k ≤ n. The action ρ of Rn descends to an action of Rn /Zρ ∼ = Tk × Rn−k on M , which we will also denote by ρ : (Rn /Zρ ) × M n → M n . We will denote by (3.10) ρT : Tk × M n −→ M n the subaction of ρ given by the subgroup Tk ⊂ Tk ×Rn−k ∼ = Rn /Zρ . More precisely, n ρT is the action of (Zρ ⊗ R)/Zρ on M induced from ρ, which becomes a Tk -action after an isomorphism from (Zρ ⊗ R)/Zρ to Tk . We will call ρT the induced torus action of ρ. Definition 3.3.12. The number k = rank Z Zρ is called the toric degree of the action ρ. If the toric degree is equal to 0, then ρ is called a totally hyperbolic action. Remark 3.3.13. If M n admits an Rn -action of toric degree k then, in particular, it must admit an effective Tk -action. When k ≥ 1, this is a strong topological condition. For example, Fintushel [37] showed (modulo Poincar´e’s conjecture which is now a theorem) that among the simply-connected four-manifolds, only the manifolds S4 , CP2 , −CP2 , S2 × S2 and their connected sums admit an effective locally smooth T1 -action. This list is the same as the list of simplyconnected four-manifolds admitting an effective T2 -action, according to Orlik and Raymond [63, 64]. A classification of non-simply-connected 4-manifolds admitting an effective T2 -action can be found in Pao [65]. Even though the toric degree is a global invariant of the action, it can in fact be determined semi-locally from the HERT-invariant of any point on M with respect to the action. More precisely, we have the following result. Theorem 3.3.14. Let ρ : Rn × M n → M n be a non-degenerate smooth action of Rn on a n-dimensional manifold M n , and let p ∈ M be an arbitrary point of M . If the HERT-invariant of p with respect to ρ is (h, e, r, t), then the toric degree of ρ on M is equal to e + t. Proof. See [91] for the proof. It consists of the following five steps, and each step is based on relatively simple topological arguments: (i) If z ∈ M is a regular point, then toric degree(ρ) ≤ t(z). (ii) If O1 and O2 are two arbitrary different regular orbits, then Zρ (O1 ) = Zρ (O2 ), where Zρ (O) ⊂ Rn denotes the isotropy group of ρ on O. (iii) Zρ = Zρ (O) for any regular orbit O; in particular, for any regular point z, the toric rank of ρ is equal to t(z), and e(z) = h(z) = 0. (iv) If z ∈ M n is a singular point then e(z) + t(z) ≥ toric degree (ρ). (v) The converse is also true: e(z) + t(z) ≤ toric degree (ρ).



The simplest case of non-degenerate systems of type (n, 0) is when the toric degree of ρ is equal to n. This is a special case of Liouville’s theorem: we have an effective action of Tn on M n , and M n itself is diffeomorphic to the torus Tn .

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Quotient space and reduced action. In general, if the toric degree t(ρ) is positive, i.e., if the action ρ : Rn × M n → M n is not totally hyperbolic, then it naturally projects down to an action of Rn /(Zρ ⊗ R) ∼ = Rr(ρ) on the quotient n n space Q = M /ρT of M by the induced torus action ρT , which we will denote by ρR : Rr(ρ) × Q → Q, after an identification of Rn /(Zρ ⊗ R) with Rr(ρ) . Here r(ρ) = dim Rn /(Zρ ⊗ R) = n − t(ρ) = dim Q. We will call ρR the reduced action of ρ. There is a small technical problem. Namely, due to the singularities and the twisting groups, in general the quotient space Q is not a manifold, but an orbifold with boundary and corners. More precisely, it follows from the normal form theorems that, for every point z ∈ Q, locally a neigborhood of z in Q is diffeomorphic to a direct product of the type (D12 /T11 ) × · · · × (De2 /T1e ) × (B h /Gz ) × B r , where (h, e, r, t) is the HERT invariant and Gz is the twisting group of z (i.e., of any point in M n whose image under the projection M n → Q is z), B r and B h are balls of dimensions r and h, respectively, and each Di2 /T1i is a half-closed interval obtained as the quotient of a two-dimensional disk Di2 by the standard rotational action of SO(2) ∼ = T1 . Due to this fact, we have to extend the notion of non-degenerate Rr -actions to the case of orbifolds: it simply means that, locally, we have a non-degenerate (infinitesimal) Rr -action on a local branched covering space, which is a manifold together with a finite group action on it so that the quotient by that finite group action is our local orbifold, and we require that the Rr -action commutes with the finite group action so that it can be projected to an Rr -action on the orbifold. In the case with boundary and corners, the boundary components are singular orbits of the action. The notions of toric degree can be naturally extended to the case of actions on orbifolds with boundary and corners, and if the toric degree is 0 we will also say that the action is totally hyperbolic. With this in mind, we have the following reduction theorem. Theorem 3.3.15. Let ρ : Rn × M n → M n be a non-degenerate action of toric degree t(ρ) on a connected manifold M n , and put r = r(ρ) = n − t(ρ). Then the quotient space Q = M n /ρT of M n by the associated torus action ρT is an orbifold of dimension r, and the reduced action ρR of Rn /(Zρ ⊗ R) ∼ = Rr on Q is totally hyperbolic. If the twisting group Gz is trivial for every point z ∈ M n , then Q is a manifold with boundary and corners. Cross multi-sections and reconstruction. In the case when (M n , ρ) has no twistings, Q is a manifold with boundary and corners, and one can talk about cross Tt(ρ)

sections of the singular torus fibration M n −→ Q over Q. We will say that an embedded submanifold with boundary and corners Qc ⊂ M n is a smooth cross section of the singular fibration M n → Q if the projection map proj . : Qc → Q is

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a diffeomorphism. The existence of a cross section is equivalent to the fact that the desingularization via blowing up of M n → Q is a trivial principal Tt(ρ) -bundle. (The blow-up process here does not change the quotient space of the action ρT on M n , but changes every singular orbit of ρT into a regular orbit, and changes M n into a manifold with boundary and corners, see Figure 3.2 for an illustration. This blow-up process is a standard one, and it was used for example by Dufour and Molino [30] in the construction of action-angle variables near elliptic singularities of integrable Hamiltonian systems).

Figure 3.2: Desingularization of M n → Q by blowing up. In the case when (M n , ρ) has twistings, a priori Q is only an orbifold and we cannot have a submanifold Qc in M n diffeomorphic to Q. In this case, instead of sections, one can talk about multi-sections. A smooth multi-section of M n → Q is a smooth embedded submanifold with boundary and corners Qc in M n , together with a finite subgroup G ⊂ (Zρ ⊗ R)/Zρ such that Qc is invariant with respect to G (i.e., if z ∈ Qc and w ∈ G, then ρ(w, z) ∈ Qc ), and Qc /G ∼ = Q via the projection. Remark that multi-sections also appear in many other places in the literature. For example, Davis and Januskiewicz [28] used them in their study of quasi-toric manifolds. They were also used in [78] in the construction of partial action-angle coordinates near singular fibers of integrable Hamiltonian systems. Proposition 3.3.16. (i) If (M n , ρ) has no twistings, then the singular torus fibration M n → M n /ρT = Q admits a smooth cross section Qc . (ii) If (M n , ρ) has twistings, then the singular torus fibration M n → M n /ρT = Q admits a smooth multi-section (Qc , G), where G ⊂ (Zρ ⊗ R)/Zρ is generated by the twisting groups Gz (z ∈ M ) of (M n , ρ). The first part of this proposition can be proved using sheaf theory, based on the existence of local cross sections and the contractibility of Q, which will be explained in Subsection 3.3.5. The second part follows from the first part and an appropriate covering of (M n , ρ). The cross (multi-)sections allow one to go back (i.e., reconstruct) from (Q, ρR ) to (M, ρ). In particular, we have the following result.

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Theorem 3.3.17. Assume that (M1n , ρ1 ) and (M2n , ρ2 ) have the same quotient space M1n /ρ1T = M2n /ρ2T = Q and, moreover, they have the same isotropy at every point of Q, i.e., Zρ1 (z) = Zρ2 (z) for all z ∈ Q, where Zρ1 (z) denotes the isotropy group of ρ1 on the ρ1T -orbit corresponding to z. Then ρ1T and ρ2T are isomorphic, i.e., there is a diffeomorphism Φ : M1n → M2n which sends ρ1T to ρ2T . Proof. Simply send a multi-section in M1n over Q to a multi-section in M2n over Q by a diffeomorphism which projects to the identity map on Q, and extend this diffeomorphism to the whole M1n in the unique equivariant way with respect to the associated torus actions. The fact that the isotropy groups are the same allows us to do so.  Beware that, even though the two torus actions ρ1T and ρ2T in the above theorem are isomorphic, and even if we assume that the two reduced actions ρ1R and ρ2R on Q are the same, it does not mean that ρ1 and ρ2 are isomorphic. The difference between the isomorphism classes of ρ1 and ρ2 can be measured in terms of an invariant called the monodromy, which will be explained in Subsection 3.3.4.

3.3.3 Systems of toric degree n − 1 and n − 2 The case of toric degree n − 1. Consider a non-degenerate action ρ of toric degree n − 1 on a compact connected manifold M n , and an orbit Op of this action, and denote by (h, e, r, t) the HERT-invariant of Op . According to Theorem 3.3.14, we have e + t = n − 1. On the other hand, the total dimension is n = h + 2e + r + t. These two equalities imply that h + e + r = 1, which means that one of the three numbers h, e, r is equal to 1 and the other two numbers are 0. So we have only three possibilities: (i) r = 1, h = e = 0, t = n − 1, and Op ∼ = Tn−1 × R is a regular orbit; the action n−1 on such an orbit is free with the orbit space diffeomorphic to an ρT of T open interval. (ii) r = e = 0, h = 1, t = n − 1, and Op ∼ = Tn−1 is a compact singular orbit of codimension one which is transversally hyperbolic; the action ρT of Tn−1 on such an orbit is locally free; it is either free (the non-twisted case) or have the isotropy group equal to Z2 (the twisted case). (iii) e = 1, h = r = 0, t = n − 2, and Op ∼ = Tn−2 is a compact singular orbit of codimension two which is transversally elbolic. The orbit space Q = M n /Tn−1 of the action ρT : Tn−1 × M n → M n is a compact one-dimensional manifold with or without boundary, on which we have the reduced R-action ρR . The singular points of this R-action on M n /Tn−1 correspond to the singular orbits of ρ. Since the toric degree is n − 1 and not n, and M is compact, ρ must have at least one singular orbit, and hence the quotient space Q = M n /Tn−1 contains at least one singular point. Topologically, Q must be a closed interval or a circle and, globally, we have the following four cases:

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Figure 3.3: The 4 cases of toric degree n − 1.

a) Q is a circle, which contains m > 0 hyperbolic points with respect to ρR . Notice that m is necessarily an even number, because the vector field which generates the hyperbolic R-action ρR on Q changes direction on adjacent regular intervals, see Figure 3.3a) for an illustration. The T(n−1) -action is free in this case, so M n is a Tn−1 -principal bundle over Q. Any homogeneous Tn−1 -principal bundle over a circle is trivial, so M n is diffeomorphic to Tn ∼ = Tn−1 × S1 in this case. d) Q is an interval, and each endpoint of Q corresponds to a transversally elbolic orbit of ρ. Topologically, in this case, the manifold M n can be obtained by gluing two copies of the solid torus D2 × Tn−2 together along the boundary. When n = 2, there is only one way to do it, and M is diffeomorphic to a sphere S2 . When n ≥ 3, the gluing can be classified by the homotopy class (up to conjugations) of the two vanishing cycles on the common boundary Tn−1 . When n = 3, the manifold M 3 is either S2 × S1 (if the two vanishing cycles are equal up to a sign) or a three-dimensional lens space. c) Q is an interval, one endpoint of Q corresponds to a twisted transversally hyperbolic orbit of ρ, and the other endpoint corresponds to a transversally elbolic orbit of ρ. Due to the twisting, the ambient manifold is non-orientable n , ρ) which belongs to  in this case. But (M n , ρ) admits a double covering (M 2 2 case b). If n = 2, then M = RP in this case. b) Q is an interval, and each endpoint of Q corresponds to a twisted transversally hyperbolic orbit of ρ. Again, in this case, M is non-orientable, but (M n , ρ) n , ρ) which is orientable and belongs to  admits a normal (Z2 )2 -covering (M 2 case a). If n = 2, then M is a Klein bottle in this case. We can classify actions of toric degree n − 1 on closed manifolds as follows. View Q as a non-oriented graph, with the singular points as vertices. Mark each vertex of Q with the vector or the vector couple in Rn associated to the corresponding orbit of ρ (in the sense of Definition 3.3.3). Then Q becomes a

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marked graph, which will be denoted by Qmarked . Note that Qmarked and the isotropy group Zρ ⊆ Rn are invariants of ρ, which satisfy the following conditions (Ci )-(Civ ): (Ci ) Q is homeomorphic to a circle or an interval; if it is a circle, then it has an even positive number of vertices. (Cii ) Each interior vertex of Q is marked with a vector in Rn . If Q is an interval then each end vertex of Q is marked with either a vector or a couple of vectors of the type (v1 , ±v2 ) in Rn (the second vector in the couple is only defined up to a sign). (Ciii ) Zρ is a lattice of rank n − 1 in Rn . (Civ ) If v ∈ Rn is the mark at a vertex of Q, then R · v ⊕ (Zρ ⊗ R) = Rn . If (v, ±w) is the mark at a vertex of Q, then we also have R · v ⊕ (Zρ ⊗ R) = Rn , while w is a primitive element of Zρ . Moreover, if vi and vi+1 are two consecutive marks (each of them may belong to a couple, e.g., (vi , ±wi )), then they lie on different sides of Zρ ⊗ R in Rn . In the case when Q is a circle, there is another invariant of ρ, called the monodromy, and defined as follows. Denote by F1 , . . . , Fm the (n−1)-dimensional orbits of (M n , ρ) in cyclical order (they correspond to vertices of Q in cyclical order). Denote by σi the reflection associated to Fi . Let z1 ∈ M be an arbitrary regular point which projects to a point lying between the images of Fm and F1 in Q. Put z2 = σ1 (z1 ) (which is a point lying on the regular orbit between F1 and F2 ), z3 = σ2 (z2 ), . . . , zm+1 = σm (zm ). Then zm+1 lies on the same regular orbit as z1 , and so there is a unique element μ ∈ Rn /Zρ such that zm+1 = ρ(μ, z1 ). This element μ is called the monodromy of the action. Notice that μ does not depend on the choice of z1 nor on the choice of F1 (i.e., which singular orbit is indexed as the first one), but only on the choice of the orientation of the cyclic ordering on Q: if we change the orientation of Q, then μ will be changed to −μ. So a more correct way to look at the monodromy is to view it as a homomorphism from π1 (Q) ∼ = Z to Rn /Zρ . Theorem 3.3.18. (i) If (Qmarked , Z) is a pair of a marked graph and a lattice which satisfies conditions (Ci )-(Civ ) above, then they can be realized as the marked graph and the isotropy group of a non-degenerate action of Rn of toric degree n − 1 on a compact n-manifold. Moreover, if Q is a circle then any monodromy element μ ∈ Rn /Z can also be realized. (ii) In the case when Q is an interval, any two such actions having the same (Qmarked , Z)-invariant are isomorphic. In the case when Q is a circle, any two actions having the same (Qmarked , Z, μ) are isomorphic. Proof. The proof of (i) is by surgery, i.e., gluing of linearized pieces given by Theorem 3.3.7. There is no obstruction to doing so.

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Figure 3.4: Monodromy μ when Q ∼ = S1 .

For (ii), if there are two different actions (M1 , ρ1 ) and (M2 , ρ2 ) with the same marked graph (Smarked , Z), then one can construct an isomorphism Φ from (M1 , ρ1 ) to (M2 , ρ2 ) as follows: take z1 ∈ M1 and z2 ∈ M2 such that z1 and z2 project to the same regular point on Smarked . Put Φ(z1 ) = z2 . Extend Φ to Oz1 by the rule Φ(ρ1 (θ, z1 )) = ρ2 (θ, z2 ). Then extend Φ to rest of M1 by the reflection principle and the continuity principle. This proves the first case; the second one is similar.  Three-dimensional case of toric degree 1 = 3 − 2. Consider an action ρ : R3 × M 3 → M 3 of toric degree 1. Let z ∈ Oz be a point in a singular orbit of ρ. Denote the HERT-invariant of z by (h, e, r, t), and the rank over Z2 of the twisting group Gz of ρ at z by k = rank Z2 Gz . According to the results of the previous subsections, we have the following constraints on the nonnegative integers h, e, r, t, k: h + 2e + r + t = 3, e + t = 1, e + h ≥ 1, k ≤ min(h, t). In particular, we must have k ≤ 1, i.e., the twisting group Gz is either trivial, or isomorphic to Z2 . Taking the above constraints into account, we have the following full list of possibilities for the singular point z, together with their abbreviated names: (I) (h):

h = 1, e = 0, r = 1, t = 1, Gz = {0};

(II) (ht ):

h = 1, e = 0, r = 1, t = 1, Gz = Z2 ;

(III) (e):

h = 0, e = 1, r = 1, t = 0, Gz = {0};

(IV) (h − h):

h = 2, e = 0, r = 0, t = 1, Gz = {0};

(V) (h − ht ): h = 2, e = 0, r = 0, t = 1, Gz = Z2 acting by the involution (x1 , x2 ) → (−x1 , x2 );

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(VI) (h − h)t : h = 2, e = 0, r = 0, t = 1, Gz = Z2 acting by the involution (x1 , x2 ) → (−x1 , −x2 ); (VII) (e − h):

h = 1, e = 1, r = 0, t = 0, Gz = {0}.

In the above list, (h) means hyperbolic non-twisted, (h − h)t means a joint twisting of a product of two hyperbolic components, and so on. The local structure of the corresponding 2-dimensional quotient space Q2 = 3 M /ρT (together with the traces of singular orbits on M 3 ) is described in Figure 3.5.

Figure 3.5: The seven types of singularities of R3 -actions of toric degree 1 on three-manifolds. Remark that, in case (VI), locally, Q ∼ = D2 /Z2 is homeomorphic, but not diffeomorphic to a disk. In the other cases, Q can be viewed locally as either a disk (without boundary), or a half-disk (with boundary), but it cannot be a corner. Globally, the quotient space Q can be obtained by gluing copies of the above seven kinds of local pieces together, in a way which respects the letters (e.g., an edge marked e will be glued to an edge marked e, an edge marked ht will be glued to an edge marked ht , etc.). Notice, for example, that cases (II) and (III) in the above list are different but have diffeomorphic quotient spaces. To distinguish such situations, we must attach letters to the singularities, which describe the corresponding types of singularities coming from (M 3 , ρ). The quotient space Q together with these letters on its graph of singular orbits will be called the typed quotient space and denoted by Qtyped .

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Theorem 3.3.19. (i) Let (Qtyped , ρR ) be the typed quotient space of (M 3 , ρ), where ρ is of toric degree 1 and M 3 is a three-manifold without boundary. Then each singularity of Qtyped belongs to one of the seven types (I)–(VII) listed above. (ii) Conversely, let (Qtyped , ρR ) be a two-orbifold together with a totally hyperbolic action ρR on it, and together with the letters on the graph of singular orbits, such that the singularities of Qtyped belong to the above list of types (I)– (VII). Then there exists (M 3 , ρ) of toric degree 1 which admits (Qtyped , ρR ) as its quotient. Moreover, the T1 -equivariant diffeomorphism type of M 3 is completely determined by Qtyped . Proof. (i) It was shown above that the list (I)–(VII) is complete in the case of dimension three, due to dimensional constraints. (ii) When the toric degree is 1, assuming that Zρ ∼ = Z is fixed in R3 , because Zρ has only dimension one and doesn’t allow multiple choices, there is a unique choice of isotropy groups in this case. The second part of the theorem now follows from Theorem 3.3.17. 

Figure 3.6: Example of Q2 for n = 3, t(ρ) = 1. Some examples of realizable Qtyped which can be obtained by gluing the above seven kinds of pieces are shown in Figure 3.6. Notice that Qtyped may be without boundary, as in Figure 3.6(a), or with boundary, as in Figure 3.6(b)-(c). The boundary components of Qtyped correspond to the orbits of type e (elbolic) and ht (hyperbolic twisted). In the interior of Qtyped , one may have edges of type h (hyperbolic non-twisted) and singular points of type h − h or (h − h)t . In Figure 3.6(c), Qtyped is not a smooth manifold (though it is homeomorphic to a disk). The branched 2-covering of Figure 3.6(c) is shown in Figure 3.7 (Z2 acts by rotating 180◦ around 0). It is easy to see that, the three-manifolds corresponding to the situations (a), (b), and (c) in this example are S2 × S1 , RP2 × S1 , and RP3 , respectively.

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Figure 3.7: Branched double covering of Figure 3.6(c).

Remark 3.3.20. If M 3 admits an action of R3 of toric degree 1, then M 3 is a graph-manifold in the sense of Waldhausen, see, e.g., [47]. As was observed by Fomenko [38], graph-manifolds are also precisely those manifolds which can appear as isoenergy three-manifolds in an integrable Hamiltonian system with two degrees of freedom. The case of dimension n ≥ 4 with toric degree n − 2. When the dimension n is at least 4 and the toric degree is n − 2 ≥ 2, we have the following three new types of singularities, in addition to the seven types listed above (See Figure 3.8): (VIII) (ht − ht ): h = 2, e = 0, r = 0, t = n − 2, Gz = Z2 × Z2 acting separately on the two hyperbolic components; (IX) (e − ht ):

h = 1, e = 1, r = 0, t = n − 3, Gz = Z2 ;

(X) (e − e):

h = 0, e = 2, r = 0, t = n − 4, Gz = {0}.

Figure 3.8: The additional three possible types of singularities for actions of toric degree n − 2 when n ≥ 4.

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Figure 3.9: Example of Q = M 4 /T2 .

Theorem 3.3.21. Let (Qtyped , ρR ) be the typed quotient space of (M n , ρ), where M n is a compact manifold and ρ : Rn × M n → M n is non-degenerate of toric degree n − 2. Then, each singularity of Qtyped belongs to one of the ten types (I)– (X) listed above. Conversely, let (Qtyped , ρR ) be a 2-orbifold with boundary and corners together with a totally hyperbolic action ρR on it, and together with the letters on the graph of singular orbits, such that the singularities of Qtyped belong to the above ten types (I)–(X). Then, for any n ≥ 4, there exists (M n , ρ) of toric degree n − 2 which admits (Qtyped , ρR ) as its quotient. Proof. The main point of the proof is to show that one can choose compatible isotropy groups, but it is a simple exercise. Remark that, unlike the case of dimension three, when n ≥ 4 the typed quotient Qtyped does not determine the diffeomorphism type of the manifold M completely, because there are now multiple choices for the isotropy groups.  An example of the quotient space Q, which can’t appear for n = 3 but can appear for n ≥ 4, is shown in Figure 3.9.

3.3.4

Monodromy

Definition of the monodromy. In the classification of actions of toric degree n−1, we have encountered a global invariant called the monodromy. It turns out that the monodromy can also be defined for any non-degenerate action ρ : Rn × M n → M n of any toric degree, and is one of the main invariants of the action. Choose an arbitrary regular point z0 ∈ (M n , ρ), and a loop γ : [0, 1] → M n , γ(0) = γ(1) = z0 . By a small perturbation which does not change the homotopy class of γ, we may assume that γ intersects the n − 1 singular orbits of ρ transversally (if at all), and does not intersect the orbits of dimension ≤ n − 2. Denote by p1 , . . . , pm (m ≥ 0) the singular points of corank 1 on the loop γ, and by σ1 , . . . , σm the associated reflections of the singular hypersurfaces which contain p1 , . . . , pm respectively as given by Theorem 3.3.5. Put z1 = σ1 (z0 ), z2 = σ2 (z1 ), . . . , zm =

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σm (zm−1 ). (The involution σ0 can be extended from a small neighborhood of p1 to z0 in a unique way which preserves ρ, and so on.) Then zm lies in the same regular orbit as z0 , so there is a unique element μ = μ(γ) ∈ Rn /Zρ such that zm = ρ(μ(γ), z0 ). It turns out that μ(γ) depends only on the homotopy class of γ, and gives rise to a group homomorphism μ : π1 (M n , z0 ) → Rn /Zρ . Moreover, due to the commutativity of Rn /Zρ , μ does not depend on the choice of z0 and can be viewed as a homomorphism from the first homology group H1 (M n , Z) to Rn /Zρ which, for simplicity, will also be denoted by μ : H1 (M n , Z) → Rn /Zρ . Definition 3.3.22. The morphisms μ : H1 (M n , Z) → Rn /Zρ and μ : π1 (M n ) → Rn /Zρ above are called the monodromy of the action ρ : Rn × M n → M n . Remark 3.3.23. The above monodromy is a continuous invariant, and is completely different from the notions of monodromy defined by Duistermaat [31] and Zung [83] for integrable Hamiltonian systems, which are discrete invariants. Monodromy and twisting groups. A simple but important observation is that the twisting groups are subgroups of the monodromy group, i.e., the image of π1 (M n ) by μ in Rn /Zρ . Theorem 3.3.24 (Twistings and monodromy). For any point z ∈ M n , we have Gz ⊆ (μ), where Gz = (Zρ (z) ∩ Zρ ⊗ R)/Zρ is the twisting group of the action ρ at z, and (μ) = μ(π1 (M n )) ⊆ Rn /Zρ is the image of π1 (M n ) by the monodromy map μ. In particular, if M n is simply-connected, then (μ) is trivial, and ρ has no twisting. Proof. Let q ∈ M n , (w mod Zρ ) ∈ Gz , and z0 be a regular point close enough to z. Consider the loop γ : [0, 1] → M n defined as follows: for 0 ≤ t ≤ 1/2, γ(t) = ρ(2tw, z0 ) and, for 1/2 ≤ t ≤ 1, γ(t) is a path from ρ(w, z0 ) to z0 in a small neighborhood of z. Then, using the definition of the monodromy and the  semi-local normal form theorem, one verifies that μ([γ]) = w mod Zρ . The monodromy map μ : H1 (M n , Z) → Rn /Zρ satisfies the following compatibility condition with the isotropy groups, as shown in the proof of Theorem 3.3.24: (∗) If [γ] ∈ H1 (M n , Z) can be represented by a loop of the type {ρ(tw, p) | t ∈ [0, 1]}, where p ∈ M n and w ∈ Zρ (p) ∩ Zρ ⊗ R, then μ([γ]) = w mod Zρ . In particular, If [γ] ∈ H1 (M n , Z) can be represented by a loop of the type {ρ(tw, p) | t ∈ [0, 1]}, where w ∈ Zρ , then μ([γ]) = 0. Remark 3.3.25. We do not know yet, in general, whether Gtorsion is completely generated by the twisting elements or not. Changing of monodromy. The torus action ρT induces a natural homomorphism τ : Zρ → H1 (M n , Z), which associates to each element w ∈ Zρ the homology class of a loop of the type {ρ(tw, z) | t ∈ [0, 1]} in H1 (M n , Z), which does not depend on the choice of z in M n . The composition of τ with μ is trivial, because the image

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of the homology class of any such loop under the monodromy map is zero. Thus, we can also view the monodromy as a homomorphism from H1 (M n , Z)/τ (Zρ ) to Rn /Zρ , which we will also denote by μ : H1 (M n , Z)/τ (Zρ ) → Rn /Zρ , abusing the language. According to the structural theorem for finitely generated Abelian groups, we can write H1 (M n , Z)/τ (Zρ ) = Gtorsion ⊕ Gfree , whereGtorsion ≤ H1 (M n ,Z)/τ (Zρ ) is its torsion part, and Gfree ∼ = Zk , where k = rank Z H1 (M n , Z)/τ (Zρ ) , is a free part complementary to Gtorsion . This decomposition of H1 (M n , Z)/Im(Zρ ) gives us a decomposition of μ, say μ = μtorsion ⊕ μfree , where μtorsion : Gtorsion → Rn /Zρ is the restriction of μ to the torsion part Gtorsion , and μfree is the restriction of μ to Gfree . Notice that μtorsion is not arbitrary, but must satisfy the above compatibility condition (∗) with the twisting groups. On the other hand, μfree can be arbitrary. More precisely, we have the following result. Theorem 3.3.26. With the above notations, assume that μfree : Gfree → Rn /Zρ is another arbitrary homomorphism from Gfree to Rn /Zρ , and put μ = μtorsion ⊕ μfree : H1 (M n , Z)/τ (Zρ ) −→ Rn /Zρ . Then there exists another non-degenerate action ρ : Rn × M n → M n , which has the same orbits as ρ and the same isotropy group at each point of M n as ρ, but whose monodromy is μ . See [91] for the proof, which is based on a covering of M trivializing the monodromy. Remark 3.3.27. In Theorem 3.3.26, it is possible to change μtorsion also to another homomorphism μtorsion : Gtorsion → Rn /Zρ . Then the construction of the proof still works, but the new action ρ will not have the same isotropy groups as ρ at twisted singular orbits in general, and even the diffeomorphism type of M  may be different from M , because the new action of π1 (M n , z0 )/Im(Zρ ) will not be isotopic to the old one. Monodromy under reduction. Even though Q = M n /ρT is just an orbifold in general, we can still define the monodromy map μρR : H1 (Q, Z) → Rn /(Zρ ⊗ R) ∼ = Rn−t(ρ) of the action ρR on Q, just like the case of actions on manifolds. The following proposition is an immediate consequence of the definition of monodromy. Proposition 3.3.28. We have the following natural commutative diagram: H1 (M n , Z)

μρ

/ Rn /Zρ

proj .

 H1 (Q, Z)

μ ρR



proj .

/ Rn /(Zρ ⊗ R),

where proj . denotes the natural projection maps.

(3.11)

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3.3.5 Totally hyperbolic actions Hyperbolic domains and their fans. Recall that a non-degenerate action ρ : Rn × M n → M n is totally hyperbolic if its toric degree is zero, or equivalently, its regular orbits are diffeomorphic to Rn . Each such orbit will be called a hyperbolic domain of the action on the manifold. Let O be a hyperbolic domain of a totally hyperbolic action ρ : Rn × M n → n M . According to the toric degree formula (Theorem 3.3.14), when the toric degree of the action is 0, then every orbit is diffeomorphic to some Rk (0 ≤ k ≤ n), and so we have a cell decomposition of M n , whose cells are the orbits of ρ. Denote ¯ also ¯ the closure of O in M n and call it a closed hyperbolic domain. Then O by O admits a cell decomposition by the orbits of ρ. ¯ we will denote by Fix an arbitrary point z0 ∈ O. For each orbit H of ρ in O, CH = {w ∈ Rn | lim ρ(−tw, z0 ) ∈ H} t→+∞

the set of all elements w ∈ Rn such that the flow of the action ρ through z0 in the direction −w tends to a point in H. It is clear that if H and K are two different ¯ then CH and CK are disjoint. Using local normal forms, one can prove orbits in O, the following. Proposition 3.3.29. With the above notations, we have: (i) CH does not depend on the choice of z0 ∈ O; ¯ (ii) CO = {0} and CFi = R>0 · vi for each (n − 1)-dimensional orbit Fi ⊂ O, n where vi ∈ R is the vector associated to Fi with respect to the action ρ; (iii) if w ∈ CH , then Xw = 0 on H, where Xw = of ρ associated to w;

d dt ρ(tw, ·)|t=0

is the generator

(iv) C¯H is a simplicial cone in Rn (i.e., a convex cone whose base is a simplex, that is a k-dimensional polytope with exactly k + 1 vertices for some k) and dim CH + dim H = n; ¯ and in that case CK is a face of C¯H ; (v) CK ⊂ C¯H if and only if H ⊂ K, ¯ is compact, then the family (CH ; H ⊂ O) ¯ is a partition of Rn . (vi) if O In the literature, a partition of Rn into a finite disjoint union of simplicial cones starting at the origin is called a complete fan, see, e.g., [34, 43]. More precisely, we have the following definition. Definition 3.3.30. A fan in Rn is a set of data (CH , vi ), where H and i are indices such that: (1) the family (CH ) is a finite family of disjoint subsets of Rn ; (2) the closure C¯K of each CK is a simplicial cone in Rn whose vertex is the origin of Rn , and C¯K \CK is the boundary of the cone C¯K ;

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129

(3) if C¯K \CK = ∅ (i.e., CK = {0}), then each face of C¯K is again an element of the family (CH ); (4) the (vi ) are vectors in Rn which lie on one-dimensional cones in the family (CH ), such that each one-dimensional CKi contains exactly one element vi : CKi = R>0 · vi ; (5) if, moreover, (CH ) is a finite partition of Rn , i.e., Rn is the disjoint union of this family (CH ), then we say that (CH , vi ) is a complete fan. Proposition 3.3.29 tells us exactly that, for each compact closed hyperbolic ¯ there is a naturally associated complete fan of Rn which is an invariant domain O, ¯ is not compact, then the cones (CH ) do not fill the whole Rn , of the action. If O and one has an incomplete fan in that case. Figure 3.10 is an illustration of the construction of the associated fan for a hyperbolic domain.

Figure 3.10: The fan at Tz0 M n ∼ = Rn . The following theorem shows that, conversely, any complete fan can be realized, and is the full invariant of the action on a compact closed hyperbolic domain. Theorem 3.3.31. (i) Let (CH , vi ) be a fan of Rn . Then, there exists a totally hyperbolic action ρ : Rn × M n → M n on a manifold M n with a hyperbolic ¯ ρ) is (CH , vi ). If the fan is domain O such that the associated fan to (O, n complete, then M can be chosen to be compact without boundary. ¯1 , ρ1 ) and (O ¯2 , ρ2 ) of two actions (ii) If there are two closed hyperbolic domains (O ρ1 and ρ2 , which have the same associated complete fan (CH , vi ), then there ¯1 to O ¯2 which intertwines ρ1 and ρ2 . is a diffeomorphism from O Proof. (i) One uses the gluing method to construct an abstract hyperbolic domain associated to the fan, and then use reflections to glue pieces isomorphic to that domain together into a closed manifold. More precisely, take an arbitrary closed ¯ and take 2k identical copies of it, indexed by numbers a ∈ hyperbolic domain O

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¯ Glue these {1, . . . , 2k }, where k is the number of (n − 1)-dimensional facets of O. identical copies together to get a global manifold together with a totally hyperbolic ¯b along the i-th facet if ¯a will be glued to O action on it by the following rule: O i−1 and only if |a − b| = 2 . (ii) Take any two points z1 ∈ O1 and z2 ∈ O2 . Define Φ(z1 ) = z2 and ¯ by Φ(ρ1 (θ, z1 )) = ρ2 (θ, z2 ) for all θ ∈ Rn , and then extend Φ to the boundary of O ¯ ¯ continuity. The fact that (O1 , ρ1 ) and (O2 , ρ2 ) have the same associated complete ¯2 is a diffeomorphism, which ¯1 → O fan ensures that the constructed map Φ : O  sends ρ1 to ρ2 . The topology of closed hyperbolic domains. Since closed hyperbolic domains are classified by fans, their topology (together with the cell decomposition by the orbits) is also completely determined by the corresponding fans, which are combinatorial objects. By starting from fans, one can show that any closed hyperbolic ¯ is contractible. It is easy to see that any convex simple polytope in Rn domain O (i.e., a convex polytope which has exactly n edges at each vertex) can be realized by a complete fan (whose vectors are orthogonal to the facets of the polytope and pointing towards the facets from inside). A natural question arises: is it true that ¯ is also difeomorphic to a convex simple any compact closed hyperbolic domain O polytope? The following theorem, which was pointed out by Ishida, Fukukawa, and Masuda [43] in the context of topological toric manifolds (see Subsection 3.3.6) gives an answer to this question. ¯ of dimension n ≤ 3 Theorem 3.3.32. Any compact closed hyperbolic domain O is diffeomorphic to a convex simple polytope. If n ≥ 4, then there exists a com¯ of dimension n which is not diffeomorphic to a pact closed hyperbolic domain O polytope. The case n = 2 of the above theorem is obvious. The case n = 3 is a consequence of the classical Steinitz theorem. When n = 4 of higher, there are counterexamples: the first known counterexample comes from the so-called Barnette’s sphere [5]. The Barnette’s sphere is a simplicial complex whose ambient space is a three-dimensional sphere S 3 , but which cannot be realized as the boundary of a convex simplicial polyhedron in R3 for some reasons of combinatorial nature. It is known [34] that Barnette’s sphere can be realized as the base of a complete fan ¯ in R4 , which we will call the Barnette fan. Take the closed hyperbolic domain O ¯ given by this Barnette fan. Then O cannot be diffeomorphic to a convex simple four-dimensional polytope, because if there is such a polytope, then the boundary of the simplicial polytope dual to it will be a realization of the Barnette’s sphere, which is a contradiction. Totally hyperbolic actions in dimension 2. The existence of totally hyperbolic actions on any closed 2-manifold was known to Camacho [16], who called them “Morse–Smale R2 -flows on a 2-manifold”. The following theorem is a slight improvement of Camacho’s result.

3.3. Geometry of integrable systems of type (n, 0)

131

Figure 3.11: Cutting S2 into 8 trigones, and cutting Σ2 into 4 domains. Theorem 3.3.33 ([16, 91]). (i) On the sphere S2 there exists a totally hyperbolic R2 action that has exactly 8 hyperbolic domains; the number 8 is also the minimal number possible: any totally hyperbolic action of R2 on S2 must have at least 8 hyperbolic domains. (ii) For any g ≥ 1, on a closed orientable surface of genus g there exists a totally hyperbolic action of R2 which has exactly 4 hyperbolic domains; the number 4 is also the minimal possible. (iii) Any non-orientable closed surface also admits a totally hyperbolic action with 4 hyperbolic domains, and the number 4 is also the minimal possible. Proof. As for the existence of totally hyperbolic actions on orientable surfaces, one can cut a sphere into 8 triangles, or a surface Σg of genus g ≥ 1 into four pieces, as shown in Figure 3.11, turn one of the pieces into a hyperbolic domain and then extend the R2 -action to the whole surface by reflections. In the case of a nonorientable surface, one can embed Σg (where g ≥ 0) into R3 in such a way that it is symmetric with respect to the three planes {x = 0}, {y = 0}, {z = 0}, and it is cut into 8 polygons by these planes (each polygon has g +3 edges). Like in the case of S2 , we turn one of these (g + 3)-gons into a hyperbolic domain and then extend it to Σg by reflections, in such a way that the R2 action is invariant with respect to the antipodal involution. Hence, this action can be projected to the quotient space of Σg with respect to this involution to become a totally hyperbolic action with exactly four hyperbolic domains on a non-orientable surface. The minimality of all these numbers is also easy to check, see [91].  There is a very interesting question of combinatorial nature: what are the necessary and sufficient conditions for a graph on a surface Σ to be the singular set of a totally hyperbolic action of R2 on Σ? We do not know much about this question, but we know that such a graph must consist of simple closed curves which intersect transversally and, besides that, there are some other obstructions. For example, there does not exist any totally hyperbolic action which contains three domains O1 , O2 , O3 as in Figure 3.12(a). Indeed, assume that there is such an

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action. Denote by v1 , v2 , v3 , v4 the vectors associated to the curves F1 , F2 , F3 , F4 , respectively. Since v1 , v2 , v3 form the fan of O1 , we must have v3 = αv1 + βv2 for some α, β < 0. Similarly, looking at the fan of O3 , we have v4 = γv1 + δv2 for some γ, δ < 0. However, looking at the fan of O2 , we have that either v3 or v4 is a positive linear combination of v1 and v2 ; this is a contradiction. Similarly, the configuration in Figure 3.12(b) is also impossible.

Figure 3.12: Impossible configurations.

Classification of totally hyperbolic actions. As of this writing, we do not know if there is any obstruction to the existence of a totally hyperbolic Rn -action on an n-manifold when n ≥ 3. We do not even know whether simple manifolds such as three-dimensional lens spaces admit a totally hyperbolic Rn -action. Nevertheless, we have the following abstract classification theorem, whose proof is straightforward. Theorem 3.3.34. Non-degenerate totally hyperbolic actions of Rn on a connected n-manifold M n (possibly with boundary and corners) are completely determined by their invariants (I1 ), (I2 ), (I3 ) listed below: (I1 ) the singular set: smooth invariant hypersurfaces of M n which intersect transversally and which cut M n into a finite number of “curved polytopes”, which are hyperbolic domains of the action; (I2 ) the family of fans: a fan for each domain, with a correspondence between the cones of the fan and the faces of the closure of the domain; (I3 ) the monodromy. In other words, assume that (M1n , ρ1 ) and (M2n , ρ2 ) are totally hyperbolic actions, such that there is a homeomorphism ϕ : M1n → M2n sending hyperbolic domains of (M1n , ρ1 ) to hyperbolic domains of (M2n , ρ2 ), and such that the monodromy and the associated fans are preserved by ϕ; then, there is a diffeomorphism Φ : M1n → M2n which sends ρ1 to ρ2 .

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133

Remark that the family of fans in the above theorem must satisfy the following compatibility condition: all the fans of the domains sharing the same boundary hypersurface must also share the same vector associated to that hypersurface. This additional condition is also sufficient for the set of invariants to be realizable.

3.3.6 Elbolic actions and toric manifolds Definition 3.3.35. A non-degenerate action ρ : Rn × M n → M n is called elbolic, if it does not admit any hyperbolic singularity, i.e., all singular points have only elbolic components. In the elbolic case, the singular set of the action is of codimension two, hence the regular part is connected, which implies that the action has just one regular orbit. The rest of the proof of the following proposition is also straightforward. Theorem 3.3.36. Let ρ : Rn × M n → M n be an elbolic action. Then we have: (i) ρ has exactly one regular open dense orbit in M n ; (ii) if the action admits a compact orbit of dimension k, then the toric degree is t(ρ) = (n + k)/2 ≥ n/2; in particular, if (M n , ρ) admits a fixed point, then the dimension n is even and t(ρ) = n/2; (iii) the monodromy of ρ is trivial, and the quotient space Q = M n /ρT of M n by the induced torus action ρT is a contractible manifold with boundary and corners, on which the reduced action ρR is non-degenerate totally hyperbolic and has only one regular orbit. The case of elbolic actions with a fixed point is of special interest in geometry, because of its connection to toric manifolds. Recall that, a toric manifold in the sense of complex geometry is a complex manifold (which is often equipped with a K¨ alerian structure, or equivalently, a compatible symplectic structure) of complex dimension m, together with a holomorphic action of the complex torus (C∗ )m which has an open dense orbit. See, e.g., [2, 26] for an introduction to toric manifolds. From our point of view, a complex toric manifold has real dimension n = 2m, and the action of (C∗ )m ∼ = Rm × Tm is an elbolic non-degenerate R2m action. Complex toric manifolds are classified by their associated fans. So, our classification of hyperbolic domains (and of the quotient spaces of elbolic actions) is very similar to the classification of complex toric manifolds, except that, unlike the complex case, the vectors of our fans are not required to lie in an integral lattice. There have been many generalizations of the notion of toric manifold to the case of real manifolds. In particular, Davis and Januskiewicz [28] introduced quasitoric manifolds in 1991. A quasi-toric manifold is a 2m-dimensional manifold with an almost-everywhere-free action of Tm such that the quotient space M 2m /Tm is homeomorphic to a simple convex polytope (like in the case of complex toric manifolds), and such that near each fixed point the action is locally isomorphic to

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a standard action of Tm on Cm . Hattori and Masuda [42] introduced torus manifolds in 2003. A torus manifold is simply a closed connected orientable smooth manifold M of dimension 2m with an effective smooth action of Tm having a fixed point. Ishida, Fukukawa, and Masuda [43] recently introduced the notion of topological toric manifolds in 2010. A topological toric manifold is a closed smooth manifold M of dimension n = 2m with an almost-everywhere-free smooth action of (C∗ )m ∼ = Tm × Rm which is covered by finitely many invariant open subsets, each equivariantly diffeomorphic to a direct sum of complex one-dimensional linear representations of Tm × Rm . According to the results of [43], topological toric manifolds are the right generalization of the notion of toric manifolds to the category of real manifolds; they have very nice homological properties similarly to toric manifolds (see [43, Sect. 8]), and they are also classified by a generalized notion of fans. It turns out that Ishida–Fukukawa–Masuda’s notion of topological toric manifolds is equivalent to our notion of manifolds admitting an elbolic action whose toric degree is half the dimension of the manifold. The proof of the following proposition is a simple verification that their conditions and our conditions are the same. Proposition 3.3.37. A closed manifold M 2m , together with a smooth action of (C∗ )m ∼ = Rm × Tm , is a topological toric manifold if and only if the action (when viewed as an action of R2m ) is elbolic of toric degree m. In [43], topological toric manifolds are classified by the so-called complete non-singular topological fans, which encode the following data: the complete fan in Rn associated to the reduced totally hyperbolic action ρR on the quotient space Q = M 2m /ρT , and the vector couples associated to corank-two transversally elbolic orbits (see Definition 3.3.3). These vector couples tell us how to build back (M 2m , ρ) from (Q, ρR ). So one can recover Ishida–Fukukawa–Masuda’s classification theorem for topological toric manifolds from our point of view of general non-degenerate Rn -actions on n-manfiolds: one can prove this theorem in the same way as Theorem 3.3.31, by gluing together local pieces equipped with canonical coordinates and adapted bases. Another very interesting proof, based on the quotient method, which represents the topological toric manifold (M 2m , ρ) as a quotient of another global object, is given in [43]. (The quotient method is also discussed in [2, 26] for the construction of toric manifolds.)

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