Symplectic and Contact Geometry - A Concise Introduction [1 ed.] 9783031562242, 9783031562259

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Latin American Mathematics Series

Anahita Eslami Rad

Symplectic and Contact Geometry A Concise Introduction

Latin American Mathematics Series

This new series aims to gather monographs, textbooks and contributed volumes based on mathematical research conducted at/with institutions based in Latin America. This series is open to subseries, which could be topical – with contributions from a myriad of institutions – or exclusively supplied by a single Latin American research institution. Standalone books are also welcome. Books published in cooperation with research institutions may carry the name and the logo of the research institute on the cover. While it is expected that many subseries will be linked to a research institute or University located in Latin America, the creation of topical subseries, not necessarily related to institutions, is also possible. All subseries should have their own editorial board comprised of distinguished international researchers with a strong academic background. Legal permission for Springer to commercially use the institution’s brand and logo may be required.

Anahita Eslami Rad

Symplectic and Contact Geometry A Concise Introduction

Anahita Eslami Rad FaMAF National University of Córdoba Córdoba, Córdoba, Argentina

ISSN 2524-6755 ISSN 2524-6763 (electronic) Latin American Mathematics Series ISBN 978-3-031-56224-2 ISBN 978-3-031-56225-9 (eBook) https://doi.org/10.1007/978-3-031-56225-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Dedicated to the memory of my parents

Preface

This book gives an introduction to symplectic and contact geometry. The topics in this book have been chosen so as to emphasize the relations between symplectic geometry with other subjects such as Lie theory, classical mechanics, contact geometry, etc. Symplectic geometry is a part of differential geometry and differential topology which studies symplectic manifolds. Symplectic geometry has its origin in classical mechanics. To formulate the equations of motion for a classical mechanical system, one needs to solve some Euler-Lagrange equations. Hamilton, using Legendre transforms, formulated the Euler-Lagrange equations into Hamiltonian systems. This was the starting point for symplectic geometry. Later, for instance, in Arnold’s works, the Hamiltonian systems were formulated in .iHf ω = −df, where .ω is a symplectic form. More precisely, a symplectic form is a non-degenerate closed twoform. A smooth manifold equipped with a symplecitc form is a symplectic manifold. The non-degeneracy property of symplectic forms implies that symplectic manifolds are even-dimensional. One of the most important facts in symplectic geometry is Darboux’s theorem, which states that, locally, any 2n-dimensional symplectic manifold is the same as .R2n with the standard symplectic form. Contact geometry is the odd-dimensional counterpart of symplectic geometry. In other words, most of the facts in symplectic geometry have analogs in contact geometry. For instance, we have Darboux’s theorem for contact manifolds, i.e., locally .(2n + 1)-dimensional contact manifolds are the same. This text is based on lectures on symplectic and contact geometry which I gave during the second semester of the year 2020 in the mathematics department of Universidad Nacional de Cordoba (FaMAF, UNC). In fact, this is the revised version of the same course which I gave in 2013 and 2015 in the mathematics department of Universidad de Santiago de Chile. There are great references on symplectic and contact geometry and I have mentioned some of them in the bibliography. In these lectures, I have tried to cover, in detail, the main topics in an introductory way so that it motivates and prepares the reader to investigate in other books and papers. In fact, I received encouraging feedback from my students to publish these lecture

vii

viii

Preface

notes, as they have told me these lectures are “ concise ” and help them to learn the subject. This book is for whomever wants to start learning symplectic and contact geometry, such as advanced undergraduate students, graduate students, and researchers who want to learn the subject independently. To study this book, knowledge about differential geometry, manifolds, algebraic topology, and de Rham cohomology and basic facts about the theory of Lie groups are required. However, in some parts where it is necessary to recall some background, some quick review is given. This book consists of four chapters. Chapter 1 presents definitions, examples of symplectic linear algebra whose objects of study are vector spaces. In particular, it explains the standard form of the symplectic form, the Lagrangian Grassmannian, symplectic linear group, and Hermitian forms. This chapter provides the foundation to understand the second chapter. Chapter 2 is about symplectic manifolds and their examples, such as cotangent bundles, reduced symplectic manifolds, complex projective manifolds, etc. Also, it talks about symplectomorphisms, Lagrangian submanifolds, and their applications such as the Moser trick and Geodesic Flows. In this chapter, we see the proof of Darboux’s theorem using neighborhood theorems, given by Weinstein. Chapter 3 introduces Hamiltonian systems by defining Hamiltonian vector fields and we see the famous formula .iHf ω = −df . It explains classical mechanical motivation describing the Legendre transform. It presents the definition of Poisson brackets and the relation to Hamiltonian systems, hence classical mechanics. Then we will see the original proof of Darboux’s theorem given by Darboux. Finally, Hamiltonian group actions, momentum mapping, and symplectic quotient are discussed. In Chap. 4, the definition of contact manifolds, examples and constructions are given. Then we have Darboux’s theorem for contact manifolds. Therefore, the same as symplectic manifolds, there is no local invariant for contact manifolds. Legendrian submanifolds, in particular Legendrian knots, are introduced. For a motivation of contact geometry, we study the geometric optic. Finally, contact surgery as a way to construct contact manifolds is explained, and briefly we will see about contact invariants. When I was preparing the first version of my lecture notes, which later turned to this text, I used my notes from symplectic geometry classes of J. J. Duistermaat in Utrecht University. I am very thankful to him for showing the beauty of symplectic geometry in his lectures. I hope in this text I can transfer a little of this beauty to the reader. Last but not least, I would like to thank the reviewers for the careful reading of the manuscript. I am very grateful to them for their comments and suggestions. Córdoba, Argentina December, 2023

Anahita Eslami Rad

Contents

1

Symplectic Linear Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Symplectic Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Annihilator of a Linear Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Orthogonal Complement of a Linear Subspace . . . . . . . . . . . . . . . . . . . . . . 1.4 Isotropic Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Symplectic Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Lagrangian Grassmannian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The Symplectic Linear Group and Its Lie Algebra . . . . . . . . . . . . . . . . . . 1.8 Hermitian Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 6 9 11 15 20 27 35

2

Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of Symplectic Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Examples and Constructions of Symplectic Manifolds . . . . . . . . . . . . . . 2.2.1 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Reduced Symplectic Manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Complex Projective Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Almost Complex Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cohomology Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Symplectomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Lagrangian Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Geodesic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Local Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Symplectic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 42 42 47 51 54 57 59 61 65 67 74

3

Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Invariant Form Under the Flow of a Vector Field . . . . . . . . . . . . . . . . . . . . 3.2 Hamiltonian Vector Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Darboux’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 84 87 91 94 98 ix

x

Contents

3.7 3.8 3.9 4

Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Hamiltonian Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Symplectic Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Contact Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition of Contact Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Examples and Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Hypersurfaces in Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . 4.2.2 Contactization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Contactomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Complex Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Darboux’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Legendrian Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Geometric Optic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Characteristic Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Legendrian Knots in Contact 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Contact Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Contact Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Legendrian Surgery Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 116 118 122 125 128 132 134 138 143 148 152 164 170 171

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Chapter 1

Symplectic Linear Algebra

A symplectic structure on a smooth manifold is defined by assigning a symplectic form to the tangent space at each point of the manifold, with additional property which we will see in the next chapter. Considering tangent spaces as vector spaces motivates us to study first about symplectic linear algebra. Hence, in this chapter we study symplectic forms over vector spaces, and we see properties, examples, and more definitions in the linear world. Symplectic linear algebra by itself is an interesting source for a lot of investigations. This chapter prepares us for the next chapter where we study symplectic manifolds and their examples. We start by the definition of a symplectic form. To define specific subspaces of a symplectic vector space, such as isotropic subspaces and Lagrangian subspaces, first we study the notion of orthogonal complements for a bi-linear form. Then, the standard form of the symplectic form is introduced, which is in fact the linear version of Darboux’s theorem. As an interesting application, we discuss the space of all Lagrangian subspaces of a symplectic vector space, known as Lagrangian Grassmannian. We see that the Lagrangian Grassmannian is a smooth manifold. After describing symplectomorphisms as morphisms between symplectic vector spaces, discussion about the group of such morphisms arises. We will see that the symplectic linear group is a Lie group, and we will compare this group with the other linear groups. Moreover, we see its relation with the Lagrangian Grassmannian. We study Hermitian forms and their relation with symplectic forms. In fact, we see how symplectic structures and complex structures are related by Hermitian forms.

1.1 Symplectic Forms Let E be a finite-dimensional vector space over a field k which usually we consider as .k = R. A bi-linear form on E is a mapping that is linear with respect to both of its components. More precisely, we have the following definition. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. Eslami Rad, Symplectic and Contact Geometry, Latin American Mathematics Series, https://doi.org/10.1007/978-3-031-56225-9_1

1

2

1 Symplectic Linear Algebra

Definition 1.1 A bi-linear form on the vector space E is a mapping .σ : E×E → k such that for every choice of .u ∈ E the mapping .E → k : v ‫ →׀‬σ (u, v) is a linear form and for every choice of .v ∈ E the mapping .E → k : u ‫ →׀‬σ (u, v) is a linear form. Changing the order of the components in a bi-linear mapping may change the sign. Definition 1.2 The bi-linear form .σ is called anti-symmetric if .σ (v, u) = −σ (u, v) for every .u, v ∈ E. Example 1.1 A simple example of an anti-symmetric bi-linear form is the twoform .dx ∧ dy in Euclidean space. If we consider x and y as vectors in the Euclidean space, then we have (dx ∧ dy)(x, y) = dx(x)dy(y) − dx(y)dy(x) = 1 − 0 = 1

.

and (dx ∧ dy)(y, x) = dx(y)dy(x) − dx(x)dy(y) = 0 − 1 = −1,

.

which shows that the two-form .dx ∧ dy is anti-symmetric. The other property which a bi-linear mapping may satisfy is non-degeneracy. Definition 1.3 The bi-linear form .σ is called non-degenerate if .σ (u, v) = 0 for every .v ∈ E implies that .u = 0. Example 1.2 In the above example, we saw that .(dx ∧dy)(x, y) = dx(x)dy(y). In other words, for .x /= 0 there is a .y /= 0 such that .dx ∧ dy /= 0. This is equivalent to the definition of non-degeneracy. Therefore, .dx ∧ dy is a non-degenerate two-form. Now we have all the ingredients to define a symplectic form. Definition 1.4 A symplectic form on E is a non-degenerate anti-symmetric bilinear form .σ on .E. The vector space E equipped with the symplectic form .σ is called a symplectic vector space and is denoted by .(E, σ ). Example 1.3 By the above examples and according to the definition, the two-form dx ∧ dy is a symplectic form on .R2 and .(R2 , dx ∧ dy) is a symplectic vector space.

.

Let .E ∗ = lin(E, k) denote the dual vector space that is the space of all linear mappings from E to .k. Remember that we can always identify a bi-linear form .σ : E × E → R with the linear mapping σ : E −→ E ∗

.

σ (u)

u ‫׀‬−→ (v ‫׀‬−→ σ (u, v)), which assigns to .u ∈ E the linear form

1.1 Symplectic Forms

3

σ (u) ‫׀‬−→ (σ (u))(v) := σ (u, v),

.

which induces a linear isomorphism from E onto .(E ∗ )∗ . Remark 1.1 The non-degeneracy property of .σ means that the linear mapping .σ : E → E ∗ has zero kernel, and since .dim(E ∗ ) = dim(E) (is finite dimensional), this is equivalent to saying that .σ : E → E ∗ is a linear isomorphism. In matrix format, the linear isomorphism .σ is a square matrix with linear independent columns and linear independent rows. Remark 1.2 The anti-symmetric property of .σ is equivalent to .σ t = −σ , where the transposed mapping .σ t is the linear mapping from .(E ∗ )∗ = E to .E ∗ . Since (σ t (v))(u) = (σ (u))(v)

.

= σ (u, v) = −σ (v, u) = −(σ (v))(u) for every .u, v ∈ E. Therefore .σ can be written in an anti-symmetric matrix format. The simplest example of a symplectic form is defined as follows. Later we use this example, and we will see that it is actually the most important example of symplectic forms. Example 1.4 Let .E = R2n = Rn ×Rn , with coordinates .(q1 , . . . , qn , p1 , . . . , pn ) ∈ E. Then .σ defined by σ ((q1 , . . . , qn , p1 , . . . , pn ), (q1' , . . . , qn' , p1' , . . . , pn' )) :=

n ∑

.

pi qi' − pi' qi

i=1

is a symplectic form on .E. Exercise 1.1 Show that .σ in the above example is a non-degenerate anti-symmetric bi-linear form on .Rn × Rn , and so it is a symplectic form on .Rn × Rn . Example 1.5 A coordinate free version of the above example is the symplectic form on .E = F × F ∗ defined by σ ((x, ξ ), (y, η)) = ξ(y) − η(x)

.

x, y ∈ F

ξ, η ∈ F ∗ ,

for any finite-dimensional vector space F over .k. Exercise 1.2 Show that the coordinate free version of the previous exercise is a symplectic form.

4

1 Symplectic Linear Algebra

1.2 Annihilator of a Linear Subspace In this section, from linear algebra, we recall the definition of annihilators or orthogonal complements in the dual space, and we recall some properties which we are going to use in the next section. Definition 1.5 Let .L ⊂ E be a linear subspace of .E. The orthogonal complement or annihilator of L in .E ∗ is the linear subspace L◦ = {α ∈ E ∗ | α(v) = 0, ∀v ∈ L} ⊂ E ∗ .

.

In other words the orthogonal complement of L in the dual space .E ∗ is the set of all elements of .E ∗ that are zero on .L. Note that dim(L◦ ) = dim(E ∗ ) − dim(L∗ )

.

= dim(E) − dim(L), the co-dimension of L in .E. Example 1.6 If we consider the subspace L as the x-axis in the linear space .R2 , then the orthogonal complement or annihilator of x-axis in .(R2 )∗ is generated by the form .dy. Definition 1.6 Let .A ⊂ E ∗ be a linear subspace of .E ∗ . The orthogonal complement or annihilator of A in .(E ∗ )∗ = E is A◦ = {v ∈ E | α(v) = 0, ∀α ∈ A} ⊂ (E ∗ )∗ = E.

.

Therefore, the orthogonal complement of A in .(E ∗ )∗ = E is the set of all ◦ is the set of kernel of elements of E such that A is zero on them. In other words, .A∩ ◦ all elements of .A. So, by Definition 1.6 we can write .A = α∈A kerα. Moreover, we have dim(A◦ ) = dim(E) − dim(A).

.

Example 1.7 Let A be the subspace generated by the form dy of .(R2 )∗ . The orthogonal complement or annihilator of A in .((R2 )∗ )∗ = R2 is x-axis. In the following we are going to study some properties of orthogonal complements in the dual space through Lemmas 1.1, 1.2, and 1.3. To visualize these properties, we can consider the above examples of x-axis and its orthogonal complement in .(R2 )∗ . Lemma 1.1 Let .L ⊂ E be a linear subspace of .E. Then .L = (L◦ )◦ . Proof By Definition 1.5, the orthogonal complement of L in .E ∗ is

1.2 Annihilator of a Linear Subspace

5

L◦ = {α ∈ E ∗ | α |L = 0}.

.

By Definition 1.6, the orthogonal complement of .L◦ in .(E ∗ )∗ is (L◦ )◦ = {v ∈ E | α(v) = 0 ∀α ∈ L◦ } ⊃ L.

.

On the other hand, dim(L◦ )◦ = dim(E) − dim(L◦ )

.

= dim(E) − (dim(E) − dim(L)) = dim(L). Therefore, we conclude that .L = (L◦ )◦ .

⨆ ⨅

Similarly, using Definitions 1.6 and 1.5, for any .A ⊂ E ∗ , we obtain that .A = (A◦ )◦ . Lemma 1.2 For subspaces .L, M ⊂ E we have L ⊂ M =⇒ M ◦ ⊂ L◦ .

.

Proof Using Definition 1.5 we obtain that M ◦ = {α ∈ E ∗ ; α |M = 0} ⊂ {α ∈ E ∗ ; α |L = 0} = L◦ .

.

⨆ ⨅ Lemma 1.3 For subspaces .L, M ⊂ E we have: (1) .(L ∩ M)◦ = L◦ + M ◦ . (2) .(L + M)◦ = L◦ ∩ M ◦ . Proof By Lemma 1.2 we get .

L ∩ M ⊂ L =⇒ L◦ ⊂ (L ∩ M)◦ L ∩ M ⊂ M =⇒ M ◦ ⊂ (L ∩ M)◦ .

Therefore, .L◦ + M ◦ ⊂ (L ∩ M)◦ . (1) Again using Lemma 1.2 we obtain .

L ⊂ L + M =⇒ (L + M)◦ ⊂ L◦ M ⊂ L + M =⇒ (L + M)◦ ⊂ M ◦ .

So, we conclude that .(L + M)◦ ⊂ L◦ ∩ M ◦ . Consider the following computation:

(2)

6

1 Symplectic Linear Algebra

L ∩ M = ((L ∩ M)◦ )◦ ⊂ (L◦ + M ◦ )◦ ⊂ (L◦ )◦ ∩ (M ◦ )◦ = L ∩ M,

.

where the first equality follows by Lemma 1.1. Applying Lemma 1.2 on (1), we obtain the first inclusion. The second inclusion is obtained from (2). In the last equality, we used Lemma 1.1. From this computation we conclude that the inclusions should be equality. Therefore applying Lemma 1.1 on .((L ∩ M)◦ )◦ = (L◦ + M ◦ )◦ , we obtain ◦ ◦ ◦ .(L ∩ M) = L + M . The next equality which we obtained is .(L◦ + M ◦ )◦ = (L◦ )◦ ∩ (M ◦ )◦ . If we replace L and M by .L◦ and .M ◦ , using Lemma 1.1, we obtain .(L +M)◦ = L◦ ∩M ◦ . ⨆ ⨅

1.3 Orthogonal Complement of a Linear Subspace An inner product g on a vector space E is a non-degenerate symmetric bi-linear form that is positive definite. In other words, g is non-degenerate and satisfies .g(u, v) = g(v, u) for all .u, v ∈ E and .g(u, u) > 0 for all non-zero .u ∈ E. If we consider the bi-linear mapping .g : E × E −→ R as the linear mapping g : E −→ E ∗

.

g(u)(v) := g(u, v), then the inner product g is represented by a non-degenerate symmetric matrix. So, the difference between an inner product and a symplectic form is that an inner product is symmetric and a symplectic form is anti-symmetric. The anti-symmetric property of symplectic forms leads to (non-zero) isotropic subspaces in symplectic vector spaces which we will study in the next section. Below, first we start with the definition of the orthogonal complements with respect to a bi-linear form, regardless of being symmetric or anti-symmetric. Definition 1.7 Let .σ be a non-degenerate bi-linear form on E (not necessarily antisymmetric) and .L ⊂ E be a linear subspace of .E, and then the .σ -orthogonal complement of L in E is Lσ := {u ∈ E | σ (u, v) = 0 ∀v ∈ L} ⊂ E.

.

Example 1.8 Consider .σ as the non-degenerate bi-linear form .dx ∧dy on .R2 . Then the .σ -orthogonal complement of the x-axis in .R2 is the x-axis itself since (dx ∧ dy)(x, x) = dx(x)dy(x) − dx(x)dy(x) = 0.

.

1.3 Orthogonal Complement of a Linear Subspace

7

For a non-degenerate bi-linear form .σ, the following Lemma describes the .σ orthogonal complement of L in .E, in terms of orthogonal complement of L in .E ∗ , i.e., .L◦ from previous section. Lemma 1.4 Let .σ be a non-degenerate bi-linear form on E; then: (1) .Lσ = σ −1 (L◦ ). (2) .σ −1 (L◦ ) = (σ t (L))◦ , if .σ is symmetric or anti-symmetric. t (3) .Lσ = Lσ , if .σ is symmetric or anti-symmetric. Proof For the part (1), consider .u ∈ Lσ . By definition of .σ -orthogonal complement of L in .E, we conclude that .σ (u, v) = 0 for every .v ∈ L. If we consider the bilinear mapping .σ : E × E −→ k as the mapping σ : E −→ E ∗

.

σ (u)(v) := σ (u, v), so we have .(σ (u))(v) = 0 for every .v ∈ L. Using definition of orthogonal complement of L in .E ∗ , we obtain that .σ (u) ∈ L◦ . The non-degeneracy of the bilinear form .σ means that .σ : E −→ E ∗ is an isomorphism, so we get .u ∈ σ −1 (L◦ ). By the same reasons we can reverse all this argument, that is, u ∈ Lσ ⇐⇒ σ (u, v) = 0, ∀v ∈ L

.

⇐⇒ (σ (u))(v) = 0, ∀v ∈ L ⇐⇒ σ (u) ∈ L◦ ⇐⇒ u ∈ σ −1 (L◦ ), which means that .Lσ = σ −1 (L◦ ). For the part (2), let .u ∈ σ −1 (L◦ ). Since .σ : E −→ E ∗ is an isomorphism, we conclude that .σ (u) ∈ L◦ . By definition of orthogonal complement of L in .E ∗ , we obtain that .(σ (u))(v) = 0 for every .v ∈ L. Assuming that .σ is symmetric or anti-symmetric means that .σ = σ t or .σ = −σ t . By definition of transpose t : (E ∗ )∗ −→ E ∗ , we conclude that .(±σ t (v))(u) = 0 for every .v ∈ L, or .σ t .(σ (v))(u) = 0 for every .v ∈ L. This means that u is in the orthogonal complement of .σ t (L) in .(E ∗ )∗ = E, i.e., .u ∈ (σ t (L))◦ . Reversing this argument using the same reasons, we get u ∈ σ −1 (L◦ ) ⇐⇒ σ (u) ∈ L◦

.

⇐⇒ (σ (u))(v) = 0, ∀v ∈ L ⇐⇒ (±σ t (v))(u) = 0, ∀v ∈ L ⇐⇒ (σ t (v))(u) = 0, ∀v ∈ L ⇐⇒ u ∈ (σ t (L))◦ . So, .σ −1 (L◦ ) = (σ t (L))◦ .

8

1 Symplectic Linear Algebra

For part (3) consider .u ∈ Lσ . From parts (1) and (2) we have .Lσ = σ −1 (L◦ ) = So, .u ∈ (σ t (L))◦ . By definition of orthogonal complement of .σ t (L) in ∗ ∗ t t .(E ) = E, we have .(σ (v))(u) = 0 for every .v ∈ L, or .σ (v, u) = 0 for every t σt .v ∈ L. This means that u is in .σ -orthogonal complement of .L, i.e., .u ∈ L . Reversing the argument, we have (σ t (L))◦ .

u ∈ Lσ = σ −1 (L◦ ) = (σ t (L))◦ ⇐⇒ (σ t (v))(u) = 0, ∀v ∈ L

.

⇐⇒ σ t (v, u) = 0, ∀v ∈ L t

⇐⇒ u ∈ Lσ . t

Therefore, we get .Lσ = Lσ .

⨆ ⨅

The analogue of Lemma 1.3 for .σ -orthogonal complements holds using the facts we saw for orthogonal complements in the dual space. Lemma 1.5 Let .σ be a non-degenerate bi-linear form on E; then: (1) (2) (3) (4) (5)

dim(Lσ ) = dim(E) − dim(L). σ ⊂ Lσ . .L ⊂ M =⇒ M σ σ σ .(L ∩ M) = L + M . σ σ σ .(L + M) = L ∩ M . σ σ .L = (L ) . .

Proof (1) By part (1) of Lemma 1.4 we have .dim(Lσ ) = dim(σ −1 (L◦ )). Since .σ is nondegenerate, we have .dim(σ −1 (L◦ )) = dim(L◦ ). In previous section we saw that .dim(L◦ ) = dim(E) − dim(L). So, in summary we have dim(Lσ ) = dim(σ −1 (L◦ ))

.

= dim(L◦ ) = dim(E) − dim(L). (2) Let .L ⊂ M. By Lemma 1.2 we obtain .M ◦ ⊂ L◦ . Since .σ is non-degenerate, we get .σ −1 (M ◦ ) ⊂ σ −1 (L◦ ). Now, using part (1) of Lemma 1.4, we obtain σ ⊂ Lσ . .M (3) We remember from Lemma 1.3 that .(L∩M)◦ = L◦ +M ◦ . The non-degeneracy of .σ implies that σ −1 ((L ∩ M)◦ ) = σ −1 (L◦ + M ◦ ).

.

By part (1) of Lemma 1.4 and linearity of .σ −1 , we obtain .(L∩M)σ = Lσ +M σ . (4) From Lemma 1.3 we have .(L + M)◦ = L◦ ∩ M ◦ . By non-degeneracy of .σ we obtain that .σ −1 ((L + M)◦ ) = σ −1 (L◦ ∩ M ◦ ). The bijectivity of .σ −1 implies that

1.4 Isotropic Subspaces

9

σ −1 ((L + M)◦ ) = σ −1 (L◦ ) ∩ σ −1 (M ◦ ).

.

Now using part (1) of Lemma 1.4, we obtain .(L + M)σ = Lσ ∩ M σ . (5) By definition, .σ -orthogonal complement of .Lσ is (Lσ )σ = {u ∈ E | σ (u, v) = 0, ∀v ∈ Lσ } ⊃ L.

.

On the other hand, by part (1) we have dim(Lσ )σ = dim(E) − dim(Lσ )

.

= dim(E) − (dim(E) − dim(L)) = dim(L). Therefore, we conclude that .L = (Lσ )σ .

⨆ ⨅

Remark 1.3 If .k = R and .σ is an inner product, then the .σ -orthogonal complement of L is usual orthogonal complement denoted by .L⊥ . For the inner product we have additional property that .L ∩ L⊥ = {0}, which implies that E is equal to the direct sum .L ⊕ L⊥ of L and .L⊥ . Exercise 1.3 For the linear subspace L in .E, and a non-degenerate bi-linear form σ on a vector space E, show that .(Lσ )σ = L.

.

1.4 Isotropic Subspaces Let .(E, σ ) be a symplectic vector space, i.e., E is a finite-dimensional vector space over a field k and .σ is a symplectic form on .E. Lemma 1.6 If the characteristic of k is not equal to 2, then anti-symmetric condition is equivalent to .σ (v, v) = 0 for any .v ∈ E. In other words ( ) σ (u, v) = −σ (v, u) ⇐⇒ σ (v, v) = 0.

.

Proof Assume that .σ is anti-symmetric, so we have .σ (v, v) = −σ (v, v). Therefore, .2σ (v, v) = 0 and .σ (v, v) = 0. Conversely, assume that for every .v ∈ E we have .σ (v, v) = 0. So, in particular for every .u, v ∈ E we have .σ (u + v, u + v) = 0. Using bi-linearity of .σ , we obtain that .σ (u, u) + σ (u, v) + σ (v, u) + σ (v, v) = 0. Once again we use assumption and we get .0 + σ (u, v) + σ (v, u) + 0 = 0 or .σ (u, v) = −σ (v, u), which means that .σ is anti-symmetric. ⨆ ⨅

10

1 Symplectic Linear Algebra

Remark 1.4 If the characteristic of k is equal to 2, then we replace the antisymmetric condition by the condition .σ (v, v) = 0, ∀v ∈ E. As we saw in the previous lemma, the anti-symmetric property of a symplectic form is equivalent to saying that .σ (v, v) = 0 for any .v ∈ E. This shows that for a symplectic form we have .L ∩ Lσ /= {0}. Definition 1.8 A linear subspace L of E is called isotropic with respect to .σ if L ⊂ Lσ , that is .σ (u, v) = 0, ∀u, v ∈ L.

.

Remark 1.5 Note that this is very different from the situation for an inner product, where .{0} is the only isotropic subspace since .L ∩ L⊥ = {0}. The simplest example of an isotropic subspace of a symplectic vector space is any one-dimensional subspace. Lemma 1.7 Every one-dimensional subspace of .(E, σ ) is isotropic. Proof Let .L =< v > be a one-dimensional subspace of .E, generated by .v ∈ E. By the previous lemma, the anti-symmetric property of the symplectic form .σ, is equivalent to .σ (v, v) = 0. This means that .σ is zero on .L =< v > so .L ⊂ Lσ . ⨅ ⨆ Example 1.9 The x-axis as a one-dimensional subspace of the symplectic vector space .(R2 , dx ∧ dy) is isotropic. Similarly, the y-axis is an isotropic subspace of 2 .(R , dx ∧ dy). Definition 1.9 Since the symplectic vector space .(E, σ ) is finite dimensional, any strictly increasing sequence of isotropic subspaces terminates at a maximal one. We call such a maximal isotropic subspace a Lagrangian subspace or a Lagrange plane. This definition shows that every isotropic subspace is contained in at least one Lagrange plane. Proposition 1.1 A subspace .L ⊂ E is a maximal isotropic linear subspace of (E, σ ) if and only if .L = Lσ .

.

Proof Let .L ⊂ Lσ and .L /= Lσ ; then .∀v ∈ Lσ \ L we have kv ⊂ Lσ

.

L ⊂ Lσ . Therefore, .kv + L ⊂ Lσ , so by Lemma 1.5, parts (2) and (4), we have L ⊂ (kv + L)σ = (kv)σ ∩ Lσ .

.

On the other hand, .

v ∈ Lσ , v ∈ (kv) ⊂ (kv)σ (by Lemma 1.7)

1.5 Symplectic Basis

11

imply that .v ∈ (kv)σ ∩ Lσ = (kv + L)σ , by Lemma 1.5, part (4). Therefore, .kv+L ⊂ (kv+L)σ , which means that .kv+L is an isotropic subspace, but L is a maximal isotropic subspace, so .L = Lσ . ⨆ ⨅ In summary, by the above definition and previous proposition, .L ⊂ E is a Lagrange plane if and only if .L = Lσ . Lemma 1.7 shows that in a symplectic vector space we always have isotropic subspaces. This fact, together with the previous proposition, shows that in a symplectic vector space we always have Lagrange planes. Remark 1.6 We saw that for a Lagrange plane L in E the condition is .L = Lσ . Hence, .dim(L) = dim(Lσ ). By part (1) of Lemma 1.5, we have .dim(Lσ ) = dim(E)−dim(L), so we get .dim(L) = dim(Lσ ) = dim(E)−dim(L). Therefore, .dim(E) = 2dim(L), which means: (1) Any symplectic vector space has even dimension, say .dim(E) = 2n. (2) Any Lagrange plane is an isotropic subspace of dimension .n. (3) For any isotropic subspace L we have .dim(L) ≤ n. Example 1.10 The x-axis or the y-axis are Lagrangian subspaces of the symplectic vector space .(R2 , dx ∧ dy), since they are isotropic subspaces and they have half dimension of .R2 . As we saw above, for a linear subspace .L ⊂ E, we have .L ∩ Lσ /= {0}. This fact leads to considering different subspaces in a symplectic vector space. Definition 1.10 Let .(E, σ ) be a symplectic vector space, and L be a linear subspace of E; then: L is said to be isotropic if .L ⊂ Lσ . L is said to be Lagrangian if .L = Lσ . L is said to be coisotropic if .Lσ ⊂ L. L is said to be symplectic if .(L, σ |L ) is a symplectic vector space. Exercise 1.4 Let .L ⊂ E be a subspace of a symplectic vector space E; show that (L is Lagrangian ) ⇐⇒ (L is isotropic and coisotropic ).

.

Exercise 1.5 Let .(E, σ ) be a symplectic vector space. Show that every 1codimensional subspace .W ⊂ E is coisotropic.

1.5 Symplectic Basis Here we are going to explain that for any symplectic form on a vector space E, we can find a basis for E such that on this basis the symplectic form is the same as the symplectic form in Example 1.4. For this reason, the symplectic form of Example 1.4 is called “ standard.”

12

1 Symplectic Linear Algebra

Definition 1.11 Recall from Example 1.4 the symplectic form defined by σst : E = Rn × Rn −→ R

.

σst ((q, p), (q ' , p' )) = pq ' − p' q =

n ∑

pi qi' − pi' qi ,

i=1

where .(q, p) is the short notation of .(q1 , . . . , qn , p1 . . . , pn ). This symplectic form is called standard symplectic form. For an inner product g and a linear subspace L in E, we have .E = L ⊕ L⊥ , where by .L⊥ we mean g-orthogonal complement of .L. But, for a symplectic form σ .σ on E, we do not have such a direct sum because .L ∩ L /= {0}. However, we can still consider E as the direct sum of two Lagrange planes such that their intersection is .{0}. Proposition 1.2 Let .(E, σ ) be a symplectic vector space of .dim(E) = 2n over a field .k. Let L be a Lagrange plane; then there exists a Lagrange plane M such that .L ∩ M = {0}, in other words .E = L ⊕ M. Proof Let M be an isotropic subspace of .(E, σ ) (i.e., .M ⊂ M σ ) such that .L ∩ M = {0}. We claim that there exists .v ∈ M σ \ M such that .L ∩ (M + kv) = {0}. Then we can continue this claim until we get a maximal isotropic subspace as in the claim. Hence, we obtain a Lagrange plane with zero intersection with .L. Proof of the Claim If for all .v ∈ M σ \ M we have .L ∩ (M + kv) /= {0}, so there is .m ∈ M and .c ∈ k such that .m + cv = l is a non-zero element of .L. So .v ∈ L + M. Therefore for all .v ∈ M σ \ M, we have .v ∈ L + M, which means that .M σ ⊂ L + M. Moreover, we have L ∩ M σ = Lσ ∩ M σ = (L + M)σ ⊂ (M σ )σ = M,

.

where in the first equality we used .L = Lσ (because L is a Lagrange plane). The second equality, the inclusion, and the last equality are followed by Lemma 1.5. So σ ⊂ L ∩ M = {0}; hence .L ∩ M σ = {0}. .L ∩ L ∩ M On the other hand, .dim(L) = n and dim(M σ ) = dim(E) − dim(M) > dim(E) − n

.

because we assumed that M is isotropic. Therefore, dim(L) + dim(M σ ) > dim(E).

.

However, dim(L ∩ M σ ) = dim(L) + dim(M σ ) − dim(E) > 0

.

is in contradiction with .L ∩ M σ = {0}.

⨆ ⨅

1.5 Symplectic Basis

13

Proposition 1.3 There is an identification of an arbitrary symplectic form S on E with the standard symplectic form .σst . Proof Let .S : E × E −→ k, be a symplectic form over .E. By Proposition 1.2, we consider E as the direct sum of two Lagrange planes L and M, i.e., .E = L ⊕ M. The restriction of the symplectic form .S : E −→ E ∗ to M as S |M : M −→ L∗

.

m ‫׀‬−→ Sm |L is an isomorphism because if .S(m) = 0, then .Sm |L = 0, which means that m is in S-orthogonal complement of .L, i.e., .m ∈ LS . Since L is a Lagrange plane, we have S .L = L, so .m ∈ L. But we know that .L ∩ M = {0}, so .m = 0 : • Let .{ei }i=1,...,n be a basis of L and .{εj }j =1,...,n be the corresponding dual basis of .L∗ , that is, .εj (ei ) = δij . .• Let .{fj }j =1,...,n be the basis of M such that .Sfj = εj . .

Therefore we obtain S(ei , ei ' ) = 0 ( because L = LS , i.e., S is zero on L)

.

S(fj , fj ' ) = 0 ( because M = M S , i.e., S is zero on M) −S(ei , fj ) = S(fj , ei ) = (S(fj ))(ei ) = εj (ei ) = δij . Now we see that the symplectic form S in the basis of∑ .ei ’s and .fj ’s has the n standard form. Any .u, v ∈ E in this basis is written as .u = i=1 pi ei + qi fi , and ∑n ' ' .v = i=1 pi ei + qi fi . Therefore S(u, v) = S(

n ∑

.

(pi ei + qi fi , pi' ei + qi' fi ))

i=1

=

n ∑

S(pi ei , pi' ei ) + S(pi ei , qi' fi ) + S(qi fi , pi' ei ) + S(qi fi , qi' fi ))

i=1

=

n ∑

pi qi' S(ei , fi ) + qi pi' S(fi , ei )

i=1

=

n ∑

pi qi' − pi' qi

i=1

= σst (q, p). Here in the second equality, we used bi-linearity of .S. In the third and fourth ⨆ ⨅ equality, we calculated S over .ei ’s and .fj ’s as we defined above.

14

1 Symplectic Linear Algebra

As a corollary, if we consider the isomorphism φ : (Rn × Rn , σst ) −→ (E, S)

.

(p, q) ‫׀‬−→

n ∑

pi ei + qi fi ,

i=1

then, by the above lemma, we have obtained .φ ∗ (S) = σst . This is the linear version of Darboux’s theorem, which states that locally, any 2n-dimensional symplectic manifold looks like .(R2n , σst ). We will study Darboux’s theorem on manifolds in Chap. 2 using neighborhood theorems, and in Chap. 3 we will see the original proof of Darboux. Definition 1.12 We call the basis for E defined in the above lemma a symplectic basis of the symplectic vector space .(E, S). The above proposition shows that any basis of a Lagrange plane can be extended to a symplectic basis. The matrix of the symplectic form .S : E −→ E ∗ in the above proposition, in the symplectic basis consisting of .{ei }i=1,...,n and .{fj }j =1,...,n , is written as ⎛

⎞ 0 I .S = σst = , −I 0 where I denotes the .(n × n)-identity matrix. Suppose .σ is an anti-symmetric bi-linear form on a finite-dimensional vector space E over a field .k, which is not non-degenerate. Since .σ is not non-degenerate, the mapping .σ : E −→ E ∗ is not isomorphism, so it has non-zero kernel. Let .N := kerσ and define the bi-linear form σ E (u + N, v + N) := σ (u, v)

.

N

E . It is an easy exercise to check that .σ E is a well-defined antion the quotient space . N N

E that is non-degenerate. Hence, .σ E is a symplectic symmetric bi-linear form on . N N

E form on . N .

Definition 1.13 The bi-linear form .σ E is called induced symplectic form on the E quotient space . N , where .N = kerσ.

N

E Let .r := dim(N), t := 12 dim( N ) and choose .n1 , . . . , nr , e1 , . . . , et , f1 , . . . , ft in E such that .n1 , . . . , nr form a basis of N and .e1 +N, . . . , et +N, f1 +N, . . . , ft + E E , σ E ), i.e., on this basis of . N we have .σ E = σst . N form a symplectic basis of .( N N N Then these .r + 2t vectors form a basis of .E, for which .

σ (ni , nj ) = σ (ni , ej ) = σ (ni , fj ) = 0

1.6 The Lagrangian Grassmannian

15

σ (ei , ej ) = σ (fi , fj ) = 0 σ (ei , fj ) = −σ (fj , ei ) = δij . Here in the first line we used the fact that .ni ’s are in the kernel of .σ. In the second line we have .σ (ei , ej ) = σ E (ei +N, ej +N) = 0 because .ei +N’s are in symplectic N

E basis of . N . Similarly, .σ (fi , fj ) = 0 and we have the third line. The matrix of .σ in the basis consisting of .{ni }i=1,...,r ,.{ei }i=1,...,t , and .{fj }j =1,...,t is written as

⎛

⎞ 0 0 0 .σ = ⎝0 0 I ⎠ . 0 −I 0 This is the standard matrix of a standard form for an arbitrary anti-symmetric bi-linear form .σ over .E. Comparing this matrix .σ with the matrix of standard symplectic form .σst , we see that for the part on which .σ is not non-degenerate we get zero row and column, while for the induced part we get the standard matrix as for .σst . Exercise 1.6 Show that if L is a Lagrangian subspace of .(E, σ ), then any basis e1 , . . . , en of L can be extended to a symplectic basis .e1 , . . . , en , f1 , . . . , fn of .(E, σ ). .

Exercise 1.7 Show that a linear subspace L in symplectic vector space E is symplectic, i.e., .(σ |L×L is non-degenerate ) if and only if L ∩ Lσ = {0} ⇔ E = L ⊕ Lσ .

.

Exercise 1.8 For an arbitrary anti-symmetric bi-linear form .σ on a finitedimensional vector space E over a field .k, show that .σ E (u + N, v + N) := σ (u, v) N

E that is non-degenerate, where .N = kerσ. is an anti-symmetric bi-linear form on . N

1.6 The Lagrangian Grassmannian In Sect. 1.4 we saw the definition of a Lagrange plane. In this section we consider the set of all Lagrange planes in a symplectic vector space .(E, σ ), and we see that actually this set is a smooth manifold. Definition 1.14 The set .L = L(E, σ ) of all Lagrange planes in the symplectic vector space .(E, σ ) is called the Lagrangian Grassmannian of .(E, σ ). The names “ Lagrange plane ” and “ Lagrangian Grassmannian ” were introduced by Arnold [2].

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1 Symplectic Linear Algebra

Remark 1.7 Recall that the set of all n-dimensional linear subspaces of a vector space E is called Grassmann manifold and is denoted by .Gn (E). The Lagrangian Grassmannian .L is a non-empty (see Sect. 1.4) subset of the Grassmann manifold .Gn (E). In the following we show that .L(E, σ ) is a smooth . 21 n(n + 1)-dimensional linear submanifold of the .n2 -dimensional manifold .Gn (E). Notation If .L ∈ L and .k ∈ Z0 (i.e., positive integers), then we denote by .LL,k the set of all Lagrange planes .M ∈ L such that .dim(L ∩ M) = k. We consider the topology over .L as the set of all subsets of the family .{LL,k }L,k . Lemma 1.8 .LL,0 is a non-empty open subset of .L. Proof .LL,0 , which is the set of all Lagrange planes .M ∈ L such that .dim(L∩M) = 0 (hence .L ∩ M = 0) is non-empty by Proposition 1.2. In particular, LL,0 = {M ∈ L(E, σ ) | L ∩ M = {0}}

.

is an element of the topology, so it is an open subset of .L.

⨆ ⨅

Remark 1.8 Note that (by Proposition 1.2) for every .M ∈ L there exists U an .L ∈ L such that .M ∈ LL,0 , and therefore we obtain the open covering .L = L∈L LL,0 . Before we show that the Lagrangian Grassmannian of .(E, σ ) is a manifold, we show below that the Grassman manifold of E is a manifold. Proposition 1.4 The Grassman manifold of E is a manifold of dimension equal to n2 .

.

Proof If .L ∈ Gn (E) and .k ∈ Z0 (i.e., positive integers), then we denote by (Gn (E))L,k the set of all n-planes .M ∈ Gn (E) such that .dim(L ∩ M) = k. We consider the topology over .Gn (E) as the set of all subsets of the family .{(Gn (E))L,k }L,k . Let .L, M ∈ Gn (E) be such that .E = L ⊕ M. For every linear mapping .A : L −→ M, we consider .

L' = {x + Ax | x ∈ L} ∈ Gn (E),

.

which has the property that .L' ∩ M = {0}. Conversely, for every .L' ∈ Gn (E) such that .L' ∩ M = {0}, there is a unique linear mapping .A : L → M such that ' .L = {x + Ax | x ∈ L}. Therefore, there is a bijection from the open subset ' .{L ∈ Gn (E) | L' ∩ M = {0}} of .Gn (E) onto the .n2 -dimensional vector space .Lin(L, M) of all linear mappings from L to .M. The matrix coefficients of A with respect to any bases in L and M define a coordination of the above open subset of .Gn (E), and these are the standard coordination of .Gn (E). In other words, for every ' ' .L ∈ Gn (E) and an open subset containing .L , we obtain the mapping .ϕ

1.6 The Lagrangian Grassmannian

17

ϕ : {L' ∈ Gn (E) | L' ∩ M = {0}} −→ Lin(L, M)

.

L' ‫׀‬−→ A. 2

In other words, .Gn (E) is locally isomorphic to .Rn and all the mappings such as .ϕ are charts. ⨆ ⨅ Proposition 1.5 The Lagrangian Grassmannian of the symplectic vector space (E, σ ) is a manifold that has dimension equal to

.

dim(L(E, σ )) =

.

1 n(n + 1). 2

Proof Assume that L and M are Lagrange planes, i.e., .L, M ∈ L(E, σ ). Consider L' ∈ Gn (E) such that .L' ∩ M = {0}. Following the previous Proposition, there is a unique linear mapping .A : L −→ M such that .L' = {x + Ax | x ∈ L}. Now we observe that .L' ∈ L(E, σ ) if and only if for all .x, y ∈ L

.

0 = σ (x + Ax, y + Ay)

.

= σ (x, y) + σ (x, Ay) + σ (Ax, y) + σ (Ax, Ay) = σ (Ax, y) − σ (Ay, x), where in the second line we used bi-linearity of .σ and in the third line we used the fact that .σ is zero over L and .M, because they are Lagrangian planes. In other words, .L' is a Lagrangian subspace if and only if σ (Ax, y) = σ (Ay, x).

.

We define a bi-linear mapping T : E × E −→ k

.

T (x, y) := σ (Ax, y). This means that the bi-linear mapping T on L is symmetric, because T (x, y) = σ (Ax, y)

.

= σ (Ay, x) = T (y, x). In other words, .L' is a Lagrangian subspace if and only if the bi-linear mapping T is symmetric. Now the linear mapping .ψ : A ‫( →׀‬TA : (x, y) ‫ →׀‬σ (Ax, y)) is an isomorphism from .Lin(L, M) onto the space of all bi-linear forms on .L. And when .L, M, L' are

18

1 Symplectic Linear Algebra

Lagrangian, this is an isomorphism from .Lin(L, M) to all symmetric bi-linear form. That is, the matrix coefficient of A is in correspondence to a symmetric matrix. So, ' .TA is in correspondence to .L by the mapping .ϕ ¯: ϕ¯ : {L' ∈ Gn (E) ∩ L(E, σ ) | L' ∩ M = {0}}

.

−→

Symm2 (L)

L' ‫׀‬−→ A − ‫ → ׀‬TA . 1

In other words .L(E, σ ) is locally the same as .R 2 n(n+1) . This shows that the set of all Lagrangian subspaces, .L(E, σ ), is a . 12 n(n + 1)-dimensional linear submanifold of the .n2 -dimensional manifold .Gn (E) and all the mappings such as .ϕ¯ are charts. ⨆ ⨅ Proposition 1.6 The Lagrangian Grassmannian .L(E, σ ) is a smooth submanifold of the Grassmann manifold .Gn (E). Proof In previous Proposition we saw that .L(E, σ ), is a submanifold of the Grassmann manifold .Gn (E). Here we are going to show that this structure of manifold on .L(E, σ ) is smooth. To show the smoothness, we have to show the compatibility of the charts which we found in previous Proposition, as in Fig. 1.1. Let .M˜ ∈ LL,0 be another Lagrange plane for which .L∩ M˜ = {0}, and let .L' ∈ L ' ˜ with .L' ∈ LL,0 (i.e., .L' ∩ L = {0}) and .L' ∈ LM,0 ˜ (i.e., .L ∩ M = {0}). Then we ˜ such that have a second coordination of .L' by the .A˜ ∈ Lin(L, M) ˜ | y ∈ L}. L' = {y + Ay

.

On the other hand there exists .B ∈ Lin(M, L) such that M˜ = {z + Bz | z ∈ M}.

.

Fig. 1.1 Compatible charts

1.6 The Lagrangian Grassmannian

19

˜ From previous Proposition, the elements of .L' are of the form .x + Ax = y + Ay ˜ = z + Bz for a unique .z ∈ M. Therefore, from for unique .x, y ∈ L, and .Ay .x + Ax = y + z + Bz, with .x, y, Bz ∈ L and .Ax, z ∈ M, it follows that .z = Ax and therefore, .y = x − Bz = x − BAx, and hence ˜ x + Ax = y + Ay

.

˜ − BAx). = (x − BAx) + A(x Calculating the symplectic form with .u ∈ L, we get ˜ − BAx), u). σ (x + Ax, u) = σ ((x − BAx) + A(x

.

Since .x, u ∈ L and L is Lagrangian, we have .σ (x, u) = 0. So, σ (x + Ax, u) = σ (x, u) + σ (Ax, u) = 0 + σ (Ax, u).

.

Hence, we get ˜ − BAx), u) σ (Ax, u) = σ (x − BAx, u) + σ (A(x

.

˜ − BAx), u) = 0 + σ (A(x ˜ u) − σ (ABAx, ˜ = σ (Ax, u), where in the second equality we used the fact that .(x − BAx), u ∈ L and L is Lagrangian. Hence, we obtain ˜ TA (x, u) = TA˜ (x, u) − σ (ABAx, u).

.

Therefore, for A that corresponds to .L' close to .L, we see that the symmetric bilinear form .TA on L defined by M differs from the symmetric bi-linear form .TA˜ on ˜ L defined by .M˜ by a term .−σ (ABAx, u). This means that there is a correspondence between .TA and .TA˜ , and this provides a smooth mapping Symm2 (L) −→ Symm2 (L)

.

˜ TA (x, u) ‫׀‬−→ TA (x, u) + σ (ABAx, u), where .Symm2 (L) ∼ = R 2 n(n+1) . 1

⨆ ⨅

Remark 1.9 By the above proposition, the tangent space .TL L of .L at the element L ∈ L is canonically identified with the space .Symm2 (L) of all symmetric bi-linear forms on .L.

.

20

1 Symplectic Linear Algebra

Exercise 1.9 Let .(E, σ ) be a symplectic vector space and .L ⊂ E a subspace. Show that there is a symplectic form .σ¯ on .L/L ∩ Lσ such that .σ¯ (pr(X), pr(Y )) = σ (X, Y ) for all .X, Y ∈ L where .pr : L −→ L/L ∩ Lσ is the quotient map. Exercise 1.10 Let .L ∈ L(E, σ ) and .I ∈ Gm (L). Prove that: (a) .σ induces a symplectic form on the .(2n − 2m)-dimensional vector space .I σ /I. (b) Prove that .L' ∈ L(E, σ ) and .L' ∩ L = I if and only if .L' /I is a Lagrange plane in .I σ /I that is transversal to .L/I , i.e., .L' /I ∩ L/I = {0}.

1.7 The Symplectic Linear Group and Its Lie Algebra In this section we study mappings between symplectic vector spaces, which preserve their symplectic structures. We see that the set of such mappings is a Lie group and it is in relation with Lagrangian Grassmannian. Definition 1.15 Let .(E, σ ) and .(F, τ ) be symplectic vector spaces over a field .k. A linear mapping .A : E −→ F is called a symplectic linear mapping from .(E, σ ) to .(F, τ ) if it preserves symplectic structure, i.e., A∗ τ = σ.

.

In other words, we have .A∗ τ (u, v) = σ (u, v) for every .u, v ∈ E, or by definition of pullback we have τ (Au, Av) = σ (u, v), ∀u, v ∈ E.

.

Remark 1.10 If we consider the symplectic forms .σ : E × E ∗ −→ k and .τ : F × F ∗ −→ k as isomorphisms .σ : E −→ E ∗ and .τ : F −→ F ∗ , then the condition .A∗ τ = σ in the above definition is equivalent to say that At τ A = σ.

.

This means that the following diagram commutes.

Since .σ and .τ are isomorphisms (and hence injective), so A is injective and dim(E) ≤ dim(F ).

.

1.7 The Symplectic Linear Group and Its Lie Algebra

21

Definition 1.16 If .dim(E) = dim(F ), we call the mapping A a symplectic linear isomorphism or symplectomorphism from .(E, σ ) onto .(F, τ ). If .dim(E) < dim(F ), then the mapping A is a symplectomorphism from .(E, σ ) onto the symplectic vector subspace .(A(E), τ |A(E)×A(E) ) of .(F, τ ). The following lemma shows that a symplectic linear mapping preserves symplectic basis, and, conversely, if we have a linear mapping that preserves symplectic basis, then it is a symplectic linear mapping. Lemma 1.9 Let .e1 , . . . , en , f1 , . . . , fn be a symplectic basis for the symplectic vector space .(E, σ ). Then a linear mapping .A : (E, σ ) −→ (F, τ ) is symplectic if and only if the elements .Ae1 , . . . , Aen , Af1 , . . . , Afn is a symplectic basis for .(F, τ ). Proof Let A be a symplectic linear mapping. Then by definition it satisfies the condition .A∗ τ = σ or .τ (Au, Av) = σ (u, v), ∀u, v ∈ E. Therefore, by Proposition 1.3, on the symplectic basis .e1 , . . . , en , f1 , . . . , fn , we have .

τ (Aei , Aei ' ) = σ (ei , ei ' ) = 0 τ (Afi , Afi ' ) = σ (fi , fi ' ) = 0 τ (Aei , Afj ) = σ (ei , fj ) = δij ,

which means that .Ae1 , . . . , Aen , Af1 , . . . , Afn is a symplectic basis, by Proposition 1.3. Reversing this argument completes the proof. ⨆ ⨅ Definition 1.17 A symplectic linear mapping from .(E, σ ) to itself is called a symplectic linear transformation in .(E, σ ). Definition 1.18 The symplectic linear transformations form an algebraic subgroup of the group .GL(E), of all linear transformations in .E, which is called the symplectic linear group and is denoted by .Sp(E, σ ). Lemma 1.10 The symplectic linear group, .Sp(E, σ ), is a closed subgroup of the Lie group .GL(E). Proof Consider the continuous mapping f : GL(E) −→ R2n×2n

.

f (A) = At σ A − σ.

.

Then .f −1 ({0}) = Sp(E, σ ) is a closed subgroup of the Lie group .GL(E).

⨆ ⨅

Remark 1.11 From Lie groups theory, by closed-subgroup theorem (also known as Cartan’s theorem) [27, 34, 48], we remember that a closed subgroup of a Lie group is equipped with a smooth structure as a submanifold and hence is Lie group. So the previous lemma shows that .Sp(E, σ ) is a Lie subgroup.

22

1 Symplectic Linear Algebra

In the following we see that the Lagrangian Grassmannian .L(E, σ ) is described by the symplectic linear group .Sp(E, σ ). Lemma 1.11 The Lagrangian Grassmannian .L(E, σ ) is a homogeneous space for the group .Sp(E, σ ). Proof From Proposition 1.3, we remember that any basis of a Lagrange plane can be extended to a symplectic basis. This fact together with Lemma 1.9 implies that the symplectic linear group .Sp(E, σ ) acts transitively on the Lagrangian Grassmannian .L(E, σ ). More precisely, if we consider the action of the group .Sp(E, σ ) on Lagrangian Grassmannian as Sp(E, σ ) × L(E, σ ) −→ L(E, σ ),

.

then for any .L, L' ∈ L(E, σ ), there exists .A ∈ Sp(E, σ ) such that .A(L) = L' .

⨆ ⨅

Lemma 1.12 .L(E, σ ) ∼ = Sp(E, σ )/Sp(E, σ )L . Proof For given .L ∈ L(E, σ ), its stabilizer group is Sp(E, σ )L := {A ∈ Sp(E, σ ) | A(L) = L}.

.

The orbit of L that is .O(L) = {A.L | A ∈ Sp(E, σ )} is equal to .L(E, σ ), since the action of the group is transitive. Therefore, we can write the Lagrangian Sp(E,σ ) ⨆ ⨅ Grassmannian as .L(E, σ ) ∼ = Sp(E,σ )L . Proposition 1.7 The dimension of Lie group .Sp(E, σ ) is equal to .dim(Sp(E, σ )) = 2n2 + n. Proof We compute the dimension of .Sp(E, σ ) in two ways: (1) If we write ⎛ A=

.

αβ γ δ

⎞ 2n×2n

on a symplectic basis, in which .α, β, γ , and .δ are .n × n-matrices, then .A ∈ Sp(E, σ ) if and only if .α t γ − γ t α = 0, β t δ − δ t β = 0 and .α t δ − γ t β = I. Because A ∈ Sp(E, σ ) ⇐⇒

.

At σst A = σst ⇐⇒ ⎛ .

αt γ t β t δt

⎞

⎞ ⎞ ⎛ ⎞ ⎛ 0 I αβ 0 I ⇐⇒ = × × −I 0 γ δ −I 0 ⎛

1.7 The Symplectic Linear Group and Its Lie Algebra

23

⎛ t ⎞ ⎛ ⎞ ⎛ ⎞ −γ I α t I αβ 0 I × . = ⇐⇒ −δ t I β t I γ δ −I 0 ⎛ t ⎞ ⎛ ⎞ −γ α + α t γ −γ t β + α t δ 0 I . = ⇐⇒ −δ t α + β t γ −δ t β + β t δ −I 0 α t γ − γ t α = 0,

.

β t δ − δ t β = 0, α t δ − γ t β = I. In the left side of the first equation, .α t γ − γ t α is anti-symmetric so we get 1 t t . n(n − 1) independent equations. Similarly in the second equation, .β δ − δ β is 2 1 anti-symmetric so we have . 2 n(n−1) independent equations. The third equation gives us .n2 equations. So in sum we have .2( 12 n(n − 1)) + n2 = 2n2 − n independent equations. Moreover, the first two equations mean that .α t γ and .β t δ are symmetric. So each one provides . 12 n(n + 1) generators, and the third equation gives us .n2 generators. So for .A ∈ Sp(E, σ ) we get .2( 12 n(n+1))+n2 = 2n2 +n generators so .dim(Sp(E, σ )) = 2n2 + n. (2) If the first n vectors of the symplectic basis span the Lagrange plane .L, then .A ∈ Sp(E, σ )L if and only if .γ = 0 and therefore by above equations we get .δ = (α t )−1 and .δ t β = β t δ = β t (α t )−1 = β t (α −1 )t = (α −1 β)t = (α −1 β), where the last equality follows by the fact that .δ t β is symmetric. So, in summary, when t t t t .A ∈ Sp(E, σ )L two equations .β δ − δ β = 0 and .α δ − β γ = I lead to a −1 t t symmetric matrix .α β and the equation .α δ − γ β = In×n ; hence, dim(Sp(E, σ )L ) =

.

1 n(n + 1) + n2 . 2

We saw in previous lesson that .dim(L(E, σ )) = 12 n(n + 1), and from previous Sp(E,σ ) lemma, we know that .L(E, σ ) = Sp(E,σ )L . Therefore dim(Sp(E, σ )) = dim(L(E, σ )) + dim(Sp(E, σ )L )

.

=

1 1 n(n + 1) + n(n + 1) + n2 2 2

= n(n + 1) + n2 = n2 + n + n2 = 2n2 + n. ⨆ ⨅

24

1 Symplectic Linear Algebra

Remark 1.12 It follows that the co-dimension of .Sp(E, σ ) in .GL(E) is equal to dim(GL(E)) − dim(Sp(E, σ )) = 4n2 − (2n2 + n) = 2n2 − n,

.

which is the number of independent equations for matrices .α, β, γ , and .δ. Recall We remember that .GLR (n) is the linear group of degree n that is the set of n × n invertible matrices (their determinant is non-zero). .GLR (n) together with the operation of matrix multiplication forms a group. It is a real group of dimension .n2 . We denote the special linear group by .SLR (n), which is the subgroup of .GLR (n) consisting of matrices with determinant equal to .1.

.

Similarly, .GL(E) is the group of all invertible matrices on E, and we have Sp(E, σ )  GL(E).

.

Lemma 1.13 If .A ∈ Sp(E, σ ), then .det (A) = 1 and so .Sp(E, σ ) is a subgroup of SL(E), that is, .Sp(E, σ )  SL(E).

.

Proof Let .A ∈ Sp(E, σ ), and therefore .At σ A = σ. Using property of determinant we have .det (At )det (σ )det (A) = det (σ ). If we identify .σ with the standard symplectic form, then its matrix is ⎛ .

⎞ 0 I , −I 0

whose determinant is equal to .1. Therefore, .det (At )det (A) = 1. We know that t 2 .det (A ) = det (A) so we get .(det (A)) = 1 or .det (A) = ±1. Later in the Sect. 1.9, we will see that .det (A) = 1. ⨆ ⨅ Below we see an interesting property of eigenvalues of symplectic matrices. Recall A scalar .λ ∈ k is an eigenvalue of a matrix A if and only if there is an eigenvector .v /= 0 such that .Av = λv. Since v is non-zero, this means the matrix .λI −A is singular (non-invertible), which means its determinant is .0. Thus, the roots of the function f (λ) = det (λI − A)

.

are the eigenvalues of .A. This function is called the characteristic polynomial of .A, a polynomial whose zeros are the eigenvalues of .A. Let .A ∈ Sp(E, σ ). Then .At σ A = σ implies that .(At )−1 = σ Aσ −1 , which means that the linear isomorphism .σ : E → E ∗ conjugates the linear mapping t −1 : E ∗ → E ∗ .This implies that A has the .A : E → E with the linear mapping .(A ) t −1 same eigenvalues as .(A ) (with the same multiplicities). Because by determinant properties, we have det (A − λI ) = det (σ (A − λI )σ −1 )

.

= det ((At )−1 − λI ).

1.7 The Symplectic Linear Group and Its Lie Algebra

25

Lemma 1.14 If .λ is an eigenvalue of .A ∈ Sp(E, σ ) (with algebraic multiplicity m), then . λ1 is also an eigenvalue of A (with algebraic multiplicity m). Proof Consider the characteristic polynomial of the matrix A as .f (λ) = det (A − λI ). Using determinant properties this is equal to det (σ (A − λI )σ −1 ) = det ((At )−1 − λI )

.

= det (A−1 − λI ) 1 = det[λA−1 ( I − A)] λ 1 = λ2n det (A−1 )det ( I − A) λ 1 = λ2n det (A − I ) λ 1 = λ2n f ( ), λ where the fifth equality follows from the fact that .det (A−1 ) = 1 (because: from −1 = I , we have .(detA)(detA−1 ) = 1 and we know that .detA = 1.) .AA ⨆ ⨅ Remark 1.13 (1) Note that .λ /= 0 since A is isomorphism. (2) If .k = R, then considering the complexification, the “ complex" eigenvalues of A are in foursomes .λ, λ¯ , λ1 , λ1¯ with .λ ∈ C, λ ∈ / R, λλ¯ /= 1, or in real pairs .λ, λ1 , or in complex conjugate pairs .λ, λ¯ on the unit circle, in each case with equal multiplicities. Remark 1.14 The above lemma shows that the Lie group .Sp(E, σ ) is not compact. Because, if we consider a sequence of elements of .Sp(E, σ ) for which some of their eigenvalues .λ go to zero, then such elements also have eigenvalues going to infinity. After studying the Lie group of symplectic linear transformations, it is interesting to know how one can define the Lie algebra of such a Lie group. Let .sp(E, σ ) be the Lie algebra of the symplectic linear group .Sp(E, σ ). Consider .M(s) as a smooth path in .Sp(E, σ ) such that .M(0) = I and .M ' (0) = A. We have .M t σ M = σ. We take derivation, so we obtain .(M t )' (s)σ M(s) + M t (s)σ M ' (s) = 0. For .s = 0 we have .At σ + σ A = 0. This is equivalent to σ A + At σ = 0

.

or .

((σ A)(u))(v) + ((At σ )(u))(v) = 0 ⇐⇒ (σ (Au))(v) + (At (σ (u)))(v) = 0 ⇐⇒

26

1 Symplectic Linear Algebra

σ (Au, v) + (σ (u))(Av) = 0 ⇐⇒ σ (Au, v) + σ (u, Av) = 0, where the third line is obtained by the definition of transpose. Definition 1.19 The Lie algebra .sp(E, σ ) of the Lie group .Sp(E, σ ) consists of the linear mappings .A : E −→ E such that .σ A+At σ = 0 or .σ (Au, v)+σ (u, Av) = 0 for all .u, v ∈ E. Such a linear mapping A is called the infinitesimal symplectic linear transformation in .(E, σ ). Remark 1.15 Consider the bi-linear form .T : E × E −→ R defined by .T (u, v) = σ (Au, v). The bi-linear form T is symmetric because T (u, v) = σ (Au, v)

.

= −σ (u, Av) = σ (Av, u) = T (v, u), where the second equality follows by the fact that A is in the Lie algebra and the third equality is obtained by anti-symmetric property of .σ. Note that the mapping .ϕ : sp(E, σ ) −→ Symm2 (E) that assigns to .A ∈ sp(E, σ ) the bi-linear form .T = σ A : (u, v) − ‫ → ׀‬σ (Au, v) is a linear isomorphism 2 from .sp(E, σ ) onto the space .Symm (E) of all symmetric bi-linear forms on E (we can consider the mapping .ϕ −1 as .T −→ σ −1 T = A).Therefore, dim(Sp(E, σ )) =

.

1 2n(2n + 1) = 2n2 + n, 2

which we obtained in dimension calculations in Proposition 1.7. The same as for elements of the Lie group .Sp(E, σ ), the eigenvalues of the elements of the Lie algebra .sp(E, σ ) have an interesting property. Lemma 1.15 If .λ is an eigenvalue of .A ∈ sp(E, σ ) (with multiplicity m), then .−λ is also an eigenvalue of A (with multiplicity m). Proof The equation .σ A+At σ = 0 or .At = −σ Aσ −1 implies that the characteristic polynomial of A is equal to f (λ) = det (A − λI ) ) ( = det σ (A − λI )σ −1 ) ( = det (σ Aσ −1 ) − λI

.

1.8 Hermitian Forms

27

= det (−At − λI ) = det (−(At + λI )) = (−1)2n det (At − (−λI )) = det (A − (−λI )) = f (−λ). Here, in the second equality we used determinant property, and the forth equality follows from the fact that A is in the Lie algebra. In the seventh equality, we used the fact that the eigenvalues of A and .At are the same. ⨆ ⨅ Remark 1.16 If .k = R, then the “complex" eigenvalues of A are in foursomes ¯ −λ, −λ¯ with .λ ∈ C, λ ∈ λ, λ, / R, λ ∈ / iR, or in real pairs .λ, −λ, or in complex conjugate pairs .λ, λ¯ on the imaginary axis.

.

Exercise 1.11 Let .ϕ : Rn × Rn → E be an isomorphism defined by .(p, q) ‫→׀‬ ∑n n n i=1 pi ei + qi fi , where .ei , fi are elements of the basis of .E. We equip .R × R with the standard symplectic form .σ and E with a symplectic form .S. Show that in a symplectic basis of .(E, S) the mapping .ϕ is symplectomorphism. Exercise 1.12 Show that .Sp(E, σ ) is a subgroup of .GL(E). Exercise 1.13 Let .(E, σ1 ) and .(F, σ2 ) be symplectic vector spaces and let .ϕ : E → F be a linear isomorphism. Show that .ϕ is a symplectomorphism if and only if the graph of .ϕ, i.e., .graph(ϕ) = {(v, ϕ(v) | v ∈ E}, is a Lagrangian subspace of .E × F with symplectic form .π1∗ σ1 − π2∗ σ2 , where .π1 : E × F → E and .π2 : E × F → F are projections.

1.8 Hermitian Forms In this section we are going to see how symplectic structures and complex structures are related by Hermitian forms. Recall Let E be an n-dimensional vector space over .C. A mapping . : E × E −→ C is an inner product on E if for every .u, v ∈ E it satisfies the following properties: (1) Conjugate symmetry: . = . (2) Linearity in the first component: . = α and . = + , .α ∈ C. (3) Positive-definiteness: . ≥ 0 with equality only for .u = 0. Definition 1.20 Let E be an n-dimensional vector space over .C. A mapping .h : E× E −→ C is a Hermitian form on E if for every .u ∈ E the mapping .v ‫׀‬−→ h(u, v) is complex anti-linear, i.e., .h(u, cv) = ch(u, ¯ v),, and for every .v ∈ E the mapping

28

1 Symplectic Linear Algebra

u ‫׀‬−→ h(u, v) is complex linear, and .h(u, u) > 0 for every non-zero element u of E. In summary

. .

(1) h(cu, v) = ch(u, v),

.

(2) h(u, cv) = ch(u, ¯ v), (3) h(u, u) > 0, (∀ 0 /= u ∈ E). In other words, the first condition in the definition is the complex linearity with respect to the first component that is .h(au + bv, w) = ah(u, w) + bh(v, w), and the second condition is the complex anti-linearity with respect to the second component that is .h(u, av + bw) = ah(u, v) + bh(u, w). Note that the conjugate symmetry property of an inner product together with linearity in the first component implies the conjugate linearity in the second component. Because: . = = α = α = α, where .α ∈ C. From now on we consider E as a 2n-dimensional vector space over .R, and we denote .g := Re(h) and .σ := I m(h). In other words we write .h(u, v) = g(u, v) + iσ (u, v). The bi-linearity of h implies that .g := Re(h) and .σ := I m(h) are bi-linear forms. If we consider the Hermitian form h as an inner product, it is conjugate symmetric, i.e., .h(v, u) = h(u, v). So .g(v, u) + iσ (v, u) = g(u, v) − iσ (u, v). This shows that g is symmetric and .σ is anti-symmetric. Below we see that for a Hermitian form h actually g is an inner product and .σ is a symplectic form. Lemma 1.16 The real part .g := Re(h) of a Hermitian form h is an inner product on .E. Proof We write .h(u, v) = g(u, v) + iσ (u, v) for any .u, v ∈ E. So we have g(u, v) = h(u, v) − iσ (u, v). As we said above the bi-linearity of h implies that g and .σ should be bi-linear. In other words, for .a ∈ R, the linearity of g with respect to the first component is obtained by

.

g(au, v) = h(au, v) − iσ (au, v)

.

= a(h(u, v) − iσ (u, v)) = ag(u, v). The linearity of g with respect to the second component is obtained by g(u, av) = h(u, av) − iσ (u, av)

.

= ah(u, ¯ v) − aiσ (u, v) = a(h(u, v) − iσ (u, v)) = ag(u, v),

1.8 Hermitian Forms

29

where in the second equality we used anti-linearity property of .h; however, since we assumed that .a ∈ R, we get the third equality. Next, g is symmetric because g(v, u) = h(v, u) − iσ (v, u)

.

= h(u, v) + iσ (u, v) = g(u, v) − iσ (u, v) + iσ (u, v) = g(u, v). Here in the second equality we used conjugate symmetric assumption for .h, and we used anti-symmetric property of .σ. (3) Since .σ is anti-symmetric and h is positive definite, we have g(u, u) = h(u, u) − iσ (u, u) = h(u, u) − 0 > 0.

.

Moreover, .h(u, u) = 0 only for .u = 0 so .g(u, u) = 0 only for .u = 0.

⨆ ⨅

In other words, in above lemma we showed that the real part .g := Re(h) of h is a non-degenerate symmetric bi-linear form. If we consider the bi-linear mappings g and .σ as the linear mappings .g : E −→ E ∗ and .σ : E −→ E ∗ , then we obtain the following lemma. Lemma 1.17 Let .J : E −→ E be the real linear transformation in E defined by the complex multiplication by the complex number .i, i.e., .J (u) = iu, u ∈ E. Then σ = −g ◦ J.

.

Proof Consider the bi-linear form .σ : E × E −→ R and then (σ (u))(v) = σ (u, v)

.

= I m(h(u, v)) = −Re(iRe(h(u, v)) − I m(h(u, v))) = −Re(ih(u, v)) = −Re(h(iu, v)) = −g(iu, v) = (−g(J (u)))(v) = ((−g ◦ J )(u))(v). Here in the fifth equality, we used linearity of h with respect to the first component. ⨆ ⨅

30

1 Symplectic Linear Algebra

Lemma 1.18 The imaginary part of h, i.e., .σ , is a symplectic form on .E. Proof In the above we saw that .σ is an anti-symmetric bi-linear form, more precisely σ (u, v) = I m(h(u, v))

.

= −Re(ih(u, v)) = −Re(h(iu, v)) = −Re(h(v, iu)) = −Re(ih(v, u)) = −Re(−ih(v, u)) = −I m(h(v, u)) = −σ (v, u), where in the fourth equality we used the symmetric property of the real part of h that is an inner product. In the fifth equality we used anti-linearity of h with respect to the second component. Since .J : E −→ E and .g : E −→ E ∗ are injective from previous lemma, we conclude that .σ : E −→ E ∗ is also injective. Since .dim(E) = dim(E ∗ ), it is isomorphism. Therefore .σ is a symplectic form on .E. ⨆ ⨅ Remark 1.17 Given a basis .{e1 , e2 , . . . , en } for the complex space, this set, together with these vectors multiplied by .i, namely .{ie1 , ie2 , . . . , ien }, forms a basis for the real space. If we consider the order of the basis as .{e1 , e2 , . . . , en , ie1 , ie2 , . . . , ien }, then the matrix for J is written as ⎛

J2n

.

⎞ 0 −In = . In 0

Note that .J 2 = −I (or .J = −J −1 ), so .σ = −g ◦ J implies that .σ J = −g ◦ J ◦ J, and hence g = σ J.

.

Therefore the Hermitian form is equal to h = σ ◦ J + iσ.

.

Definition 1.21 In general if E is a vector space over .R, then a linear mapping J : E −→ E such that .J 2 = −I is called a complex structure in .E. The linear mapping .J : E −→ E u ‫׀‬−→ iu is an example of a complex structure that is called standard complex structure.

.

1.8 Hermitian Forms

31

Remark 1.18 A complex structure in E makes E into a complex vector space if we define .(a + ib)v := av + J (bv) for any .a, b ∈ R and .v ∈ E, and it follows that the real dimension of E is equal to 2n if n denotes the dimension of E as a complex vector space. We saw that having a Hermitian form means that we have a symplectic form. Lemma 1.19 Every symplectic form is equal to the imaginary part of a Hermitian form with respect to a suitable complex structure. ∑ Proof Consider the standard Hermitian form written as .hst (z, z' ) = n zj z¯' in Cn , where .zj = qj + ipj and .zj' = qj' + ipj' . Thus we have

j =1

j

.

hst (z, z' ) =

n ∑

.

zj z¯j'

j =1

=

n ∑ (qj + ipj )(qj' + ipj' ) j =1

=

n ∑ (qj + ipj )(qj' − ipj' ) j =1

=

n ∑

qj qj' − iqj pj' + ipj qj' + pj pj'

j =1

=

n ∑ (qj qj' + pj pj' ) + i(pj qj' − qj pj' ) j =1

=

n n ∑ ∑ (qj qj' + pj pj' ) + i (pj qj' − qj pj' ) j =1

j =1

= gst (z, z' ) + iσst (z, z' ). So we see that from the standard Hermitian form we get the standard inner product and standard symplectic form. From Sect. 1.5 we remember that there is an identification of an arbitrary symplectic form on E with the standard symplectic form .σst in a symplectic basis for E. This means that we can write every symplectic form in the standard form, and by the above calculation, we can consider it as the imaginary part of a Hermitian form (with respect to a suitable complex structure). ⨆ ⨅ Definition 1.22 If .(E, ω) is a symplectic vector space, then a complex structure J on E is called .ω-compatible if it satisfies:

32

1 Symplectic Linear Algebra

1. .ω(J u, J v) = ω(u, v) for all .u, v ∈ E. 2. .ω(v, J v) > 0 for all non-zero .v ∈ E. As we saw, linear mappings that preserve symplectic structure make the Lie group of symplectic linear group. Below we see that linear mappings that preserve Hermitian structure are symplectomorphisms as well. Moreover, they preserve the inner product and complex structure. Definition 1.23 .U (E, h) denotes the unitary group of all complex linear transformations .A : E −→ E such that .A∗ (h) = h, in which the Hermitian form .A∗ h on E is defined by .(A∗ h)(u, v) := h(Au, Av) for all .u, v ∈ E, i.e., U (E, h) = {A : E −→ E | A∗ (h) = h}.

.

In other words, .A ∈ U (E, h) if .h(Au, Av) = h(u, v) for all .u, v ∈ E. Note that from .h(Au, Av) = h(u, v) we have .h(AA∗ u, v) = h(u, v), for all ∗ = I. Therefore the above definition of the .u, v ∈ E. This implies that .AA unitary group turns into its common definition that is the group of all complex linear transformations .A : E −→ E such that .AA∗ = I. Recall Let .GLC (E) denote the group of complex linear transformations in .E. A real linear transformation A in E is complex linear if and only if .A ◦ J = J ◦ A. Let .O(E, g) denote the group of g-orthogonal real linear transformations in .E, i.e., O(E, g) = {A : E −→ E | A∗ (g) = g}.

.

Remark 1.19 From .g = σ ◦ J and .h = g + iσ we conclude that: (1) The equality .A ◦ J = J ◦ A implies that .U (E, h) = GLC (E) ∩ O(E, g). (2) .g = σ ◦ J implies that .U (E, h) = GLC (E) ∩ Sp(E, σ ). (3) .h = g + iσ implies that .U (E, h) = Sp(E, σ ) ∩ O(E, g). Therefore .U (E, h) is equal to the intersection of any two of .Sp(E, σ ) and .O(E, g) with .GLC (E), i.e., U (E, h) = GLC (E) ∩ O(E, g) = GLC (E) ∩ Sp(E, σ ) = Sp(E, σ ) ∩ O(E, g).

.

Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible. In other words any two of the three equations .A∗ g = g, A∗ σ = σ, AJ = J A imply the third one. For instance, a compatible orthogonal and complex structure induces a symplectic structure. In summary we have the following Fig. 1.2 which we have borrowed from [7]. In the previous section, we saw how one can describe the Lagrangian Grassmannian .L(E, σ ) by symplectic linear group .Sp(E, σ ). Now that we know there is a

1.8 Hermitian Forms

33

Fig. 1.2 Unitary group as the intersection of linear groups

relation between symplectic linear group and unitary group .U (E, h), it is natural to ask how we can describe the Lagrangian Grassmannian by unitary group. Lemma 1.20 The vectors .e1 , . . . , en ∈ E form an h-orthonormal basis of the complex vector space .E, if and only if they form a g-orthonormal basis of a Lagrange plane .L. Proof Let .{e1 , . . . , en } be an h-orthonormal basis of the complex vector space .E, i.e., .h(ei , ej ) = δij . More precisely, h(ei , ej ) =

.

⎛ 0, if i /= j 1, if i = j

implies that g(ei , ej ) = Re(h(ei , ej )) =

.

⎛ 0, if i /= j 1, if i = j

.

So .{e1 , . . . , en } is g-orthonormal. On the other hand .σ (ei , ej ) = I m(h(ei , ej )) = 0, which means .σ on .{e1 , . . . , en } is zero, i.e., .L = is a Lagrange plane generated by the g-orthonormal basis .{e1 , . . . , en }. We can converse the above argument. ⨆ ⨅ Lemma 1.21 .U (E, h) acts transitively on the set of all h-orthonormal bases of .E. Proof Let M be the set of all h-orthonormal bases of E on which the unitary group U (E, h) acts by

.

.

U (E, h) × M −→ M (A, {e1 , . . . , en }) − ‫ → ׀‬A({e1 , . . . , en }).

34

1 Symplectic Linear Algebra

Note that .A({e1 , . . . , en }) is also an h-orthonormal basis, since from assumption that A ∈ U (E, h) we have .h(Aei , Aej ) = h(ei , ej ) = δij . Therefore for an arbitrary element .{e1 , . . . , en } in M, we have

.

Orbit ({e1 , . . . , en }) = {A.{e1 , . . . , en } | A ∈ U (E, h)} = M.

.

This means that the group action has exactly one orbit that implies that .U (E, h) acts transitively over .M. ⨆ ⨅ Lemma 1.22 .U (E, h) acts transitively on the Lagrangian Grassmannian .L(E, σ ). Proof By previous lemma, .U (E, h) acts transitively on the set of all h-orthonormal bases of .E. Moreover, using Lemma 1.20, we have the following correspondence. (h-orthonormal basis of the complex vector space ).←→ (g-orthonormal basis of a Lagrange plane) This implies that the (compact) Lie group .U (E, h) acts transitively on the Lagrangian Grassmannian. ⨆ ⨅ Lemma 1.23 .U (E, h)/O(L, g) ∼ = L(E, σ ). Proof By previous lemma, since the group action is transitive, the orbit space is the whole space .L(E, σ ), i.e., .

U (E, h) ∼ = L(E, σ ). U (E, h)L

On the other hand, we have .U (E, h)L ∼ = O(L, g). Because the stabilizer group of an L is written as U (E, h)L = {A ∈ U (E, h) | A(u) ∈ L, ∀u ∈ L}

.

= {A : E → E | h(Au, Av) = h(u, v), A(u) ∈ L ∀u ∈ L} = {A : L → L | g(Au, Av) = (u, v), ∀u, v ∈ L} = O(L, g), where the third equality is because of the fact that for any .A ∈ U (E, h) we have that A(L) = L if and only if A is the complex linear extension to E of a g-orthogonal (E,L) ∼ transformation in .L. Therefore, . UO(L,g) ⨆ ⨅ = L(E, σ ).

.

Remark 1.20 As a corollary, the above lemma shows that the Lagrangian Grassmannian is a compact and connected manifold. Exercise 1.14 Let .J : E −→ E; u ‫׀‬−→ iu be the linear transformation in .E. Show that: (a) .J ∈ Sp(E, σ ). (b) If L is a Lagrange plane, then .J (L) is a Lagrange plane. (c) .J (L) is g-orthogonal to .L, and therefore .J (L) ∩ L = {0}.

1.9 Exterior Algebra

35

Exercise 1.15 Show that for any complex structure J we can prove previous exercise if we have compatibility condition of g and .σ that is .g(u, v) = σ (J u, v).

1.9 Exterior Algebra Remember that a symplectic form is a two-form. In this section we are going to see if we consider the n-time exterior products of a symplectic form with itself, then we get a volume form. To this end, first we recall some basic definitions and facts from exterior algebra [49]. Let E be an n-dimensional vector space over a field .k. Consider the vector space of anti-symmetric multi-linear forms of degree p (i.e., p-forms) on .E, and denote it by .Λp E ∗ . Hence, .α ∈ Λp E ∗ means that the mapping α : E p = E × · · · × E −→ R

.

is linear with respect to each component and α(v1 , . . . , vi , . . . , vj , . . . , vp ) = −α(v1 , . . . , vj , . . . , vi , . . . , vp ),

.

where .1 ≤ i < j ≤ p. The exterior product of .α ∈ Λp E ∗ and .β ∈ Λq E ∗ is denoted by .α ∧ β ∈ Λp+q E ∗ and is defined by ∑

(α ∧ β)(v1 , . . . , vp+q ) :=

.

sgn(σ )α(vσ (1) , . . . , vσ (p) )β(vσ (p+1) , . . . , vσ (p+q) )

σ

for all .v1 , . . . , vp+q ∈ E. Here the sum is taken over all equivalence classes of permutations .σ of the indices .{1, . . . , p + q}, where .σ and .σ ◦ ψ are equivalent if .ψ is a mapping that maps the subsets .{1, . . . , p} and .{p + 1, . . . , p + q} of indices to themselves. Note that terms with equivalent permutations are equal, so the expression is well-defined. The signature .sgn(σ ) of the permutation is .+1 when .σ consists of an even number of transpositions, or is .−1 when .σ consists of an odd number of transpositions. The space  ∗ .ΛE = Λp E ∗ p≥0

with the exterior product becomes an algebra and is called the exterior algebra of E ∗ . Here we used .Λ0 E ∗ = k by convention; moreover, .Λ1 E ∗ = E ∗ is a linear subspace of .ΛE ∗ . The exterior product is associative, i.e.,

.

(α ∧ β) ∧ γ = α ∧ (β ∧ γ ),

.

and it is anti-commutative, i.e.,

36

1 Symplectic Linear Algebra

α ∧ β = (−1)pq β ∧ α,  p ∗ for .α ∈ Λp E ∗ and .β ∈ Λq E ∗ . Thus .ΛE ∗ = p≥0 Λ E becomes a graded associative algebra. It is clear that .Λ1 E ∗ = E ∗ generates .ΛE ∗ as an associative algebra. Note that the even degree part of .ΛE ∗ is a commutative subalgebra of ∗ .ΛE . Let .ei be a basis of E and .ɛj the dual basis of .E ∗ , so .ɛj (ei ) = δij . For strictly increasing functions .I, J : {1, . . . , p} −→ {1, . . . , n}, we write .

eI = (eI (1) , . . . , eI (p) ) ∈ E p ,

.

ɛJ = ɛJ (1) ∧ · · · ∧ ɛJ (p) ∈ Λp E ∗ , p ∗ and ∑ so we have .ɛJ (eI ) = δI J . Therefore, for .α ∈ Λ E , we obtain .α = sides to .eI and observing that .α is determined by the J α(eJ )ɛJ , by applying both ∑ numbers .α(eI ). Conversely, if . J cJ ɛJ = 0, then application to .eI yields .cI = 0 for every .I. Hence, the .ɛJ forms a basis of .Λp E ∗ , and so

⎛ ⎞ n .dim(Λ E ) = , dim(ΛE ∗ ) = 2n . p p

∗

In particular, .Λp E ∗ = 0 for .p > n, and the space .Λn E ∗ of volume forms on E is one dimensional. In the following we assume that E is a 2n-dimensional vector space equipped with a symplectic form .σ ∈ Λ2 E ∗ . Lemma 1.24 Let .σ be a symplectic form on E and .(e1 , . . . , en , f1 , . . . fn ) be a symplectic basis, with corresponding dual basis .(ε1 , . . . , ε∑ n , φ1 , . . . , φn ), and then using the standard symplectic form, we obtain that .σ = − ni=1 εi ∧ φi . Proof Consider two vectors .u, v in E written by .u = p1 e1 + · · · + pn en + q1 f1 + · · · + qn fn and .v = p1' e1 + · · · + pn' en + q1' e1 + · · · + qn' en . Then we have σ (u, v) = σ ((p, q), (p' , q ' ))

.

=

n ∑

pi' qi − pi qi'

i=1

=−

n ∑

pi qi' − pi' qi

i=1

=−

n ∑

εi (u)φi (v) − εi (v)φi (u)

i=1

=−

n ∑ (εi ∧ φi )(u, v), i=1

1.9 Exterior Algebra

37

where in the second equality we calculated .σ in the symplectic basis (e1 , . . . , en , f1 , . . . fn ), i.e., .σ (ei , ei ' ) = σ (fi , fi ' ) = 0 and .σ (ei , fj ) = δij . In the fourth equality we use dual basis as .εi (ej ) = φi (fj ) = δij . ⨆ ⨅

.

Lemma 1.25 Let .σ be a symplectic form on .E. The n-th exterior power of .σ is a non-zero volume form on E, i.e., σ n = σ ∧ · · · ∧ σ = n!ε1 ∧ φ1 ∧ ε2 ∧ φ2 ∧ · · · ∧ εn ∧ φn .

.

Proof σn = σ ∧ ··· ∧ σ n n ∑ ∑ =( εi ∧ φi ) ∧ · · · ∧ ( εi ∧ φi )

.

i=1

i=1

= (ε1 ∧ φ1 + · · · + εn ∧ φn ) ∧ (ε1 ∧ φ1 + · · · + εn ∧ φn ) ∧ . . . ∧ (ε1 ∧ φ1 + · · · + εn ∧ φn ) = ε1 ∧ φ1 ∧ ε1 ∧ φ1 ∧ · · · + ε1 ∧ φ1 ∧ ε2 ∧ φ2 ∧ · · · + . . . +ε2 ∧ φ2 ∧ ε1 ∧ φ1 · · · + . . . = n!ε1 ∧ φ1 ∧ ε2 ∧ φ2 ∧ · · · ∧ εn ∧ φn . Here we have used the associative property of the exterior product and the anticommutative property of the exterior product. Note that the exterior product is commutative on the subalgebra generated by the two-forms .ɛi ∧φi . The last equality is non-zero because of the non-degeneracy property of .σ. So .σ n is a volume form on .E. ⨆ ⨅ Remark 1.21 In Sect. 1.7 in Lemma 1.13, we saw that if A is a symplectomorphism, then we obtain .detA = ±1. Now we show that actually we have .detA = 1. If .A ∈ Sp(E, σ ), then .σ (Au, Av) = σ (u, v). We have σ ∧ · · · ∧ σ (e1 , . . . , e2n ) = σ n (Ae1 , . . . , Ae2n )

.

= (detA)σ n (e1 , . . . , e2n ),

(1.1)

where the second equality is because of the fact from exterior algebra that a linear transformation .A : E −→ E induces a linear mapping Λ2n A : Λ2n E −→ Λ2n E

.

e1 ∧ e2 ∧ · · · ∧ e2n ‫׀‬−→ Ae1 ∧ Ae2 ∧ · · · ∧ Ae2n , which has the property that .(Λ2n A)(e1 ∧ · · · ∧ e2n ) = det (A)(e1 ∧ · · · ∧ e2n ). On the other hand, .A ∈ Sp(E, σ ) implies that

38

1 Symplectic Linear Algebra

σ ∧ · · · ∧ σ (e1 , . . . , e2n ) = σ n (Ae1 , . . . , Ae2n )

.

= (A∗ σ )n (e1 , . . . , e2n ) = σ n (e1 , . . . , e2n ).

(1.2)

By above lemma, .σ n is a non-zero volume form on .E. Comparing (1.1) and (1.2), we conclude that .det (A) = 1. Exercise 1.16 Let .ω be a two-form on a 2n-dimensional vector space; show that if ωn /= 0, then .ω is symplectic.

.

Chapter 2

Symplectic Manifolds

In this chapter, we are going to see the definition of a symplectic manifold, several examples, and constructions of symplectic manifolds. We will see that cotangent bundle of a manifold is equipped with a symplectic structure. In fact, cotangent bundle is the most important example in the theory, since symplectic geometry started from solving equations of motion in a classical mechanical system in cotangent bundle. We study reduction, a procedure to obtain a symplectic manifold. Then, as an example of reduction, we see how complex projective manifolds are equipped with symplectic structure. In the previous chapter, we saw the relation between symplectic structure and complex structure, using Hermitian form, on vector spaces. Here we generalize this notion to symplectic manifolds. In Sect. 2.1, a symplectic manifold is defined by assigning a symplectic structure to the tangent space at each point of the manifold. Similarly, in Sect. 2.3, an almost complex structure is defined on a manifold by assigning a complex structure to the tangent space at each point of the manifold. Since a symplectic manifold carries a symplectic form, it is interesting to study the cohomological behavior of such a form. In particular, we will see that there are manifolds that never admit symplectic structure, by cohomological reason. Then, symplectomorphisms between symplectic manifolds, and their examples are discussed. We will explain the important applications of symplectomorphisms such as Moser theorem, their relations with Lagrangian submanifolds, and geodesic flows. Finally, we study neighborhood theorems, and as an application, we will see a proof of Darboux’s theorem, given by Weinstein.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. Eslami Rad, Symplectic and Contact Geometry, Latin American Mathematics Series, https://doi.org/10.1007/978-3-031-56225-9_2

39

40

2 Symplectic Manifolds

2.1 Definition of Symplectic Manifold In the previous chapter, we studied symplectic vector spaces, which are the first examples of symplectic manifolds. Considering tangent spaces at each point of an even-dimensional manifold as vector spaces, we can assign a symplectic structure to each of these tangent spaces. Definition 2.1 Let M be a finite-dimensional smooth manifold. A symplectic form on M is a smooth differential two-form .σ on M such that: (1) For every .m ∈ M the bi-linear form .σm on .Tm M is non-degenerate. (2) .σ is closed, i.e., .dσ = 0. Remark 2.1 Condition (1) of the definition means that for every .m ∈ M, σm is a symplectic form on .Tm M. Therefore, .dimM = dimTm M = 2n. Condition (2) forces all symplectic manifolds to be locally indistinguishable. (Locally, every closed form is exact, the same as in de Rham cohomology.) Definition 2.2 A symplectic manifold is defined as a pair .(M, σ ) in which M is a finite-dimensional smooth manifold and .σ is a symplectic form on .M. Example 2.1 The simplest example of a symplectic is .R2n with the ∑n manifold ' ' ' ' standard symplectic form .σst ((q, p), (q , p )) = j =1 pj qj − pj qj , which in differential form notation is equal to σst =

n ∑

.

dpj ∧ dqj .

j =1

Here .pj and .qj are viewed as coordinate functions on .R2n , and .εj = dpj , .φj = dqj , where .dpj and .dqj represent differential forms and .εj , φj are in the same notation as in Sect. 1.9. Moreover, for all .m ∈ R2n , the set {

.

∂ ∂ ∂ ∂ (m), . . . , (m), (m), . . . , (m)} ∂p1 ∂pn ∂q1 ∂qn

is a symplectic basis for .(σst )m = σst (u, v) =

n ∑

.

∑n

j =1 dpj (m) ∧ dqj (m).

Therefore

dpj (u)dqj (v) − dpj (v)dqj (u)

j =1

=

n ∑

pj qj' − pj' qj .

j =1

From the linearity and the multiplicative property of the exterior ∑derivative, we see that .σst is a closed form. Moreover, it is exact because .σst = d nj=1 pj dqj .

2.1 Definition of Symplectic Manifold

41

Example 2.2 Let .M = Cn with linear coordinates .z1 , . . . , zn . The form i ∑ dzj ∧ d z¯ j 2 n

ω=−

.

j =1

is symplectic. Under the identification .Cn  R2n by .zj = xj + iyj , the form .ω is equal to .σst in the previous example because i ∑ dzj ∧ d z¯ j 2 n

ω=−

.

j =1

i ∑ =− d(xj + iyj ) ∧ d(xj − iyj ) 2 n

j =1

i ∑ dxj ∧ dxj − idxj ∧ dyj + idyj ∧ dxj + dyj ∧ dyj 2 n

=−

j =1

i ∑ 2idyj ∧ dxj =− 2 n

j =1

=

n ∑

dyj ∧ dxj

j =1

= σst . Remark 2.2 We saw in Sect. 1.9 that the n-th exterior power .σ n is a top degree non-vanishing form, i.e., it is a volume form. Hence, .(M, σ ) is canonically oriented by the symplectic structure. Therefore, as an example, the Mobius band does not admit a symplectic structure. Similarly, any non-orientable surface does not admit a symplectic structure. Example 2.3 Any oriented surface .∑ with any volume form .ω on .∑, that is .(∑, ω), is symplectic. Since .ω is a volume form, so it is non-degenerate. Moreover, .ω is a closed form because it is a top form. Example 2.4 Consider the sphere .S 2 as the set of unit vectors in .R3 , i.e., S 2 = {p ∈ R3 ; | p |= 1}.

.

Then for all .p ∈ S 2 , we have Tp S 2 = {q ∈ R3 | p.q = 0}.

.

42

2 Symplectic Manifolds

Fig. 2.1 Unit sphere .S 2 in .R

3

In other words, tangent vectors .u, v ∈ Tp S 2 at a point .p ∈ S 2 are identified with vectors orthogonal to .p, see Fig. 2.1. The two-form .ω given by ωp (u, v) := p.(u × v)

.

is a symplectic form on .S 2 . The form .ω is closed because it is of top degree, i.e., since .S 2 is a two-manifold, all two-forms are automatically closed. It is nondegenerate because it is a volume form. Exercise 2.1 Let .(M1 , ω1 ) and .(M2 , ω2 ) be symplectic manifolds. Let πi : M1 × M2 → Mi , i = 1, 2

.

be the corresponding projections. Then for every .α1 , α2 ∈ R, α1 .α2 /= 0, (M1 × M2 , α1 π1∗ ω1 + α2 π2∗ ω2 )

.

is a symplectic manifold.

2.2 Examples and Constructions of Symplectic Manifolds 2.2.1 The Cotangent Bundle Let X be an arbitrary n-dimensional smooth manifold, and in the following, we see how we can obtain a generalization of the first example, i.e., .(R2n , σst ).

2.2 Examples and Constructions of Symplectic Manifolds

43

Definition 2.3 The cotangent bundle .T ∗ X of X is defined as the vector bundle over X of which the fiber at the point .x ∈ X is equal to the dual space, .Tx∗ X := (Tx X)∗ , of .Tx X, which is the space of all linear forms .ξ on the tangent space .Tx X of X at the point .x. Moreover, if we describe coordinate charts in X by .(U, x1 , . . . , xn ), where .xi : U → R, then at any .x ∈ U, the differentials .(dx1 )x , . . ∑ . (dxn )x form a basis of .Tx∗ X. So if we have .ξ ∈ Tx∗ X, then we can write it as .ξ = nj=1 ξj (dxj )x for some real coefficient .ξ1 , . . . , ξn . This induces the mapping T ∗ U −→ R2n

.

(x, ξ ) −→ (x1 , . . . , xn , ξ1 , . . . , ξn ). The chart .(T ∗ U, x1 , . . . , xn , ξ1 , . . . , ξn ) is a coordinate chart for .T ∗ X. Let .π be the projection from .M := T ∗ X onto .X, which projects every element of .Tx∗ X to .x. π : Tx∗ X −→ X

.

ξ ‫׀‬−→ x. Thus, for every .ξ ∈ Tx∗ X, the tangent map .Tξ π is a linear mapping from .Tξ M onto .Tx X, Tξ π : Tξ M = Tξ (Tx∗ X) −→ Tπ(ξ ) X.

.

Applying the linear form .ξ ∈ (Tx X)∗ to .Tξ π, we obtain the linear form τξ := ξ ◦ Tξ π

.

on .Tξ M. So we have the mapping τ : Tx∗ X −→ T ∗ (Tx∗ X)

.

ξ ‫׀‬−→ τξ , where .τξ : Tξ (Tx∗ X) −→ R. This defines a “special” smooth differential form .τ of degree one on .M = T ∗ X as follows. Let .α be any smooth differential form of degree one on .X, and then it can be considered as a smooth mapping α : X −→ T ∗ X

.

x ‫׀‬−→ αx ,

44

2 Symplectic Manifolds

and hence, .π ◦ α is equal to the identity on .X. Therefore we have (α ∗ τ )x = τα(x) ◦ Tx α

.

= α(x) ◦ Tα(x) π ◦ Tx α = α(x) ◦ Tx (π ◦ α) = α(x), here in the first equality we used the definition of pullback, in the second equality we used the definition of .τ , and in the third equality we used chain rule for differentiation. Therefore we obtained α = α ∗ τ,

.

which means that every one-form on X is the pullback of .τ, where we consider the one-form as a mapping from X to .T ∗ X. For this reason .τ is called the tautological one-form on the cotangent bundle. The exterior derivative of the tautological one-form, i.e., .σ := dτ , is a two-form on .T ∗ X, which is closed because .d(dτ ) = 0. Lemma 2.1 In local coordinates .(x1 , . . . , xn ) on X with corresponding dual coordinates .(ξ1 , . . . , ξn ), the equation .τξ := ξ ◦ Tξ π is written as τ=

n ∑

.

ξi dxi

i=1

and therefore σ =

n ∑

.

dξi ∧ dxi .

i=1

Proof Consider the local coordinates .p = (x, ξ ) = (x1 , . . . , xn , ξ1 , . . . , ξn ) on M = T ∗ X, and the projection .π : T ∗ X → X with .(x, ξ ) ‫ →׀‬x, where .ξ ∈ Tx∗ X. Then the differential of the mapping .π is written as .dπp : Tp T ∗ X → Tπ(p)=x X. The dual of this mapping is .(dπp )∗ : Tx∗ X → Tp∗ T ∗ X. Consider .α : T ∗ X → T ∗ T ∗ X with .p = (x, ξ ) ‫ →׀‬αp , where .αp : Tp=(x,ξ ) T ∗ X −→ R. We define

.

αp := (dπp )∗ ξ = ξ ◦ dπp .

.

So .αp (v) = (ξ ◦ dπp )(v) when .v ∈ Tp T ∗ X. For .(x1 , . . . , xn , ξ1 , . . . , ξn ) ∈ T ∗ X, we have

2.2 Examples and Constructions of Symplectic Manifolds

(

.

45

∂ ∂ ∂ ∂ ,..., , ,..., ) ∈ T T ∗ X. ∂x1 ∂xn ∂ξ1 ∂ξn

Then αp = α(x,ξ ) =

n ∑

.

αp (

j =1

∂ ∂ )p dxj + αp ( )p dξj . ∂xj ∂ξj

Let .τ : T ∗ X −→ T ∗ T ∗ X be the mapping defined by .τ (x, ξ ) = τξ = ξ ◦ Tξ π , where τξ : Tξ (Tx∗ X)) −→ R.

.

For the basis .( ∂x∂ 1 , . . . , ∂x∂ n , ∂ξ∂ 1 , . . . , ∂ξ∂ n ) ∈ T T ∗ X the dual basis is as (dx1 , . . . , dxn , dξ1 , . . . , dξn ) ∈ T ∗ T ∗ X.

.

Therefore if .ξ ∈ Tx∗ X, then .ξ =

∑n

i=1 ξi dxi ;

hence

∑ ∂ ∂ )= ξi dxi ( ) = ξj . ∂xj ∂xj n

ξ(

.

i=1

On the other hand we have Tξ π(

.

∂ ∂ ∂ )= and Tξ π( ) = 0. ∂xj ∂xj ∂ξj

Since .τξ = ξ ◦ Tξ π , we obtain τξ (

.

∂ ∂ ∂ ) = ξ ◦ Tξ π( ) = ξ( ) = ξj , ∂xj ∂xj ∂xj τξ (

.

∂ ∂ ) = ξ ◦ Tξ π( ) = 0. ∂ξj ∂ξj

So we write τ (x, ξ ) = τξ =

n ∑

.

j =1

and we get

τξ (

∂ ∂ )ξ dxj + τξ ( )ξ dξj , ∂xj ∂ξj

46

2 Symplectic Manifolds

τξ =

n ∑

.

ξj dxj + 0

j =1

=

n ∑

ξj dxj .

j =1

Taking differential, we obtain .σ =

∑n

i=1 dξi

∧ dxi .

⨆ ⨅

Definition 2.4 If we substitute .xi = qi and .ξi = pi , then by above lemma .σ is equal to the standard symplectic form. This shows that .σ = dτ is a symplectic form on .T ∗ X, which is called canonical symplectic form of the cotangent bundle. Definition 2.5 Let X and Y be smooth manifolds and let .φ : X −→ Y be a local diffeomorphism. Define the induced transformation Ф : T ∗ X −→ T ∗ Y

.

Ф(x, ξ ) := (φ(x), ((Tx φ)∗ )−1 (ξ )),

x ∈ X, ξ ∈ (Tx X)∗ .

This induced mapping is a canonical transformation in the sense that it preserves the tautological one-form and so the canonical symplectic form, i.e., Ф∗ τT ∗ Y = τT ∗ X

.

Ф∗ σT ∗ Y = σT ∗ X . Example 2.5 Let .X = Y = S 1 . Then .T ∗ S 1 is an infinite cylinder .S 1 × R. The canonical two-form .ω is the area form .ω = dθ ∧ dξ. If .φ : S 1 −→ S 1 is a diffeomorphism, then the induced transformation Ф : S 1 × R −→ S 1 × R

.

is a symplectomorphism, i.e., is an area-preserving diffeomorphism of the cylinder. Exercise 2.2 Consider a local diffeomorphism on .φ : X −→ Y between smooth manifolds X and .Y. Prove that the induced transformation Ф : T ∗ X −→ T ∗ Y

.

Ф(x, ξ ) := (φ(x), ((Tx φ)∗ )−1 (ξ )), x ∈ X, ξ ∈ (Tx X)∗ preserves the tautological one-form and the canonical symplectic forms, i.e., prove that: (a) .Ф∗ τT ∗ Y = τT ∗ X . (b) .Ф∗ σT ∗ Y = σT ∗ X .

2.2 Examples and Constructions of Symplectic Manifolds

47

Exercise 2.3 Let .π : T ∗ X −→ X be the cotangent bundle with canonical symplectic form .σ = dτ. Show that for a closed two-form .β on .X, the two-form ∗ ∗ .σ + π β is again a symplectic form on .T X.

2.2.2 Reduced Symplectic Manifold In this section we are going to see having a closed two-form on a smooth manifold and a fibration, we can obtain a symplectic manifold. The original idea of this construction comes from the construction of induced symplectic form on linear vector spaces which we explained in Sect. 1.5. In fact, what comes below is the analogy of that construction for manifolds. There, the two-form was not everywhere non-degenerate, and we considered the quotient over kernel of the form. Let N be a smooth manifold, and let .ω be a smooth two-form on .N. More precisely, we have ω : N −→ Ω2 T ∗ N

.

n ‫׀‬−→ ωn , where .ωn : Tn N × Tn N −→ R, which we can write as .ωn : Tn N −→ Tn∗ N. In Sect. 1.5 we considered the projection pr : Tn N −→ Tn N/ker(ωn ).

.

Below, instead of this projection we consider a fibration. Moreover, to have such a quotient, we need that .ker(ω) ⊂ T N as a subbundle, so we require the integrability condition. Assume that the smooth two-form .ω on N is closed and the kernel of .ω has constant rank. In other words, there exists a non-negative integer k such that, for every .n ∈ N, .dim(kerωn ) = k. Therefore, the .Kn := kerωn , n ∈ N, define a smooth vector subbundle K of the tangent bundle T N of .N. Remark 2.3 A smooth vector subbundle of the tangent bundle of N is also called a distribution on .N. In the nineteenth century literature, it is called a Pfaffian system in .N. For later purpose, the subbundle K should be integrable. Below, we recall the definition of integrability and next the criterion of integrability of a subbundle, given in Frobenius theorem. Definition 2.6 Let N be a finite-dimensional smooth manifold. Then a smooth vector subbundle K of the tangent bundle T N is called integrable if for each .n0 ∈ N there exists an open neighborhood .N0 of .n0 in N and a smooth fibration of .N0 , such that, for each .n ∈ N0 , Kn is equal to the tangent space of the fiber through .n.

48

2 Symplectic Manifolds

Theorem 2.1 (Frobenius Theorem) The subbundle K of the tangent bundle T N is integrable if and only if .[X, Y ] ⊂ K for any smooth vector fields .X, Y on N such that .X ⊂ K and .Y ⊂ K. Now using the Frobenius theorem, we show that the subbundle K which we defined above is integrable. Lemma 2.2 The subbundle .K := kerω is integrable. Proof Let .X ⊂ K and .Y ⊂ K; therefore X ⊂ K = kerω =⇒ ω(X) = 0 =⇒ iX ω = 0

.

Y ⊂ K = kerω =⇒ ω(Y ) = 0 =⇒ iY ω = 0, where .iX ω and .iY ω denote the interior products of the form .ω with the vector fields X and Y , respectively. Leibniz formula for Lie derivatives is .LX (iY ω) = iLX Y ω + iY LX ω. Using .[X, Y ] = LX Y , we obtain i[X,Y ] ω = LX (iY ω) − iY LX ω.

.

Since .iY ω = 0 (because .Y ⊂ K), we get i[X,Y ] ω = 0 − iY LX ω.

.

(2.1)

On the other hand the Cartan formula is .LX ω = iX dω + d(iX ω). Since we assumed that .ω is a closed form and .iX ω = 0 (since .X ⊂ K), we obtain .LX ω = 0 + 0 = 0. Therefore the equality in (2.1) turns to .i[X,Y ] ω = 0, which implies that .[X, Y ] ⊂ K. Therefore by Frobenius theorem K is integrable. ⨆ ⨅ Now, as it is shown in Fig. 2.2, consider another smooth manifold M and suppose that .π : N −→ M is a (surjective) fibration with connected fibers, such that for each .n ∈ N, kerωn is equal to the tangent space .kerTn π of the fiber through the point .n, i.e., kerTn π = kerωn .

.

(2.2)

In the following, we want to assign a symplectic form over M using the two-form ω and the fibration .π. In order to have a symplectic form on .M, first we should find a two-form on .Tm M for every .m ∈ M.

.

Lemma 2.3 For fixed .m ∈ M there exists, for each .n ∈ π −1 ({m}), a unique twoform .σm,n on .Tm M such that .(Tn π )∗ σm,n = ωn where .Tn π : Tn N −→ Tπ(n)=m M. Proof If we define a two-form as σm,n (Tn π(X)) := ωn (X),

.

2.2 Examples and Constructions of Symplectic Manifolds

49

Fig. 2.2 Reduced symplectic manifold M of the pair .(N, ω)

then by definition of pullback .σm,n is a unique two-form such that .(Tn π )∗ σm,n = ωn . Now we show that this definition is for any .n ∈ π −1 ({m}); in other words .σm,n does not depend on the choice of .n ∈ π −1 ({m}). For every smooth vector field X such that .X ⊂ K, we saw, in the proof of Lemma 2.2, that .LX ω = 0. This implies that .(etX )∗ ω = ω because .

LX ω = 0 =⇒ limt→0

(etX )∗ ω − ω = 0 =⇒ t

d tX ∗ (e ) ω |t=0 = 0. dt

50

2 Symplectic Manifolds

Therefore .(etX )∗ ω is a constant mapping. If .t = 0, then .(etX )∗ ω = (e0X )∗ ω = (id)∗ ω = ω. Hence .(etX )∗ ω = ω, where .etX : N −→ N. On the other hand we have .π ◦ etX = π : N −→ N −→ M. If we set .n' := tX e (n), then this implies that .σm,n' = σm,n , because: .π ◦etX (n) = π(n), so .π(n' ) = π(n). Moreover, we have σm,n' (Tn' π(X)) = σm,n' (Tn' (π ◦ etX )(X))

.

= σm,etX (n) (TetX (n) (π ◦ etX )(X)) = (etX )∗ σm,n (Tn π(X)) = (etX )∗ ωn (X) = ωn (X) = ((Tn π )∗ σm,n )(X) = σm,n (Tn π(X)), where in the third equality we used definition of pullback and in the fifth equality we used the fact that the flows .etX preserve .ω. Note that the compositions of the flows .etX (when .X ⊂ K) act locally transitively on the fibers, for instance, we have tX (n ) = etX ◦ etX (n ). Hence the flows .etX act transitively because the .n1 = e 2 3 fibers are connected. The conclusion is that .σm,n = σm does not depend on the choice of .n ∈ π −1 ({m}). Therefore the .σm for .m ∈ M define a smooth two-form on M such that .ω = π ∗ σ. ⨆ ⨅ Lemma 2.4 The two-form .σm is a symplectic form on .M. Proof First we note that .σm is non-degenerate because kerTn π = kerωn = Tn π −1 (kerσm ),

.

(2.3)

where the first equality is because of assumption ( 2.2 ), and we have the second equality because from .ωn = (Tn π )∗ σm = σm ◦ Tn π , and we get kerωn = ker(σm ◦ Tn π ) = Tn (π −1 (kerσm )).

.

Therefore from ( 2.3 ), we obtain Tn π(Tn π −1 (kerσm )) = 0;

.

hence, kerσm = 0,

.

which means that .σm is injective so .σm is non-degenerate.

2.2 Examples and Constructions of Symplectic Manifolds

51

On the other hand, we have .π ∗ (dσ ) = d(π ∗ σ ) = dω = 0, so we have .0 = = dσ (Tn π ), and since .Tn π, (n ∈ N) is surjective, we conclude that .dσ = 0. So .σm is a closed form, therefore it is a symplectic form. ⨆ ⨅ π ∗ (dσ )

Definition 2.7 .(M, σ ) is a symplectic manifold and is called the reduced symplectic manifold of the pair .(N, ω). In the next section, we see an example of reduction in which we obtain symplectic structure on complex projective manifolds. Also, in Sect. 3.9, we will study symplectic quotient as another example of reduction.

2.2.3 Complex Projective Manifolds Other interesting examples of symplectic manifolds are complex projective manifolds. Below, using reduction which we explained in the previous section, we see how we can equip a complex projective manifold with a symplectic form. Let E be an n-dimensional complex vector space with Hermitian form h for which we have the real inner product .g = Re(h) and the symplectic form .σ = I m(h), where .g = σ ◦ J, for a complex structure J on .E. Consider the unit sphere S := {z ∈ E | h(z, z) = g(z, z) = 1}

.

in E with respect to the inner product g (here we used .h(z, z) = g(z, z)+iσ (z, z) = 1 + 0 = 1). Then for each .z ∈ S, we have Tz S = {v ∈ E | g(z, v) = 0} = {v ∈ E | σ (iz, v) = 0}.

.

(2.4)

Here, we have the second equality because if we write .h(iz, v) = g(iz, v) + iσ (iz, v), then by linearity of h with respect to the first component we have .ih(z, v) = g(iz, v) + iσ (iz, v), and equivalently, we get i(g(z, v) + iσ (z, v)) = g(iz, v) + iσ (iz, v),

.

or 0 + i 2 σ (z, v) = g(iz, v) + iσ (iz, v),

.

or (g(iz, v) + σ (z, v)) + iσ (iz, v) = 0,

.

so we obtain .σ (iz, v) = 0.

52

2 Symplectic Manifolds

From the first equality in (2.4) it follows that .iz ∈ Tz S because we have g(z, iz) = h(z, iz) − iσ (z, iz)

.

= −ih(z, z) − iσ (z, iz) = −i − iσ (z, iz). Therefore .g(z, iz) = 0 (because g is the real part). From the second equality in ( 2.4 ), it follows that iz belongs to the kernel of the restriction of .σ to .Tz S since S is .(n − 1)-dimensional so it is .Tz S or in other words .Tz S has co-dimension one in .E. Therefore, its symplectic orthogonal complement, i.e., .(Tz S)σ , is one dimensional and therefore is equal to .R(iz), i.e., (Tz S)σ = R(iz) = kerσ |Tz S .

.

As it is shown in Fig. 2.3, consider the circle Cz := {cz | c ∈ C, | c |= 1} = (Cz) ∩ S

.

passing through the point .z. Then .R(iz) is equal to the tangent space of the circle through the point .z. If we denote the identity mapping from S to E by .ı : S ᶜ→ E, then .ı ∗ σ is the restriction of the two-form .σ to .S, i.e., .ı ∗ σ = σ |S .

Fig. 2.3 .Cz element of projective space .CP (E)

2.2 Examples and Constructions of Symplectic Manifolds

53

By above explanation and previous lesson, the fibration of S by the integral curves of the kernels of .ı ∗ σ is equal to the fibration of S by the circles .Cz , for .z ∈ S. We denote the space of these circles .Cz by M := {Cz | z ∈ S}.

.

In fact one can see that M is a smooth manifold. Then by previous lesson we have the reduced symplectic form .σˆ , i.e., the unique two-form .σˆ on M such that .ı ∗ σ = π ∗ σˆ where .π : S −→ M denotes the projection defined by .π(z) = Cz , z ∈ S. .Cz for .z ∈ S are the complex one-dimensional linear subspaces l of E that form the elements of the complex .(n − 1)-dimensional projective space .CP (E) of .E. The mapping CP (E) −→ M

.

Cz = l ‫׀‬−→ l ∩ S = Cz is a diffeomorphism from .CP (E) onto M; therefore we can identify M with .CP (E) that is .M ∼ = CP (E). Hence, we can consider the fibration .π from S over .CP (E). Moreover, if we consider complex lines as real planes, then .M ∼ = CP (E) has .2(n − 1) real dimension. Therefore .(CP (E), σˆ ) is a symplectic manifold with the reduced symplectic form .σˆ . Note that .CP (E) is a complex analytic manifold with a complex multiplication .Jl by i in each tangent space .Tl (CP (E)). We define .g ˆ := σˆ ◦ Jl . Since .σˆ and .Jl are isomorphisms, so we conclude that .gˆ is non-degenerate and therefore is an inner product on .Tl (CP (E)) that corresponds to the restriction of g to the g-orthogonal complement of iz in .Tz S because π ∗ g(u, ˆ v) = π ∗ σˆ ◦ Jl (u, v)

.

= π ∗ σˆ (iu, v) = ı ∗ σ (iu, v) = σ (iu, v). On the other hand we remember that .(Tz S)σ = R(iz) = kerσ |Tz S and .σ = −g ◦ J. Definition 2.8 Let .E = Cn and let h be the standard Hermitian structure on .Cn . The Hermitian inner product .hˆ := π1 (gˆ + i σˆ ) is called the Fubini–Study metric on .CP (E) = CPn−1 . Remark 2.4 In the above definition, . π1 (where .π = 3.14) is written to obtain the integral of .ω := π1 σˆ over any complex projective line in .CPn−1 equal to one.

54

2 Symplectic Manifolds

Definition 2.9 In general, if M is a complex analytic manifold, then a smooth Hermitian inner product h on the tangent bundle of M with the imaginary part .σ := I mh, as a closed two-form, is a Kähler structure on .M. Hence, .σ is a symplectic form on M and is called the Kähler form of the Kähler manifold .(M, h). Lemma 2.5 The Fubini–Study metric is a Kähler structure on the complex projective space. Proof Let .i : V ᶜ→ M be a complex analytic submanifold of a Kähler manifold (M, h). So, the restriction .i ∗ h of h to T V is a Kähler structure on .V . Hence, every smooth complex projective space is a Kähler manifold if we consider the restriction of the Fubini–Study metric of the projective space to its tangent bundle, i.e., if we consider .CP (E) as the submanifold of its tangent bundle, ∗ ˆ is a Kähler structure on .CP (E). .i : CP (E) ᶜ→ T CP (E), then .i h ⨆ ⨅ .

2.3 Almost Complex Structure In the previous chapter where we were talking about symplectic vector spaces, in Sect. 1.8 we saw that from any symplectic form on a vector space, we obtain a Hermitian form, using a complex structure J on the vector space. Here we want to say a similar statement for symplectic manifolds; hence we consider a complex structure on tangent space on each point of a manifold. Definition 2.10 An almost complex structure J on a smooth manifold M is a complex structure Jm : Tm M −→ Tm M, Jm2 = −I d

.

on each tangent space .Tm M, depending smoothly on .m ∈ M (i.e., the mapping M → End(T M); m ‫ →׀‬Jm is smooth).

.

Remark 2.5 By above definition, each tangent space is a complex vector space because we can define .(a + ib)v := av + J (bv) for any .a, b ∈ R and .v ∈ Tm M. If n is the complex dimension of .Tm M with respect to the complex structure .Jm , then the real dimension is equal to .2n. Definition 2.11 An almost complex manifold is a pair .(M, J ) in which M is a smooth manifold and J is an almost complex structure on .M. Definition 2.12 The almost complex structure is called integrable if for every m0 ∈ M there is coordinate system in an open neighborhood of .m0 in .M, in which the mapping

.

ϕ : M −→ End(T M)

.

m ‫׀‬−→ Jm is constant.

2.3 Almost Complex Structure

55

Remark 2.6 We can identify .R2n with .Cn , using the constant complex structure, and we obtain a system of local coordinates for which the coordinate changes are complex analytic mappings. In these coordinates, the .Jm ’s are the multiplications by i in the tangent spaces and M is a complex analytic manifold. Definition 2.13 Let .(M, J ) be an almost complex manifold. For each .m ∈ M, its Nijenhuis tensor is the anti-symmetric bi-linear mapping [J, J ]m : Tm M × Tm M −→ Tm M

.

defined by [J, J ](v, w) := [J v, J w] − J [J v, w] − J [v, J w] − [v, w]

.

in which v and w are smooth vector fields on M and the brackets in the right-hand side are the Lie brackets of vector fields. Theorem 2.2 (Newlander and Nirenberg [30, 39, 43]) The almost complex structure J is integrable if and only if its Nijenhuis tensor is zero, i.e., .[J, J ] = 0. Recall from the previous chapter that a complex structure J on a symplectic vector space .(E, ω) is .ω-compatible if it satisfies .ω(J u, J v) = ω(u, v) for all .u, v ∈ E, and .ω(v, J v) > 0 for all non-zero .v ∈ E. A compatible almost complex structure J on a smooth manifold M is a compatible complex structure .Jm on each tangent space .Tm M, at .m ∈ M. Definition 2.14 A Kähler manifold is a symplectic manifold .(M, ω) equipped with an integrable compatible almost complex structure. The symplectic form .ω is called a Kähler form. Proposition 2.1 Let .σ be a symplectic form on .M. Then there exists an almost complex structure J on M such that .h = σ ◦ J + iσ is a Hermitian structure on .T M. Proof First note that there exists a Riemannian structure g in .M. Such a Riemannian structure exists in local coordinates. Consider locally finite covering of M consisting of open subsets .Mj on which we have a Riemannian structure .gj . Let .ηj be a smooth partition of unity subordinate to the covering of .M. In other words, .ηj are smooth real-valued functions on M∑ such that .ηj ≥ 0, the support of .ηj is contained in .Mj , ∑ and . j ηj = 1. Then .g = j ηj gj is Riemannian structure on .M. Since .σ is a symplectic form on .M, the mapping .σm : Tm M −→ Tm∗ M is an isomorphism and so is the mapping .(σm )−1 : Tm∗ M −→ Tm M. Moreover, .gm is an inner product on .Tm M by which we consider the Riemannian structure on .M. Hence, .gm is non-degenerate, and we have the isomorphism .gm : Tm M −→ Tm∗ M. We define for each .m ∈ M, Am := σm−1 gm . Then .gm ◦ Am = gm ◦ σm−1 ◦ gm is anti-symmetric because

56

2 Symplectic Manifolds t t (gm ◦ σm−1 ◦ gm )t = gm ◦ (σm−1 )t ◦ gm

.

t t = gm ◦ (σmt )−1 ◦ gm

= −gm ◦ (σm )−1 ◦ gm , where in the last equality we used symmetric property of g and anti-symmetric property of the symplectic form .σ. So .Am is .gm -anti-symmetric, and there exists a .gm -orthonormal basis in .Tm M on which the matrix of .Am consists of .2 × 2-matrices (

0 −aj aj 0

.

)

along the diagonal, with .aj > 0. Let .Bm be the linear transformation in .Tm M of which the matrix consists of the .2 × 2-matrices ) (1 aj 0 .

0

1 aj

along the diagonal. Claim: .Jm := Am ◦ Bm = σm−1 ◦ gm ◦ Bm is a complex structure on .Tm M because Jm2 = (Am ◦ Bm ) ◦ (Am ◦ Bm ) ( )( ) 0 −I d 0 −I d = Id 0 Id 0 ( ) −I d 0 = 0 −I d

.

= −I d. Also, we define .gˆ m := gm ◦ Bm = σm ◦ Jm , which is an inner product on .Tm M. Because .gm and .Bm are both non-degenerate, so .gˆ m is non-degenerate. Moreover, .g ˆ m is symmetric, since .gm and .Bm are both symmetric and .Bm is diagonal, or we can say t t t gˆ m = (gm ◦ Bm )t = Bm ◦ gm = Bm ◦ gm = gm ◦ Bm = gˆ m ,

.

where the fourth equality follows from the fact that .Bm is diagonal. Therefore .σm ◦ Jm + iσ is a Hermitian form on .Tm M. Up to now this construction depends on the choice of the .gm -orthonormal basis; 2 = −A−2 , which shows that .B is equal to the unique however, note that .Bm m m

2.4 Cohomology Classes

57

positive-definite square root of the positive-definite linear transformation .−A−2 m (all with respect to the inner product .gm ). Therefore, .Jm is globally well-defined and depends smoothly on .m ∈ M. ⨆ ⨅ Remark 2.7 (1) Comparing definitions 2.14 and 2.9, the above lemma shows that a symplectic form can lead to a Kähler form. (2) However, in the above lemma, it cannot always be arranged that in addition J is integrable. In other words, not every symplectic form is equal to a Kähler form on a complex analytic manifold. Lemma 2.6 If .σ in previous lemma is invariant under the action of a group G on M, and M carries a G-invariant Riemannian structure, then J can be chosen to be G-invariant as well.

.

Proof Consider the group action .G × M −→ M by .(h, m) − ‫ → ׀‬h.m. If .σ and g are G-invariant, so we have .σm = σh.m and .gm = gh.m for any .h ∈ G. Using the previous lemma and uniqueness of the .Bm , m ∈ M, it follows that B and therefore ⨆ ⨅ J are also G-invariant, i.e., .Jm = Jh.m . Exercise 2.4 Let J be an almost complex structure on the manifold .M : (a) Prove that .[J, J ](v, J v) = 0 for any smooth vector field v on .M. (b) Prove that J is integrable if .dimM = 2.

2.4 Cohomology Classes In Sect. 1.9, we saw that the n-th exterior power of a symplectic form on a 2ndimensional vector space is a non-zero 2n-form; hence, it is a volume form. Here, using de Rham cohomology theory, we see that every even-dimensional manifold is not symplectic, and we see under which conditions a manifold can accept a symplectic structure. Let M be a smooth manifold, and we denote the set of k-forms on M by .Ωk (M). Consider the mapping .dk : Ωk (M) −→ Ωk+1 (M) as the exterior derivative that acts on k-forms. Since .dk ◦ dk−1 = 0, we obtain the following co-chain complex (where the indices go up) with the exterior derivative as co-boundary operator. d

dk−1

dk

dk+1

0 −→ Ω0 (M) −→ . . . −→ Ωk−1 (M) −→ Ωk (M) −→ Ωk+1 (M) −→ . . . .

.

This complex is called the de Rham complex of .M. We know that for every differential form .α we have .d(dα)) = 0 or .dk (dk−1 (α)) = 0. This means that .im(dk−1 ) ⊂ ker(dk ). Hence the de Rham cohomology is defined by

58

2 Symplectic Manifolds

k HdeRham (M) :=

.

ker(dk ) . im(dk−1 )

Here .ker(dk ) is the set of cocycles that are closed forms. Moreover, .im(dk−1 ) is k (M) is the set of the set of co-boundaries that are exact forms. Therefore, .HdeRham closed forms that are not exact. Theorem 2.3 (de Rham [21, 51]) (1) If M is a closed (compact, without boundary) manifold of dimension m and M m is orientable, then .HdeRham (M) ∼ = R. m (2) For non-compact manifolds .HdeRham (M) = 0. m (3) For non-orientable manifolds .HdeRham (M) = 0. Example 2.6 Let M be a compact, connected manifold and .σ be a symplectic form on .M. So .dimM = m = 2n. We have σn = σ ∧ ··· ∧ σ n n ∑ ∑ =( dξi ∧ dxi ) ∧ · · · ∧ ( dξi ∧ dxi )

.

i=1

i=1

= n!(dξ1 ∧ dx1 ) ∧ (dξ2 ∧ dx2 ) ∧ · · · ∧ (dξn ∧ dxn ) /= 0. So .σ n is a non-zero volume form at each point of manifold. Therefore M is 2n orientable. Therefore the de Rham cohomology class .[σ n ] ∈ HdeRham (M) of the n nowhere vanishing volume form .σ is non-zero and therefore generates the onedimensional vector space 2n HdeRham (M) ∼ = R.

.

2k Since .[σ n ] = [σ ]n /= 0, this implies that the element .[σ ]k ∈ HdeRham (M) is nonzero for every .1 ≤ k ≤ n.

Example 2.7 Let M be a 2n-dimensional sphere. We have ⎧ k 2n .HdeRham (S )

=

0

0 < k < 2n

R

k = 0, 2n

.

Therefore the 2n-sphere (.n > 1) does not have symplectic form. Because, if it has a symplectic form, then the second cohomology has such a symplectic form as the representative class and so it is non-zero. This is in contradiction with 2 2n .H deRham (S ) = 0. 2 So only for .S 2 , we have symplectic form because .HdeRham (S 2 ) /= 0.

2.5 Symplectomorphisms

59

2k+1 Example 2.8 Let .M = CP n . We know that .HdeRham (CP n ) = 0 and 2k n ∼ .H deRham (CP ) = R, where .0 ≤ k ≤ n. On the other hand (from previous lesson), we know that there is a symplectic form on .CP n that is a Kähler form. 2k 0 /= [σ ]k ∈ HdeRham (CP n ) ∼ = R, 0 ≤ k ≤ n

.

2 0 /= [σ ] ∈ HdeRham (CP n ) ∼ = R.

.

In other words, in this case the whole cohomology ring is generated by the cohomology class .[σ ] of the Kähler form .σ defined by the Fubini–Study metric.

2.5 Symplectomorphisms Consider the set of symplectic manifolds. It is natural to ask how one can define mappings between symplectic manifolds. In other words, such mappings should preserve symplectic structures in the following sense. Definition 2.15 Let .(M1 , ω1 ) and .(M2 , ω2 ) be 2n-dimensional symplectic manifolds, and let the mapping .ϕ : M1 −→ M2 be a diffeomorphism. Then .ϕ is a symplectomorphism if .ϕ ∗ ω2 = ω1 . Remark 2.8 By definition of pullback, we have .(ϕ ∗ ω2 )p (u, v) = (ω2 )ϕ(p) (dϕp (u), dϕp (v)), where .u, v ∈ Tp M1 . So the condition .ϕ ∗ ω2 = ω1 in above definition turns into (ω2 )ϕ(p) (dϕp (u), dϕp (v)) = ω1 (u, v).

.

Remark 2.9 Let W be a compact symplectic manifold. An embedding ∑ of W cannot be symplectic into .R2n with the standard symplectic form .ωst = ni=1 dxi ∧ dyi because if the mapping .e : W → R2n is a symplectic embedding, then the cohomology class of the pullback of symplectic form .ωst is zero, i.e., .[e∗ ωst ] = e∗ ([ωst ]) = e∗ (0) = 0. However, a symplectic form on a compact manifold is never exact because its top exterior power of symplectic form is a volume form; therefore ∗ .e ωst can never be a symplectic form. Let M be a compact manifold with symplectic forms .ω0 and .ω1 . In the following, we are going to see if .(M, ω0 ) and .(M, ω1 ) are symplectomorphism, in other words, if there exists a diffeomorphism .ϕ : M −→ M such that .ϕ ∗ ω0 = ω1 . Recall Let M be a manifold and .ρ : M ×R −→ M a mapping. If we set .ρ(m, t) := ρt (m), then .ρ is an isotopy if each .ρt : M −→ M is a diffeomorphism and .ρ0 = I dM .

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2 Symplectic Manifolds

Theorem 2.4 (Moser Theorem—Version I [42]) Let M be a compact manifold with symplectic forms .ω0 and .ω1 . Suppose that their cohomology classes are equal, i.e., .[ω0 ] = [ω1 ], and suppose that the twoform .ωt = (1 − t)ω0 + tω1 is symplectic for each .t ∈ [0, 1]. Then there exists an isotopy .ρ : M × R −→ M such that .ρt∗ ωt = ω0 for all .t ∈ [0, 1]. In particular, ∗ .ϕ := ρ1 : M −→ M satisfies .ϕ ω1 = ω0 . Proof The proof is known as the Moser trick. The trick is that first we assume that such an isotopy exists and we find a corresponding vector field. Then using the obtained vector field, we define the desired isotopy that is in fact the flow of the vector field. Suppose that there exists an isotopy .ρ : M × R −→ M such that .ρt∗ ωt = ω0 , for .t ∈ [0, 1]. Given an isotopy .ρ, we obtain a time-dependent vector field, i.e., a family of vector fields .vt , t ∈ R, which at .p ∈ M satisfy vt (p) =

.

d ρs (q) |s=t , ds

t where .q = ρt−1 (p), i.e., . dρ dt = vt ◦ ρt . (Conversely, given a time-dependent vector field .vt , if M is compact or if the .vt ’s are compactly supported, there exists an isotopy .ρ satisfying the above ordinary differential equation. If M is compact, then we have a one-to-one correspondence between isotopies of M and time-dependent −1 ∗ t vector fields on M). So let .vt = dρ dt ◦ ρt , t ∈ R. From .ρt ωt = ω0 , we have d ∗ ∗ ∗ . dt (ρt ωt ) = 0. On the other hand, if we consider .ρt ωt = f (ρt , ωt ) as a real function of two variables, then by chain rule we have

.

d d d f (t, t) = f (x, t) |x=t + f (t, y) |y=t dt dx dy

or .

d ∗ d ∗ d (ρt ωt ) = ρx ωt |x=t + ρt∗ ωy |y=t dt dx dy = ρx∗ Lvx ωt |x=t +ρt∗ = ρt∗ (Lvt ωt +

dωy |y=t dy

dωt ), dt

which means that we have the following equation, known as Moser’s equation. Lvt ωt +

.

dωt = 0. dt

(2.5)

In the above calculation, the first part of the second equality is obtained by the following calculation using Cartan formula.

2.6 Lagrangian Submanifolds

61

ρx∗ (Lvx ωt ) = ρx∗ (ivx

.

= 0+ =

d d ωt ) + ρx∗ ( ivx ωt ) dx dx

d ∗ ρ ωt (vx , −) dx x

d ∗ ρ ωt , dx x

where the second equality follows from the fact that .ωt is a closed form, i.e., .dωt = d 0 and so . dx ωt = 0. We can find a smooth time-dependent vector field .vt , t ∈ R such that the Eq. (2.5) holds for .t ∈ [0, 1]. Since M is compact, we can integrate .vt to an isotopy .ρ : d M × R −→ M with . dt (ρt∗ ωt ) = 0, and therefore .ρt∗ ωt = ρ0∗ ω0 = ω0 . So it remains to show that (2.5) has a solution as .vt . From .ωt = (1 − t)ω0 + tω1 , t we have . dω dt = ω1 − ω0 . On the other hand, since .[ω0 ] = [ω1 ], there exists a one-form .α such that .ω1 − ω0 = dα. Therefore we obtain .

dωt = dα. dt

(2.6)

On the other hand, we have 0=

.

dωt d ∗ ) ρt ωt = ρt∗ (Lvt ωt + dt dt = ρt∗ (ivt dωt + divt ωt +

dωt ) dt

= ρt∗ (divt ωt + dα) = ρt∗ (d(ivt ωt + α)), where in the second equality we used Cartan formula, and in the third equality we t used (2.6) and .dωt = 0 since .ωt ’s are symplectic forms. So, if .ρt∗ (Lvt ωt + dω dt ) = 0, we conclude that .ivt ωt + α = 0. However, since .ωt ’s are non-degenerate, so we can solve the equation .ivt ωt = −α pointwise to obtain a unique solution .vt . ⨆ ⨅ Exercise 2.5 Show that a convex combination of two symplectic forms that induce the same orientation on a surface (dimension two) is itself a symplectic form.

2.6 Lagrangian Submanifolds In Sect. 1.4 we described the Lagrangian subspaces of vector spaces as maximal isotropic subspaces, which in particular have half dimension. Here we see the definition of Lagrangian submanifolds, their examples, and their applications, especially together with symplectomorphisms.

62

2 Symplectic Manifolds

Definition 2.16 Let .(M, ω) be a 2n-dimensional symplectic manifold. A submanifold Y of M is a Lagrangian submanifold if at each point .p ∈ Y, .Tp Y is a Lagrangian subspace of .Tp M, that is, .ωp |Tp Y = 0 and .dimTp Y = 12 dimTp M. Equivalently, if .i : Y ᶜ→ M is the inclusion map, then Y is Lagrangian if and only if .i ∗ ω = 0 and .dimY = 12 dimM. Example 2.9 Let .(M, ω) be an oriented surface with an area form. Then every curve is Lagrangian since the tangent space to a curve is one-dimensional and we remember that every one-dimensional vector subspace is isotropic that means that .ω vanishes on the tangent space to a curve. Example 2.10 The circle .S 1 is Lagrangian under the inclusion map .i : S 1 ᶜ→ R2 , where .R2 is equipped with the standard symplectic form. In other words the tangent space at any point .p ∈ S 1 is the real line .R that is a Lagrangian subspace of .R2 . The torus .T n = S 1 × · · · × S 1 is Lagrangian under the inclusion T n ᶜ→ R2 × · · · × R2 = R2n .

.

Since at any point .p = (p1 , . . . , pn ) ∈ T n , we have .Tp T n = Tp1 S 1 ×· · ·×Tpn S 1 ∼ = Rn , which is a Lagrangian subspace of .R2n . Example 2.11 Let X be an n-dimensional manifold, with .M = T ∗ X its cotangent . . , ξn ) on .T ∗ X. bundle. Consider the coordinates .(x1 , . . . , xn ) and .(x1 , . . . , xn , ξ1 , .∑ ∗ Then we remember that the tautological one-form on .T X is .τ = ξi dxi and the ∑ canonical one-form on .T ∗ X is .σ = dτ = dξi ∧ dxi . Consider the zero section of .T ∗ X as X0 := {(x, ξ ) ∈ T ∗ X | ξ = 0 in Tx∗ X},

.

∗ which is an n-dimensional have .ξ1 = ∑ submanifold of .T X. Moreover, on .X0 we∑ · · · = ξn = 0. So .τ = ξi dxi vanishes on .X0 and therefore .σ = dτ = dξi ∧dxi vanishes on .X0 , since considering the inclusion map .i : X0 ᶜ→ T ∗ X, we have ∗ ∗ ∗ ∗ .i σ = i dτ = di τ = 0 on .X0 . Hence .X0 is a Lagrangian submanifold of .T X.

Definition 2.17 Let S be any k-dimensional submanifold of an n-dimensional manifold .X. The conormal space at .x ∈ S is Nx∗ S := {ξ ∈ Tx∗ X | ξ(v) = 0, for all v ∈ Tx S}.

.

The conormal bundle of S is N ∗ S := {(x, ξ ) ∈ T ∗ X | x ∈ S, ξ ∈ Nx∗ S}.

.

Lemma 2.7 For any submanifold .S ⊂ X, the conormal bundle .N ∗ S is a Lagrangian submanifold of .T ∗ X.

2.6 Lagrangian Submanifolds

63

Proof Let .(U, x1 , . . . , xn ) be a coordinate system on X centered at .x ∈ S so that U ∩ S is described by .xk+1 = · · · = xn = 0. Let .(T ∗ U, x1 , . . . , xn , ξ1 , . . . , ξn ) be the associated cotangent coordinate system. The submanifold .N ∗ S ∩ T ∗ U is then described by .xk+1 = · · · = xn = 0 and .ξ1 = · · · = ξk = 0 (by definition of .N ∗ S). This shows that .N ∗ S is an n-dimensional submanifold of .T ∗ X. ∗ S ᶜ→ T ∗ X and the tautological one-form on .T ∗ U Consider ∑ the inclusion .i : N ∗ as .τ = ξi dxi . So at .p ∈ N S we obtain

.

(i ∗ τ )p = τp |Tp (N ∗ S) ∑ = ξi dxi |
k

∂ ∂xi

,i≤k>

= 0. Here in the summation we consider .i > k since on conormal bundle of S we have ξ1 = · · · = ξk = 0, and also for .Tp (N ∗ S) =< ∂x∂ i , i ≤ k > we consider .i ≤ k since on .N ∗ S we have .xk+1 = · · · = xn = 0. Hence we get .(i ∗ σ )p = (i ∗ dτ )p = d(i ∗ τ )p = 0. Therefore, the conormal bundle .N ∗ S is a Lagrangian submanifold of .T ∗ X. ⨅ ⨆

.

Remark 2.10 (1) Let .S = {x} in the above lemma; then the conormal bundle .N ∗ S = Tx∗ X is a cotangent fiber. Therefore we conclude that the fibers of the cotangent bundle are Lagrangian submanifolds of the cotangent bundle. (2) Let .S = X in the above lemma; then the conormal bundle .N ∗ S = X0 is the zero section of .T ∗ X. Hence by the above lemma, we conclude that the zero section of the cotangent bundle is a Lagrangian submanifold of the cotangent bundle (as we saw also in the above example). Definition 2.18 Let .(M, ω) be a 2n-dimensional symplectic manifold. Let .ϕ : N −→ M be an embedding of an n-manifold N in .M. Then .ϕ is called a Lagrangian embedding if .ϕ ∗ ω = 0. The condition .ϕ ∗ ω = 0 is equivalent to say that (by definition of pullback) ωp |Tp (ϕ(N )) = 0

.

for .p ∈ ϕ(N ). So the definition implies that .im(ϕ) is isotropic and half dimensional, so it is Lagrangian submanifold of .M. Example 2.12 Let .s : X −→ T ∗ X be a section of .T ∗ X. We remember that we have .s ∗ τ = s, where .τ is the tautological one-form on .T ∗ X. Moreover, .s ∗ (dτ ) = d(s ∗ τ ) = ds. Then by above definition, s is a Lagrangian embedding of X if and only if .ds = 0. Therefore .im(s) is a Lagrangian submanifold if and only if s is a closed form.

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2 Symplectic Manifolds

So in general if .α is a one-form on X, then the .graph(α) is a Lagrangian submanifold of .T ∗ X if and only if .α is closed. In other words there is correspondence between Lagrangian submanifolds of .T ∗ X that project smoothly onto X and closed one-forms on .X. We leave as an exercise to check that the product of two symplectic manifolds (M1 , ω1 ) and .(M2 , ω2 ) is the symplectic manifold .(M1 ×M2 , (pr1 )∗ ω1 −(pr2 )∗ ω2 ), where .pr1 and .pr2 indicate the projections on the first and second components, receptively. Below we see the interesting relation between Lagrangian submanifolds and symplectomorphisms.

.

Lemma 2.8 Let .ϕ : (M1 , ω1 ) → (M2 , ω2 ) be a diffeomorphism between two 2ndimensional symplectic manifolds. Then .graph(ϕ) is a Lagrangian submanifold of (M1 × M2 , (pr1 )∗ ω1 − (pr2 )∗ ω2 )

.

if and only if .ϕ is a symplectomorphism. Proof The submanifold .graph(ϕ) = {(p, ϕ(p)) | p ∈ M1 } is an embedded image of .M1 in .M1 × M2 , where the embedding is the mapping γ : M1 −→ M1 × M2

.

p ‫׀‬−→ (p, ϕ(p)). The .graph(ϕ) is Lagrangian if and only if .γ ∗ ((pr1 )∗ ω1 − (pr2 )∗ ω2 ) = 0. So we obtain equivalently 0 = γ ∗ pr1∗ ω1 − γ ∗ pr2∗ w2

.

= (pr1 ◦ γ )∗ ω1 − (pr2 ◦ γ )∗ ω2 . However, .pr1 ◦ γ is the identity mapping on .M1 and .pr2 ◦ γ = ϕ. So we get 0 = ω1 − ϕ ∗ ω2 and .ϕ ∗ ω2 = ω1 , which means that .ϕ is a symplectomorphism. ⨅ ⨆

.

Let .X1 and .X2 be n-dimensional manifolds and .τ1 , τ2 be tautological oneforms on their cotangent bundle. Under the natural identification .T ∗ X1 × T ∗ X2 ∼ = T ∗ (X1 ×X2 ), the tautological one-form on .T ∗ (X1 ×X2 ) is .τ = (pr1 )∗ τ1 +(pr2 )∗ τ2 , where pri : T ∗ X1 × T ∗ X2 −→ T ∗ Xi , i = 1, 2,

.

are projections. The canonical two-form on .T ∗ (X1 × X2 ) is σ = dτ

.

= dpr1∗ τ1 + dpr2∗ τ2 = pr1∗ dτ1 + pr2∗ dτ2 .

2.7 Geodesic Flow

65

We saw in the above example that the graph of a closed one-form is a Lagrangian submanifold of .T ∗ X. This leads to the following definition. Definition 2.19 Consider the closed one-form df on .X1 ×X2 , where f is a smooth function on .X1 × X2 . The Lagrangian submanifold generated by f is Lf := {((x, y), (df )(x,y) ) | (x, y) ∈ X1 × X2 }.

.

If we denote .(df )(x,y) projected to .Tx∗ X1 × {0} by .dx f and we denote .(df )(x,y) projected to .{0} × Ty∗ X2 by .dy f , then we can write Lf := {(x, y, dx f, dy f ) | (x, y) ∈ X1 × X2 }.

.

We use the above definition of Lagrangian submanifold generated by f in the next section where we see that geodesic flows are represented by symplectomorphisms. Exercise 2.6 Show that n-torus, i.e., .T n = S 1 × · · · × S 1 (n times), is a Lagrangian submanifold of .R2n . Exercise 2.7 Which de Rham cohomology classes of .S 2 × S 2 , T 4 are represented by symplectic forms? Exercise 2.8 Let .ω be an area form on .S 2 , and using exercise 2, find a symplectic form on .S 2 × S 2 . Exercise 2.9 Consider .ω as an area form on .S 2 and .S 2 × S 2 with the symplectic forms Ω0 = pr1∗ ω + pr2∗ ω

.

Ω1 =

1 ∗ pr ω + 2pr2∗ ω, 2 1

where .pri , i = 1, 2, is the projection on the i-th factor of .S 2 × S 2 . Show that there is no diffeomorphism .φ of .S 2 × S 2 such that .φ ∗ Ω1 = Ω0 although the volumes of 2 2 2 2 .(M, Ω ) and .(M, Ω ) are equal. Is there a symplectic structure on .S × S so that 0 1 2 2 .S × {p} is a Lagrangian submanifold for some .p ∈ S ?

2.7 Geodesic Flow Geodesic flows are always important subject of study in Riemannian geometry. Here, we are going to study geodesic flows from symplectic geometry point of view. Suppose .(X, g) is a Riemannian manifold, so the Riemannian structure is given by an inner product .gx : Tx X × Tx X −→ R on the tangent space .Tx X varying

66

2 Symplectic Manifolds

smoothly with the base point .x ∈ X. Since .gx is non-degenerate, we can consider the linear isomorphism gx : Tx X −→ Tx∗ X

.

gx (u)(v) = gx (u, v) for all .u, v ∈ Tx X. We use also g to denote the natural vector bundle isomorphism between the tangent and cotangent bundle, i.e., .g : T X −→ T ∗ X. Let .π : T ∗ X −→ X be the cotangent bundle equipped with its canonical symplectic form .dτ, where .τ is its tautological one-form. The pullback bundle .g ∗ π : T X −→ X is the projection map for the tangent bundle.

b The arc-length of a smooth curve .γ : [a, b] −→ X is defined by . a | where / dγ dγ dγ ). |:= gγ (t) ( , . | dt dt dt

dγ dt

| dt,

Assume that any two points .x, y ∈ X are joined by a unique (up to reparametrization) minimizing geodesic whose arc-length .d(x, y) is called the Riemannian distance between x and .y. Let .(X, g) be geodesically complete, i.e., every minimizing geodesic can be extended indefinitely. Given .(x, v) ∈ T X, let .γxv : R −→ X be the unique minimizing geodesic of constant velocity with initial conditions γxv (0) = x and

.

d |t=0 γxv (t) = v. dt

Consider the function .f : X × X −→ R given by 1 f (x, y) = − d(x, y)2 . 2

.

∗ (X × X) ∼ T ∗ X × Let .dx f and .dy f be components of .d(x,y) f with respect to .T(x,y) = x ∗ Ty X. Therefore

graph(df ) = {(x, y, dx f, −dy f ) | (x, y) ∈ X × X}

.

is the Lagrangian submanifold of .T ∗ X × T ∗ X generated by the function .f. If the Lagrangian submanifold .graph(df ) is the graph of a diffeomorphism .ϕ : T ∗ X −→ T ∗ X, then by above lemma .ϕ is a symplectomorphism and is called the

2.8 Local Theorems

67

symplectomorphism generated by .f. In this case .ϕ is defined as .ϕ(x, ξ ) = (y, η) if and only if .ξ = dx f and .η = −dy f. In fact, finding .ϕ leads to solving a Hamiltonian system which we are going to study in the next chapter. Under the identification of T X with .T ∗ X by .g, the symplectomorphism .ϕ generated by f coincides with the mapping .ϕ˜ : T X −→ T X, where (x, v) ‫׀‬−→ (γxv (1),

.

d |t=1 γxv (t)). dt

The mapping .ϕ˜ is called the geodesic flow on .T X.

2.8 Local Theorems One of the important facts in symplectic geometry is given by Darboux’s theorem, which indicates that all 2n-dimensional symplectic manifolds are locally the same as Euclidean symplectic manifold .(R2n , σst ) with the standard symplectic form. In the previous chapter, in Sect. 1.5 we saw the linear version of Darboux’s theorem as the normal form in symplectic linear algebra. On manifolds, the original proof given by Darboux in [11] (see also [3], p.230 and [50]) uses induction on the dimension. We will study this classical proof given by Darboux in Sect. 3.6. In this section we explain the proof of this theorem given by Weinstein [52] (see also [25, 37]) that is based on deformation argument that has been introduced in normal form theory by Moser [42]. Hence, here we describe normal neighborhoods of a point (the Darboux’s theorem) and of a Lagrangian submanifold (the Weinstein’s theorems), inside a symplectic manifold. We use Moser trick which we explained in Sect. 2.5. Moser trick leads to Moser theorems which we explain below. First we need to recall some neighborhood theorems. Let M be an n-dimensional manifold and .X ⊂ M be a k-dimensional submanifold. Let .i : X ᶜ→ M be the inclusion mapping. The differential of the inclusion mapping is the mapping .dix : Tx X ᶜ→ Tx M, which is an inclusion of the tangent space of X at the point .x ∈ X into the tangent space of M at the point .x. Definition 2.20 The normal space to X at the point .x ∈ X is the .(n − k)dimensional vector space defined by .Nx X := Tx M/Tx X. The normal bundle of X is NX := {(x, v) | x ∈ X, v ∈ Nx X}.

.

By natural projection, NX is a vector bundle over X of rank .n − k, and it is an n-dimensional manifold. The zero section of .NX, i.e., i0 : X ᶜ→ NX,

.

x ‫׀‬−→ (x, 0), is an embedding of X as a closed submanifold of .NX.

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2 Symplectic Manifolds

Definition 2.21 We say that a neighborhood .U0 of the zero section X in NX is convex if the intersection of .U0 with each fiber, i.e., .U0 ∩ Nx X, is convex. Theorem 2.5 (Tubular Neighborhood Theorem) There exists a diffeomorphism ϕ : U0 −→ U, from a convex neighborhood .U0 of X in NX onto a neighborhood U of X in .M, such that the following diagram commutes.

.

If we consider the subset .U0 ⊆ NX, we obtain a submersion .π0 : U0 −→ X whose all fibers, .π0−1 (x), are convex. If we define .π := π0 ◦ ϕ −1 , we can extend π0

π

this fibration to .U, i.e., if .NX ⊇ U0 −→ X is a fibration, then .M ⊇ U −→ X is a fibration that is called tubular neighborhood fibration. Proposition 2.2 Let U be a tubular neighborhood of a smooth submanifold X in M. If a closed one-form .ω on U has restriction .i ∗ ω = 0 (i is the inclusion mapping), then .ω is exact, i.e., .ω = dμ, for some .μ ∈ Ωl−1 (M). Moreover, we can choose .μ such that .μx = 0 for every .x ∈ X.

.

Proof First note that for the inclusion mapping .i : X ᶜ→ U we have .i ∗ : Ωl (U ) −→ Ωl (X) defined as (i ∗ ω)x (v2 , . . . , vl ) = ωx (Tx (iv2 ), . . . , Tx (ivl )).

.

By tubular neighborhood theorem, we know that .ϕ : U0 −→ U is a diffeomorphism, so we show the proof in .U0 instead of .U. So we have the following commutative diagram.

If .ω ∈ Ωp (M) is closed, then .ϕ ∗ ω is also a closed form because .d(ϕ ∗ ω) = = 0. If .i ∗ ω = 0, then .i0∗ (ϕ ∗ ω) = 0 because .i0∗ (ϕ ∗ ω) = (ϕ ◦ i0 )∗ (ω) = ∗ i ω = 0. So we have .ϕ ∗ ω = dη. X is a deformation retract of .U0 . Let .t ∈ [0, 1], and define

ϕ ∗ (dω)

ρt : U0 −→ U0

.

(x, v) ‫׀‬−→ (x, tv). This mapping is well-defined because .U0 is convex. Moreover .ρ is an isotopy, and we have ρ0 (x, v) = (x, 0);

.

∀(x, v) ∈ U0 ,

2.8 Local Theorems

69

which means that .ρ0 = i0 ◦ π0 , where .π0 (x, v) = (x, 0). Also we have ρ1 (x, v) = (x, v),

.

i.e. ρ1 = idU0 .

Therefore .ρt is a deformation retract with retraction .π0 . This means that .ρt is a homotopy for .ρ0 and .ρ1 . In other words, we have .i0 ◦ π0 ∼ idU0 and .π0 ◦ i0 = idX . Moreover, ρt ◦ i0 (x) = ρt (x, 0) = (x, 0) = i0 (x),

∀x ∈ X.

.

Now we are going to consider homotopy between co-chain (de Rham) complexes. A homotopy between .ρ0∗ and .ρ1∗ is a family of linear mappings .Qp : Ωp (U0 ) −→ Ωp−1 (U0 ) such that it satisfies the equation I dp − (i0 ◦ π0 )∗p = dp−1 ◦ Qp + Qp+1 dp .

.

Let .{vt }t∈[0,1] be a family of vector fields generated by .{ρt }t∈[0,1] and define  Qω :=

1

ρt∗ (ivt ω)dt,

.

0

ω ∈ Ω∗ (U0 ).

Thus we have  Qdω + dQω =

1

.

0

 =

1

0

 =

1

0

 = 0

1

ρt∗ (ivt dω)dt + d

 0

1

ρt∗ (ivt ω)dt

ρt∗ (ivt dω + d(ivt ω))dt ρt∗ (Lvt ω)dt d ∗ (ρ ω)dt dt t

= ρ1∗ ω − ρ0∗ ω where in the third equality we have used Cartan formula and in the fourth equality we have used ρt∗ (Lvt ω) = ρt∗ (ivt

.

= 0+ =

d d ω) + ρt∗ ( ivt ω) dt dt

d ∗ ρ ω(vt , −) dt t

d ∗ ρ ω, dt t

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2 Symplectic Manifolds

where the first equality is by Cartan formula and second equality follows from the fact that .ω is a closed form. If .ω is closed and .i0∗ ω = 0, then Qdω + dQω = (id − (i0 ◦ π0 )∗ )ω

.

= 0 + ω − π0∗ ◦ i0∗ (ω). Therefore we obtain .dQω = ω. So if we set .Qω = μ, then .ω is exact. On the other hand, .ρt (x, 0) = (x, 0) for all .t ∈ [0, 1]. .ρt is constant on .X, so .vt = 0. Therefore  μ = Qω =

.

0

1

ρt∗ (iv ω)dt = 0 ⨆ ⨅

on .X.

In Sect. 2.5 we studied Moser trick in the version I of Moser’s theorems. Now, in the following we state the version II of Moser’s theorems and the local version. Theorem 2.6 (Moser—Version II [42]) Let M be a compact manifold with symplectic forms .ω0 and .ω1 . Suppose that .(ωt )t∈[0,1] is a smooth family of closed two-forms joining .ω0 and .ω1 and satisfying: (1) (Cohomology assumption) .[ωt ] is independent of .t, i.e., .

d d [ωt ] = [ ωt ] = 0. dt dt

(2) (Non-degeneracy condition) .ωt is non-degenerate for all .t ∈ [0, 1]. Then there exists an isotopy .ρ : M × R −→ M such that ρt∗ ωt = ω0 , ∀t ∈ [0, 1].

.

Theorem 2.7 (Moser—Local Version [42]) Let M be a manifold, X a compact submanifold of .M, .i : X ᶜ→ M the inclusion mapping, .ω0 , and .ω1 two symplectic forms on .M. If .ω0 |q = ω1 |q for all .q ∈ X, we obtain that there exist neighborhoods ∗ .U0 , U1 of X in M and a diffeomorphism .ϕ : U0 −→ U1 such that .ϕ ω1 = ω0 and the following diagram commutes.

Proof We choose a tubular neighborhood .U0 of .X. The two-form .ω1 − ω0 is closed on .U0 , and by hypothesis .(ω1 − ω0 )q = 0 for all .q ∈ X. By Proposition 2.2, there exists a one-form .μ on .U0 such that .ω1 − ω0 = dμ and .μq = 0 at all .q ∈ X. We consider a family

2.8 Local Theorems

71

ωt = (1 − t)ω0 + tω1 = ω0 + tdμ

.

of closed two-forms on .U0 . By shrinking .U0 (if it is necessary), we can assume that ωt is symplectic for .t ∈ [0, 1]. We can solve the equation .iXt ωt = −μ and note that .Xt |X = 0 because in X .

iXt ωt = 0 =⇒ iXt ωt (v) = 0 =⇒ ωt (Xt , v) = 0 =⇒ Xt = 0.

.

Remember that a time-dependent vector field .vt generates local isotopy dρt = vt (ρt ), ρt : U0 −→ U0 . dt

.

Again shrinking .U0 if it is necessary, there is an isotopy .ρ : U0 × [0, 1] −→ M with .ρt∗ ωt = ω0 for all .t ∈ [0, 1]. Since .Xt |X = 0, we have .ρt |X = idX (because: dρt . dt = 0 in .X =⇒ ρt = ρ0 in .X (ρ0 = idX )). Then we can set .ϕ := ρ1 and ∗ .U1 := ρ1 (U0 ). So we obtain .ϕ ω0 = ω1 . ⨆ ⨅ Let .(M, σ ) be a symplectic manifold. The Darboux’s theorem ∑n states that locally the form .σ can be brought into the canonical form .σst = i= dξi ∧ dxi . This means∑ that if we obtain any local property invariant under symplectomorphisms for n 2n .(R , i= dξi ∧ dxi ), then any symplectic manifold has that local property as well. Theorem 2.8 (Darboux’s Theorem) For every point .p ∈ M, there exists an open neighborhood U of p in M and a diffeomorphism .ϕ from U onto an open subset of n .R × R such that ϕ∗(

n ∑

.

dξi ∧ dxi ) = σ on U.

i=1

Proof Let .(x1' , . . . , xn' , y1' , . . . , yn' ) be a local coordinate on a neighborhood .U ' of p such that {

.

∂ ∂ ∂ ∂ |p , . . . , ' |p , ' |p , . . . , ' |p } ' ∂x1 ∂xn ∂y1 ∂yn

form a symplectic basis for .Tp M, i.e.,

Then .ωp =

∑n

' i=1 dxi

.

ω(

∂ ∂ |p , ' |p ) = 0 ' ∂yi ∂yj

ω(

∂ ∂ |p , ' |p ) = 0 ∂xi' ∂xj

ω(

∂ ∂ |p , ' |p ) = δij . ∂xi' ∂yj

∧ dyi' |p .

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We have two symplectic forms .ω0 = ω and .ω1 = Moser’s theorem (local version) with:

∑n

' i=1 dxi

∧ dyi' on .M. By

• .U ' manifold .• X={p} ' .• .ω0 and .ω1 symplectic forms on .U .

there exist neighborhoods .U0 and .U1 of .p, and a diffeomorphism .ϕ : U0 −→ U1 such that .ϕ(p) = p. And ∑ ϕ ∗ (ω1 ) = ϕ ∗ ( dxi' ∧ dyi' ) ∑ = (dxi' ∧ dyi' ) ∑ = ϕ ∗ (dxi' ) ∧ ϕ ∗ (dyi' ) ∑ = d(xi' ◦ ϕ) ∧ d(yi' ◦ ϕ)

.

= ω0 and if we set .xi = xi' ◦ ϕ and .yi = yi' ◦ ϕ, the coordinates system is .(U0 , x1 , . . . , xn , y1 , . . . , yn ). ⨅ ⨆ In local version of the Moser’s theorems, consider X as a Lagrangian submanifold for .ω0 and .ω1 , i.e., as an n-dimensional submanifold with .i ∗ ω0 = i ∗ ω1 = 0, where .i : X ᶜ→ M is inclusion. Weinstein showed that under this assumption the statement is still true. Below we see the Weinstein Lagrangian neighborhood theorem. Before that, we recall the following theorem. Theorem 2.9 (Whitney Extension Theorem, [56]) Let M be an n-dimensional manifold and X a k-dimensional submanifold with .k < n. Suppose that at each .q ∈ X we are given a linear isomorphism .Lq : Tq M −→ Tq M such that .Lq |Tq X = idTq X and .Lq depends smoothly on .q. Then there exists an embedding .h : N −→ M of some neighborhood N of X in M such that .h |X = idX and .dq h = Lq for all .q ∈ X. Theorem 2.10 (Weinstein Lagrangian Neighborhood Theorem [52]) Let M be a 2n-dimensional manifold, X a compact n-dimensional submanifold, .i : X ᶜ→ M the inclusion mapping, and .ω0 and .ω1 symplectic forms on M such that .i ∗ ω0 = i ∗ ω1 = 0, i.e., X is a Lagrangian submanifold of both .(M, ω0 ) and .(M, ω1 ). Then there exist neighborhoods .U0 and .U1 of X in M and a diffeomorphism .ϕ : U0 −→ U1 such that .ϕ ∗ ω1 = ω0 and the following diagram commutes.

2.8 Local Theorems

73

Proof Consider a Riemannian metric g on .M, i.e., at each .q ∈ M we have that gq is a positive-definite inner product. For a .q ∈ X set .V := Tq M, U := Tq X and .W := U ⊥ as the orthogonal complement of U in V with respect to .gq . By ∗ ∗ .i ω0 = i ω1 = 0, we conclude that the space U is a Lagrangian subspace of both .(V , ω0 |q ) and .(V , ω1 |q ). From symplectic linear algebra, we canonically obtain from .U ⊥ a linear isomorphism .Lq : Tq M −→ Tq M such that .Lq |Tq X = idTq X and ∗ .Lq ω1 |p = ω0 |q . As the construction is canonical, .Lq varies smoothly with respect to .q. Using the Whitney extension theorem, there exist a neighborhood N of X and an embedding .h : N ᶜ→ X with .h |X = idX and .dq h = Lq for .q ∈ X. Therefore, at any point .q ∈ X, we have .

(h∗ ω1 )q = (dq h)∗ ω1 |q = L∗q ω1 |q = ω0 |q .

.

If we use the local version of Moser’s theorems to .ω0 and .h∗ ω1 , we find a neighborhood .U0 of X and an embedding .f : U0 −→ N such that .f |X = idX and .f ∗ (h∗ ω1 ) = ω0 on .U0 . Hence, we can define .ϕ := h ◦ f. ⨆ ⨅ Theorem 2.11 (Coisotropic Embedding Theorem [23, 26, 53]) Let M be a 2ndimensional manifold, X a k-dimensional submanifold with .k < n, .i : X ᶜ→ M the inclusion mapping, and .ω0 and .ω1 two symplectic forms on M such that ∗ ∗ .i ω0 = i ω1 with X being coisotropic for both .(M, ω0 ) and .(M, ω1 ). Then there exist neighborhood .U0 and .U1 of X in M and a diffeomorphism .ϕ : U0 −→ U such that .ϕ ∗ ω1 = ω0 and the following diagram commutes.

Theorem 2.12 (Weinstein Tubular Neighborhood Theorem [52]) Let .(M, ω) be a symplectic manifold, X a compact Lagrangian submanifold of .M, .ω0 the canonical symplectic form on .T ∗ X, i0 : X ᶜ→ T ∗ X the Lagrangian embedding as the zero section, and .i : X ᶜ→ M the Lagrangian embedding given by the inclusion. Then there exist neighborhoods .U0 of X in .T ∗ X, U of X in M and a diffeomorphism ∗ .ϕ : U0 −→ U such that .ϕ ω = ω0 and the following diagram commutes.

Proof For the proof, we use the tubular neighborhood theorem and the Weinstein Lagrangian neighborhood theorem. Since .NX ∼ = T ∗ X, by the tubular neighborhood theorem, we can find a neighborhood .N0 of X in .T ∗ X, a neighborhood N of X in M, and a diffeomorphism .ψ : N0 −→ N such that the following diagram commutes.

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2 Symplectic Manifolds

Consider the canonical symplectic form .ω0 on .T ∗ X and .ω1 := ψ ∗ ω. Both .ω0 and .ω1 are symplectic forms on .N0 . For both .ω0 and .ω1 , the submanifold X is Lagrangian. Now, use the Weinstein Lagrangian neighborhood theorem. There exist neighborhoods .U0 and .U1 of X in .N0 and a diffeomorphism .θ : U0 −→ U1 such that .θ ∗ ω1 = ω0 and the following diagram commutes.

Hence, we can consider .ϕ = ψ ◦ θ and .U1 = ϕ(U0 ). One can check that .ϕ ∗ ω = θ ∗ ψ ∗ ω = θ ∗ ω1 = ω0 . ⨆ ⨅ A similar statement for isotropic submanifolds was also proved by Weinstein in [53] and [54]. The Weinstein tubular neighborhood theorem classifies Lagrangian embeddings, i.e., up to symplectomorphism, the set of Lagrangian embeddings is the set of embeddings of manifolds into their cotangent bundle as zero section.

2.9 Symplectic Invariants By Darboux’s theorem, there is no local invariant besides of the dimension in symplectic geometry. This is in contrast with Riemannian geometry where the curvature provides a local invariant. There are several ways to associate global invariants to symplectic manifolds. Many of them are symplectic invariants associated to symplectic embeddings. In general, the symplectic embeddings of the form .U −→ M are studied, where U is open in .R2n and .(M, ω) is a symplectic manifold of dimension .2n. The following definition was introduced by Gromov in [24] about symplectic embeddings. Definition 2.22 Given a symplectic manifold .(M, ω) of dimension .2n, the symplectic radius of .M, denoted by .r(M), is defined as follows. r(M) = sup{r > 0 | ∃ a symplectic embedding B 2n (r) → M},

.

where .B 2n (r) is the unit open ball centered at 0 in .R2n . The symplectic radius is a symplectic invariant, and it can be considered as the symplectic equivalent of the radius of injectivity in Riemannian geometry.

2.9 Symplectic Invariants

75

The following theorem is a very important theorem in symplectic geometry that was also proved by Gromov in [24]. Theorem 2.13 (Gromov’s Non-squeezing Theorem) For the positive numbers r, R let

.

B 2n (r) = {(x, y) ∈ Rn × Rn = R2n | | x |2 + | y |2 < r 2 }

.

be the unit open ball centered at 0 in .R2n , and Z(R) = {(x, y) ∈ Rn × Rn = R2n | x12 + y12 < R 2 }

.

be an open cylinder. If there exists a symplectic embedding .ϕ : B 2n (r) −→ R2n such that .ϕ(B 2n (r)) ⊆ Z(R), then .r ≤ R. It is easy to verify that a ball of any radius can be placed into a cylinder of any radius in such a way that the volume is preserved (just “squeeze” the ball in a suitable direction). Therefore, the above theorem tells us that although symplectic mappings are volume preserving, it is much more restrictive to be a symplectic mapping than a volume-preserving mapping.

Chapter 3

Hamiltonian Systems

In this chapter we explain the origin of symplectic geometry, i.e., classical mechanics. For this, first we review some background about flows of vector fields and Lie derivatives. Then we see that if a symplectic form .σ is invariant under the flow of a vector field .v, we obtain the equation .iv σ = −df. Such a vector field v satisfying this equation is actually unique and is called Hamiltonian vector field. In coordinates, this equation turns to a Hamiltonian system. In classical mechanics, to solve an equation of motion, one obtains an Euler–Lagrange equation that can be interpreted as a Hamiltonian system, by the Legendre transform. Hence, we will see how the Legendre transform relates an Euler–Lagrange equation to a Hamiltonian system. Next, we talk about Poisson brackets and their relation with symplectic forms and, so, symplectic geometry. One can obtain a Poisson bracket from a closed, nondegenerate two-form, i.e., a symplectic form. Such a two-form appears naturally on a phase space containing the information for the base coordinates .qi ’s and the corresponding fiber coordinates .pi ’s on the vector bundle given by cotangent bundle. We see Poisson brackets in the proof of Darboux’s theorem given by Darboux. Also, Poisson brackets are used to define Hamiltonian group actions and momentum mapping. We close this chapter by explaining the construction of symplectic quotient, using momentum mapping.

3.1 Invariant Form Under the Flow of a Vector Field In this section, we recall some basic notions and facts about flows of vector fields. Then, we review the definition of Lie derivative and its properties which we will use in the next sections. We see that a form is invariant under the flow of a vector field if its Lie derivative with respect to the vector field vanishes.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. Eslami Rad, Symplectic and Contact Geometry, Latin American Mathematics Series, https://doi.org/10.1007/978-3-031-56225-9_3

77

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3 Hamiltonian Systems

Let M be a smooth manifold and v be a smooth vector field on .M. From the theory of systems of ordinary differential equations, it follows that for every .m ∈ M there is a unique maximal solution (curve) .γ = γm : Im → M of the ordinary differential equation .

dγ (t) = v(γ (t)) dt

such that .γ (0) = m (see for instance [10]). Here, the domain .Im of the maximal solution .γ is an open interval in .R containing .0. If we set .s := supIm < ∞ or .i := inf Im > −∞, then for every compact subset K of .M, there exists an .ε > 0 such that .γ (t) ∈ / K for every .t ∈]s − ε, s[ or .t ∈]i, i + ε[. Hence, the only way the maximal solutions do not exist for all time is that solutions run out of every compact subset of M in a finite time. Therefore, if .γm (t) stays within a compact subset of M for all .t ∈ Im , then .Im = R. Consider the set .D := {(t, m) ∈ R × M | t ∈ Im }, which is an open subset of .R × M and contains .{0} × M. Define the smooth mapping .A : D −→ M by .(t, m) ‫ →׀‬γm (t). For .s ∈ Im and .t ∈ Iγm (s) , we have .s + t ∈ Im and .γm (s + t) = γγm (s) (t) because .γm (s + t) and .γγm (s) (t), as functions of .t, satisfy the same differential equation and have the same initial value; hence, the equality is obtained by the uniqueness of the solutions. Definition 3.1 The vector field v is said to be complete if .D = R × M, and .A : R × M −→ M is a smooth action of the additive group .(R, +) on .M. Definition 3.2 The mapping .M → M; m ‫ →׀‬γm (t) is called the time t flow of the vector field v, and we denote it by .etv . Remark 3.1 The notation “.etv ” recalls the equation .

detv = v ◦ etv dt

and the group homomorphism property .e(s+t)v = etv ◦ esv . Since .etv ◦ e−tv = I d = e−tv ◦ etv , the time t flow is a diffeomorphism of .M. In other words, it is bijective from M to M and has a smooth inverse equal to .e−tv . Remark 3.2 If D is a proper subset of .R × M, i.e., .D /= R × M, then the flow etv is defined on the open subset .Mt := {m ∈ M | (t, m) ∈ D} of .M, and we have (s+t)v (m) = etv ◦ esv (m) when .m ∈ M and .esv (m) ∈ M , where .m ∈ M .e s t s+t . Moreover, the mapping .etv is a diffeomorphism from the open subset .Mt of M onto the open subset .M−t of .M, with inverse equal to .e−tv . So, if .D /= R × M, we do not have a group action. .

Definition 3.3 (Lie [36]) The mapping .R −→ M; t ‫׀‬−→ etv is called the oneparameter group of transformations generated by the vector field v.

3.1 Invariant Form Under the Flow of a Vector Field

79

We denote the space of smooth p-forms on the smooth manifold .M, by .Ωp (M), where .Ω0 (M) = F(M) denotes the space of smooth real-valued functions on M and .Ωp (M) = {0} if .p > dimM. Definition 3.4 Let M and N be smooth manifolds of any dimensions, and .ϕ : M → N be a smooth mapping. For any .ω ∈ Ωp (N ) the pullback .ϕ ∗ ω of .ω under the mapping .ϕ is defined by (ϕ ∗ ω)m (v1 , . . . , vp ) := ωϕ(m) (Tm ϕv1 , . . . , Tm ϕvp ).

.

Remark 3.3 (1) By the above definition, .ϕ ∗ is a continuous linear operator from .Ωp (N ) to p ∗ .Ω (M). When .p = 0, we have .ϕ ω = ω ◦ ϕ, where ϕ ∗ : F(N ) −→ F(M)

.

ω ‫׀‬−→ ω ◦ ϕ. (2) Pullback is an anti-homomorphism with respect to composition, i.e., (ψ ◦ ϕ)∗ = ϕ ∗ ◦ ψ ∗ .

.

(3) For the exterior derivative .d : Ωp (M) −→ Ωp+1 (M) of differential forms, we have ϕ ∗ (dω) = d(ϕ ∗ ω),

.

where .ω ∈ Ωp (N ) and .ϕ : M −→ N is a smooth mapping, because the following diagram commutes.

Definition 3.5 Let .χ (M) denote the vector space of smooth vector fields on .M. If v ∈ χ (M), then the Lie derivative .Lv ω of .ω ∈ Ωp (M) with respect to the vector field v is defined as

.

Lv ω :=

.

d tv ∗ (e ) ω |t=0 . dt

Definition 3.6 A p-form .ω ∈ Ωp (M) is called invariant under the flow of a vector field .v, if for every .t ∈ R, (etv )∗ ω = ω in .Mt .

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3 Hamiltonian Systems

Lemma 3.1 A p-form .ω ∈ Ωp (M) is invariant under the flow of a vector filed v if and only if the Lie derivative of .ω with respect to v is equal to zero, i.e., .Lv ω = 0. Proof We consider ((etv )∗ ◦ (esv )∗ )ω = (esv ◦ etv )∗ ω = (e(t+s)v )∗ ω.

.

We take derivative with respect to s at .s = 0 from both sides: .

d d (s+t)v ∗ ((etv )∗ ◦ (esv )∗ )ω |s=0 = (e ) ω |s=0 ds ds d d ((etv )∗ ((esv )∗ ω)) |s=0 = ( e(s+t)v |s=0 )∗ ω ds ds d d (etv )∗ ( (esv )∗ ω) |s=0 = ( (esv ◦ etv ) |s=0 )∗ ω ds ds (etv )∗ (Lv ω) = (v ◦ etv )∗ ω (etv )∗ (Lv ω) =

d tv ∗ (e ) ω. dt

d (esv ) |s=0 = v. So we have Here in the fourth line we used the initial condition . ds obtained

(etv )∗ Lv ω =

.

d tv ∗ (e ) ω. dt

This implies that .Lv ω = 0 if and only if the derivative in the right-hand side is zero. In other words .Lv ω = 0 if and only if the mapping .t ‫׀‬−→ (etv )∗ ω is constant. At tv ∗ .t = 0 we have .(e ) ω = ω; hence, it is equal to .ω. ⨆ ⨅ Definition 3.7 If .v ∈ χ (M) and .ω ∈ Ωp (M), then the interior product .iv ω ∈ Ωp−1 (M) of .ω with v is defined by (iv ω)m (v2 , . . . , vp ) := ωm (v(m), v2 , . . . , vp ).

.

In other words, .iv is the continuous linear operator from .Ωp (M) to .Ωp−1 (M) of inserting the vector field v at the first slot. iv : Ωp (M) −→ Ωp−1 (M)

.

ω ‫׀‬−→ iv ω. In the following lemma, we see that for a smooth function .f, the interior product of df with a vector field v is just directional derivative of f along the direction of .v. Lemma 3.2 If .p = 0, then .Lv ω = iv (dω).

3.1 Invariant Form Under the Flow of a Vector Field

81

Proof For .p = 0 we denote .ω by the function f ; then d tv ∗ (e ) f |t=0 dt d = (f ◦ etv (m)) |t=0 dt = (df )(v)

Lv f =

.

= (iv df ), which is the derivative of the function f in the direction of the vector field .v.

⨆ ⨅

Recall For general .p, i.e., for any p-form .ω ∈ the Cartan formula gives the relation between exterior product, interior product, and Lie derivative: Ωp (M),

Lv ω = iv (dω) + d(iv ω).

.

Definition 3.8 Let .v ∈ χ (M) and let .ϕ : M → M be a smooth mapping for which, at each point .m ∈ M, the mapping .Tm ϕ : Tm M → Tϕ(m) M is invertible. Then the pullback .ϕ ∗ v ∈ χ (M) on M of v under .ϕ is defined by (ϕ ∗ v)(m) := (Tm ϕ)−1 v(ϕ(m)), m ∈ M.

.

As we mentioned in Remark 3.3, one can change the order of pullback and differential. In the following lemma we see that we can change the order of pullback with interior product. Lemma 3.3 For any .v ∈ χ (M) and .ω ∈ Ωp (M), we have .ϕ ∗ (iv ω) = iϕ ∗ v ϕ ∗ ω. Proof We start from the left-hand side. Since .ω is a p-form, the interior product .iv ω and its pullback by .ϕ is a .(p − 1)-form. So for .p − 1 vectors .v2 , . . . , vp in .χ (M), we have .ϕ

∗

(iv ω)(v2 , . . . , vp ) = (iv ω)ϕ(m) ((Tm ϕ)v2 , . . . , (Tm ϕ)(vp )) = ωϕ(m) (v(ϕ(m)), (Tm ϕ)(v2 ), . . . , (Tm ϕ)(vp )) = ωϕ(m) ((Tm ϕ)(Tm ϕ)−1 (v(ϕ(m)), (Tm ϕ)(v2 ), . . . , (Tm ϕ)(vp )) = ωϕ(m) ((Tm ϕ)ϕ ∗ v(m), (Tm ϕ)(v2 ), . . . , (Tm ϕ)(vp )) = (ϕ ∗ ω)(ϕ ∗ v(m), v2 , . . . , vp ) = (iϕ ∗ v ϕ ∗ ω)(v2 , . . . , vp ).

Here in the first equality we used the definition of pullback of the form .iv ω, in the second equality we used the definition of interior product, in the fourth equality we used the definition of pullback of vector field .v, in the fifth equality we used

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3 Hamiltonian Systems

the definition of pullback of .ω, and in the sixth equality we used the definition of interior product. ⨅ ⨆ Above we saw the definition of Lie derivative of a form with respect to a vector field. Now, in a similar way, we are going to define the Lie derivative of a vector field .v ∈ χ (M) with respect to a vector field .w ∈ χ (M). This is the Lie bracket of v and w denoted by .[v, w], which is also in .χ (M). Definition 3.9 Let .v, w ∈ χ (M). We define the Lie brackets .[v, w] ∈ χ (M) of the vector fields v and w on M by [v, w] = Lv w :=

.

d tv ∗ (e ) w |t=0 . dt

In the following lemma we see the relation between Lie brackets with Lie derivative and interior product. This relation somehow pictures the Leibniz rule for derivations. Lemma 3.4 Let .v, w ∈ χ (M) be vector fields on M and .ω ∈ Ωp (M) be a p-form. Then Lw (iv ω) = i[w,v] ω + iv (Lw ω).

.

Proof Let w be a smooth vector field on .M, with the flow .ϕ = etw . We differentiate from both sides of .ϕ ∗ (iv ω) = iϕ ∗ v ϕ ∗ ω, which we obtained in Lemma 3.3. So we get .

d tw ∗ d (e ) (iv ω) |t=0 = (i(etw )∗ v (etw )∗ ω) |t=0 dt dt Lw (iv ω) = L(etw )∗ v (etw )∗ ω |t=0 −i(etw )∗ v

d tw ∗ (e ) ω |t=0 dt

d s(etw )∗ v ∗ tw ∗ (e ) ((e ) ω) |s=0 |t=0 −(i(etw )∗ v Lw ω) |t=0 ds d tw ∗ = ( es(e ) v |s=0 )∗ ((etw )∗ ω) |t=0 −(Lw ω)((etw )∗ v, . . . ) |t=0 ds =

= ((etw )∗ v)∗ ((etw )∗ ω) |t=0 −(Lw ω)((Tm etw )−1 v, . . . ) |t=0 d tw ∗ (e ) v, . . . ) |t=0 −(Lw ω)(Tm e−tw )v, . . . ) |t=0 dt = ω([w, v], . . . ) + (Lw ω)(v, . . . ) = ((etw )∗ ω)(

= i[w,v] ω + iv (Lw ω). Here in the right-hand side of the second line, we used Cartan formula, and in the third line, we used the definition of Lie derivative. In the fifth equality, we used the initial condition for the flow of the vector field .(etw )∗ v and the definition of pullback

3.1 Invariant Form Under the Flow of a Vector Field

83

of a vector. In the first part of the sixth line, we used the definition of pullback of a form, and in the second part, we used the derivative of inverse function. In the seventh line, we used the definition of the Lie brackets, and in its second part, we used the fact that .Tm e−tw |t=0 = −Tm etw |t=0 = −Tm (etw |t=0 ) = −Tm (id) = −id. ⨆ ⨅ Remark 3.4 Using the definition of Lie brackets, i.e., d d d tv ∗ (e ) w |t=0 := ( e−tv ◦ esw ◦ etv |s=0 ) |t=0 , dt dt ds

[v, w] = Lv w :=

.

the Lie brackets of the vector fields v and .w, in local coordinates, can be written as [v, w](m) = (Dw)(m)v(m) − (Dv)(m)w(m).

.

If the vector fields are linear in the local coordinates, then this implies [v, w] = w ◦ v − v ◦ w,

.

which is equal to commutator of w and v with the opposite sign. In general, the commutator of the linear operators A and B is denoted by .[A, B] := A◦B −B ◦A. Lemma 3.5 If .ω = df in which f is a smooth function, then .L[w,v] f = [Lw , Lv ]f. Proof In the formula of Lemma 3.4, we set .ω = df. Lw (iv df ) = i[w,v] df + iv (Lw df ).

.

From Lemma 3.2 we remember that .iv df = Lv f, and using this in the left-hand side and Cartan formula in the right-hand side, we obtain Lw Lv f = L[w,v] f + iv (iw ddf + diw df ).

.

On the other hand, the second part in the right-hand side is written as iv (iw ddf + diw df ) = 0 + iv (diw df ) = Lv iw df = Lv Lw f,

.

where the second and third equalities are again followed from Lemma 3.2. So we get Lw Lv f = L[w,v] f + Lv Lw f,

.

or we have .L[w,v] f = Lw Lv f − Lv Lw f, and therefore, we obtain that .L[w,v] f = ⨆ ⨅ [Lw , Lv ]f.

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3 Hamiltonian Systems

Exercise 3.1 Let M be a smooth manifold. Show that for every .u, v ∈ χ (M) and ω ∈ Ωp (M), we have

.

L[u,v] ω = [Lu , Lv ]ω.

.

3.2 Hamiltonian Vector Fields Below, we see that if a symplectic form is invariant under the flow of a vector field, then we obtain a Hamiltonian system. Let .(M, σ ) be a symplectic manifold. Assume that the flow .etv of the smooth vector field .v ∈ χ (M) leaves the symplectic form .σ invariant, i.e., .(etv )∗ σ = σ , which is equivalent to .Lv σ = 0 by Lemma 3.1. Using Cartan formula, we have .0 = Lv σ = iv (dσ ) + d(iv σ ). But .σ is a closed form, i.e., .dσ = 0; therefore .d(iv σ )) = 0, which means that .iv σ is a closed form. So the symplectic form .σ is invariant under the flow of v if and only if .iv σ is a closed form. The condition that .iv σ is closed is locally equivalent to the condition that .iv σ is equal to the total derivative of a smooth function (because locally every closed form is exact by de Rham cohomology). Therefore, we can formulate this condition in the following equality: iv σ = −df

.

for locally defined smooth functions .f, where the minus sign is a matter of convention. Remark 3.5 If .H 1 (M) = 0, then there is a globally defined function f on M (i.e., .f : M −→ R) such that .iv σ = −df. If M is connected, then f is uniquely determined up to an additive constant. Conversely, if we have a smooth function f , then the following lemma shows that there is a unique vector field such that .iv σ = −df, which means that the symplectic form under the flow of this vector field is invariant. Lemma 3.6 If f is a smooth function on M, then there is a unique vector field v on M such that .iv σ = −df. Proof Consider the smooth function .f : M −→ R and .(df )m : Tm M −→ R. Therefore .(df )m ∈ Tm∗ M. On the other hand we know that for the symplectic form ∗ .σ the mapping .σm : Tm M −→ Tm M is an isomorphism, and therefore .

for (df )m ∈ Tm∗ M, ∃!v ∈ Tm M such that σm (v) = −(df )m ,

which means .iv σm = −(df )m . Moreover v is smooth because df and .m ‫׀‬−→ σm−1 are smooth. ⨅ ⨆

3.2 Hamiltonian Vector Fields

85

Definition 3.10 The smooth vector field v on M such that .iv σ = −df is called the Hamiltonian vector field on M defined by the function f , and we denote it by .Hf , and f is called the Hamiltonian function of the vector field .v. Remark 3.6 Since f in previous lemma can be any smooth function on .M, there are so many smooth vector fields whose flows leave .σ invariant. Definition 3.11 A system of coordinates∑.(x1 , . . . , xn , ξ1 , . . . ξn ) in M is called a canonical system of coordinates if .σ = ni=1 dξi ∧ dxi . In a canonical coordinate system, .x˙i and .ξ˙i denote the coordinates of the vector field .v = Hf = (x˙1 , . . . , x˙n ; ξ˙1 , . . . , ξ˙n ). v = Hf =

n ∑

.

x˙i

i=1

∂ ∂ + ξ˙i . ∂xi ∂ξi

Therefore we have iv σ = i v (

∑

.

=

∑

dξi ∧ dxi )

i

iv (dξi ∧ dxi )

i

=

∑ (iv dξi )dxi − (iv dxi )dξi i

∑ = (dξi )(v)dxi − (dxi )(v)dξi i

=

∑

ξ˙i dxi − x˙i dξi ;

i

on the other hand .−df = −

∑

∂f i ∂xi dxi

⎧ .

+

∂f ∂ξi dξi .

(x,ξ ) x˙i = ∂f ∂ξ i ξ˙i = − ∂f (x,ξ )

Now from .iv σ = −df we obtain

.

∂xi

This system of ordinary differential equations is the Hamiltonian system defined by the function f in coordinates .xi and .ξi . d If . dt m(t) = v(m(t)), m ∈ M, is the differential equation for the flow defined by the vector field .v = Hf , then in canonical local coordinates we get the above system of ordinary differential equations, (x, ˙ ξ˙ ) = Hf

.

86

3 Hamiltonian Systems

d d xi (t) and . dt in which we replace .x˙i and .ξ˙i by . dt ξi (t), and in the right-hand side we take the partial derivative of f at .xi = xi (t), ξi = ξi (t).

Example 3.1 Let X be an n-dimensional smooth manifold and .v ∈ χ (X). The momentum function of the vector field v in the base manifold X is the function .μv on the cotangent bundle .M := T ∗ X, defined by .

μv : T ∗ X −→ R μv (x, ξ ) = ξ(v(x)),

x ∈ X, ξ ∈ (Tx X)∗ .

T ∗ X is a symplectic manifold with canonical symplectic form .σ , and .μv is a function on it so there is the unique Hamiltonian vector field .Hμv correspondence to this function such that .iHμv σ = −dμv . The symplectic form .σ is invariant under the flow of .Hμv , i.e., .(etHμv )∗ σ = σ, because by Cartan formula we have

.

LHμv σ = d(iHμv σ ) + iHμv (dσ ) = d(−dμv ) + 0 = 0.

.

Also .μv satisfies the Hamiltonian system so we have .x˙ = .μv this turns to x˙ =

.

∂μv (x,ξ ) . ∂ξ

By definition of

∂ξ(v(x)) = v(x) ∈ Tx X = (Tx∗ X)∗ . ∂ξ

Let Y be a smooth manifold and let .ϕ : X → Y be a local diffeomorphism. As we saw in Chap. 2, the induced transformation .Ф : T ∗ X → T ∗ Y is defined by Ф(x, ξ ) := (ϕ(x), ((Tx ϕ)∗ )−1 (ξ )); x ∈ X, ξ ∈ Tx∗ X.

.

In this example we have .ϕ = etv : X → X and Ф = etHμv : T ∗ X −→ T ∗ X

.

etHμv (x, ξ ) = (etv (x), ((Tx etv )∗ )−1 (ξ )). Hence, the flow in .T ∗ X of the Hamiltonian system defined by the function .μv is equal to the flow in .T ∗ X, which is induced by the flow in X of the vector field .v. Therefore the projection .π : T ∗ X −→ X; (x, ξ ) − ‫ → ׀‬x intertwines the .Hμv flow in .T ∗ X with the v-flow in .X, i.e., .π ◦ etHμv = etv ◦ π. In other words the following diagram commutes.

3.3 The Legendre Transform

87

Exercise 3.2 Remember in example 3.1, we defined the momentum function by .

μv : T ∗ X −→ R x ∈ X, ξ ∈ (Tx X)∗ .

μv (x, ξ ) = ξ(v(x)), Prove that .{μv , μw } = μ[v,w] .

3.3 The Legendre Transform In the previous section we saw that having a symplectic form invariant under the flow of a vector field, we get a Hamiltonian system. In this section, we explain how a Hamiltonian system is transformed to Euler–Lagrange equation, by a mapping that is called the Legendre transform. Consider a smooth real-valued function L on an open subset U of the tangent bundle T X of an n-dimensional smooth manifold .X, i.e., L : U −→ R

.

(x, v) ‫׀‬−→ L(x, v), x ∈ X, v ∈ Tx X. Let .γ : [a, b] → X be a smooth curve such that .(γ (t), γ ' (t)) ∈ U for all .t ∈ [a, b], and define  I (γ ) :=

.

b

L(γ (t), γ ' (t))dt (∗).

a

Lemma 3.7 If .γ = γε depends smoothly on a parameter .ε, then the variational formula of Euler and Lagrange is dI (γε ) =− . dε



b

[L](t)δ(t)dt + μ(γ (b), γ ' (b))δ(b) − μ(γ (a), γ ' (a))δ(a) (∗∗),

a

where: ε (t) (1) .δ(t) := ∂γ∂ε ∈ Tγ (t) X denotes the “variation with respect to .ε” of the curve .γε (t),. (2) .[L](t) is the linear form on .Tγ (t) X

[L](t) =

.

∂L d ∂L ( )− , dt ∂v ∂x

which in local coordinates is given by

88

3 Hamiltonian Systems

[L](t) =

.

∑ [L]i dxi ,

[L]i :=

dμi (γ (t), γ ' (t)) ∂L(x, γ ' (t)) − |x=γ (t) . dt ∂xi

(3) The linear form .μ(x, v) = μL (x, v) on .Tx X is defined by μ(x, v) :=

.

∂L(x, v) ∈ Tx∗ X. ∂v

This linear form .μ(x, v) is called the momentum vector assigned to the velocity vector v by means of the function .L. Proof We calculate the derivative of .I (γ ) in .(∗) with respect to .ε. .

dI (γε ) dε

 = 

∂L ∂γε' (t) ∂L ∂γε (t) . + . dt ∂γε (t) ∂ε ∂γε' (t) ∂ε  b ∂L ∂γε (t) ∂L ∂ 2 γε (t) . dt + . dt ∂x ∂ε ∂ε∂t a ∂v  b ∂L ∂γε (t) b ∂L ∂γε (t) d ∂L ∂γε (t) . dt + . |a − ( ) dt ∂x ∂ε ∂v ∂ε ∂ε a dt ∂v

b a b

= 

b a

= 

d L(γε (t), γε' (t))dt dε

a

= 

b

a b

=

( a

d ∂L ∂γε (t) ∂L − ( )) dt + μ(γ (b), γ ' (b))δ(b) ∂x dt ∂v ∂ε

− μ(γ (a), γ ' (a))δ(a)  b [L](t)δ(t)dt + μ(γ (b), γ ' (b))δ(b) − μ(γ (a), γ ' (a))δ(a). = − a

Here in the fourth equality, we used integral by part (choose: .f = ∂ 2 γε (t) ∂ε∂t

∂L ∂v

and .gdg = ⨆ ⨅

dt).

Definition 3.12 The velocity–to–momentum mapping is the mapping defined by Ф = ФL : U −→ T ∗ X

.

(x, v) ‫׀‬−→ (x, μ(x, v)). In linear coordinates in .Tx X, the bi-linear form . ∂ .

∂ 2 L(x,v) ∂vi ∂vj

2 L(x,v)

∂v 2

has symmetric matrix

for .1 ≤ i, j ≤ n, which is the Hessian of the function .v ‫ →׀‬L(x, v).

Definition 3.13 The Legendre’s condition is that “the symmetric bi-linear form 2 ∂μ(u,v) = ∂ L(x,v) on .Tx X is non-degenerate.” ∂v ∂v 2

.

3.3 The Legendre Transform

89

Lemma 3.8 The velocity-to-momentum mapping .Ф is a local diffeomorphism if and only if the Legendre’s condition holds. Proof .Ф is a local diffeomorphism .⇐⇒ the tangent map of .Ф is invertible, i.e., ( TФ =

.

I

0

)

∂μ(x,v) ∂μ(x,v) ∂x ∂v

is invertible .⇐⇒ the determinant of the triangular matrix .T Ф is non-zero .⇐⇒ ∂μ(x,v) /= 0 ⇐⇒ Legendre’s condition. ⨆ ⨅ ∂v

.

Definition 3.14 For the integral I in (.∗), a curve .γ (t) is called a stationary curve if it satisfies the Euler–Lagrange equations .[L](t) ≡ 0, and in other words, if the integral in (.∗∗) vanishes for any variation .δ(t) of .γ (t). Remark 3.7 If the Legendre condition holds, then for the stationary curve .γ (t), the Euler–Lagrange equations, .[L](t) ≡ 0, can be written in local coordinates as a second-order system of ordinary differential equations .

d 2 γi (t) = ai (γ (t), γ ' (t)) 1 ≤ i ≤ n, dt 2

where the components of the acceleration .ai (x, v) are smooth functions of x and .v. In fact, we can consider this second-order system as a first-order system x ' (t) = v(t),

.

v ' (t) = a(x(t), v(t)),

which is defined by the vector field .(v, a(x, v)) in the tangent bundle. Definition 3.15 Define the function H on .U ⊂ T X by .H : U −→ R H (x, v) := − L(x, v),

.

where . = (μ(x, v))(v). Then, the function .h := H ◦ Ф−1 defined on the open subset V of the cotangent bundle .T ∗ X of X is called the Legendre transform of the function .L. In previous lemma we saw that under the Legendre’s condition, the velocity-tomomentum mapping .Ф is a local diffeomorphism. By definition of the mapping .Ф, we have h(x, μ(x, v)) = H ◦ Ф−1 (x, μ(x, v))

.

= H (x, v) = − L(x, v).

90

3 Hamiltonian Systems

We denote .v = v(x, ξ ) as the solution of the equation .μ(x, v) = ξ. Then for (x, ξ ) ∈ V ⊂ T ∗ X, we get

.

.

h : V −→ R h(x, ξ ) = − L(x, v(x, ξ )).

Taking partial derivative of h with respect to .ξi ’s, we obtain .

) ∂ ∂ ∂ ( h(x, ξ ) = − L x, v(x, ξ ) ∂ξi ∂ξi ∂ξi ∂ ∂ v(x, ξ ), ξ > + ∂ξi ∂ξi ( ) ( ∂L(x, v(x, ξ )) ∂ )) ∂L x, v(x, ξ ) ( ∂ ( x) + v(x, ξ ) − ∂x ∂ξi ∂v ∂ξi ( ) ∂ ∂ =< v(x, ξ ), ξ > + vi (x, ξ ) − 0 − < v(x, ξ ), μ x, v(x, ξ ) > ∂ξi ∂ξi =
+ ∂xi ∂xi ( ) ( ∂L(x, v(x, ξ )) ∂ )) ∂L x, v(x, ξ ) ( ∂ ( − x) + v(x, ξ ) ∂x ∂xi ∂v ∂xi ( ) ( ) ∂L x, v(x, ξ ) ∂ ∂ v(x, ξ ), ξ > + 0 − −< v(x, ξ ), μ x, v(x, ξ ) > =< ∂xi ∂xi ∂xi ( ) ∂L x, v(x, ξ ) =− , ∂xi =
−V ∂v = 0−V

= .

In canonical local coordinates, the symplectic form is written as σ(

.

∂ ∂ ∂ ∂ ∂ ∂ , ) = 0, σ ( , ) = 0, σ ( , ) = δij . ∂xi ∂xj ∂ξi ∂ξj ∂xi ∂ξj

∑ Consider a vector .v ∈ T M as .v= x˙i ∂x∂ i + ξ˙i ∂ξ∂ i or as .v=(x˙1 , . . . , x˙n , ξ˙1 , . . . , ξ˙n ). Therefore Hamiltonian vector fields .Hf and .Hg are written by Hf =

∑

.

f

∂ f ∂ + ξ˙i , ∂xi ∂ξi

g

∂ g ∂ + ξ˙i . ∂xi ∂ξi

x˙i

and Hg =

.

∑

x˙i

96

3 Hamiltonian Systems

Hence, the Poisson brackets of the functions f and g in canonical local coordinates are ∑ f g f g ξ˙i x˙i − x˙i ξ˙i . .{f, g}(x, ξ ) = σ (Hf , Hg ) = From Hamiltonian systems defined by .Hf and .Hg , we have f

∂f (x, ξ ) ∂ξi

f ξ˙i = −

g

∂g(x, ξ ) ∂ξi

g ξ˙i = −

x˙i =

.

∂f (x, ξ ) ∂xi

and x˙i =

.

∂g(x, ξ ) . ∂xi

Therefore, the Poisson brackets of the functions f and g in canonical local coordinates are written as {f, g}(x, ξ ) =

.

n ∑ ∂f (x, ξ ) ∂g(x, ξ ) ∂f (x, ξ ) ∂g(x, ξ ) − ). ( ∂ξi ∂xi ∂xi ∂ξi i=1

Definition 3.18 A function g is a constant of motion for the Hamiltonian system defined by the function f if g is invariant under the .Hf -flow. In applications this condition is called Noether’s principle for Hamiltonian systems [45]. Lemma 3.9 The following conditions are equivalent: (a) Function g is a constant of motion for the Hamiltonian system defined by the function f . (b) .{f, g} = 0. (c) .{g, f } = 0. (d) Function f is a constant of motion for the Hamiltonian system defined by the function .g. Proof (b) .⇐⇒ {f, g} = 0 ⇐⇒ LHf g = 0 ⇐⇒ (etHf )∗ g = g ∀t ⇐⇒ (a). Here we used the Lemma that indicates that .LHf g = 0 is equivalent to say that the 0-form g is invariant under the flow of the vector .Hf . Moreover, (d) .⇐⇒ (c) .⇐⇒ (b). ⨆ ⨅ Remark 3.11 Usually, in examples, M is considered as the cotangent bundle .T ∗ X and .g = μv , the momentum function of a smooth vector field v in the base manifold .X. Remark 3.12 (1) From anti-symmetric property of Poisson brackets, it follows that .{f, f } = 0, which means that the function f is a constant of motion for the Hamiltonian system defined by the function .f.

3.5 Poisson Brackets

97

(2) If we consider f is equal to the total energy of a classical mechanical system as in previous section, then .{f, f } = 0 is the law of conservation of the total energy, i.e., .(etHF )∗ F = F. As we mentioned above, the Poisson bracket of two functions is also a function. In the following lemma, we calculate the differential of Poisson brackets. Lemma 3.10 Let f and g be two smooth functions on the symplectic manifold (M, σ ). Then we have

.

d{f, g} = −i[Hf ,Hg ] σ.

.

Proof We start from left-hand side, and we use the definition of Poisson brackets. So we get d{f, g} = dLHf g

.

= LHf dg = LHf (−iHg σ ) = −(i[Hf ,Hg ] σ + iHg (LHf σ )) = −(i[Hf ,Hg ] σ + 0) = −i[Hf ,Hg ] σ. Here in the second equality, we used Cartan formula, and in the third equality, we used .iHg σ = −dg. The fourth equality is by Lemma 3.4, and for the 5th equality, we used Cartan formula, i.e., .LHf σ = iHf dσ + diHf σ = 0 + d(−df ) = 0. ⨆ ⨅ Remark 3.13 The Hamiltonian equations for the function .{f, g} are iH{f,g} σ = −d{f, g}.

.

Hence, by the above lemma, we obtain .iH{f,g} σ = i[Hf ,Hg ] σ, and therefore σ (H{f,g} , −) = σ ([Hf , Hg ], −). Since .σ is isomorphism, we get

.

[Hf , Hg ] = H{f,g} ,

.

which means that the Lie brackets of the Hamiltonian vector fields of the functions f and g are again a Hamiltonian vector field (of the Poisson brackets of f and g). This implies that the set of Hamiltonian vector fields is a Lie algebra. Lemma 3.11 Let .f, g and h be smooth functions on .(M, σ ); then the Jacobi identity for Poisson brackets is hold as {{f, g}, h} + {{g, h}, f } + {{h, f }, g} = 0.

.

98

3 Hamiltonian Systems

Proof Using the definition of Poisson brackets several times, we obtain {{f, g}, h} = H{f,g} .h

.

= [Hf , Hg ].h = Hf .(Hg .h) − Hg .(Hf .h) = Hf .{g, h} − Hg .{f, h} = {f, {g, h}} − {g, {f, h}}, and now we use the anti-symmetric property of the Poisson brackets.

⨆ ⨅

The Jacobi identity for the Poisson brackets was mentioned in his article [31] where he conclude the theorem of Poisson that states that if g and h are constant of motion for the Hamiltonian system defined by the function .f, then .{g, h} is also a constant of motion for the Hamiltonian system defined by .f. Theorem 3.1 The space .F(M) of smooth functions on .M, is a Lie algebra with the Poisson brackets. Moreover, the mapping that assigns to a smooth function its Hamiltonian vector field is a homomorphism of Lie algebras from .F(M) to the Lie algebra .χ (M) of smooth vector fields on .M. The kernel of this homomorphism is equal to the space of functions that are constant on the connected components of .M. Proof In the above we saw that the Poisson bracket satisfies Lie bracket properties, i.e., it is anti-symmetric and it satisfies the Jacobi identity. Therefore the space .F(M) equipped with Poisson bracket is a Lie algebra. By the above lemmas, the mapping ψ : F(M) −→ χ (M)

.

f ‫׀‬−→ Hf {f, g} ‫׀‬−→ H{f,g} = [Hf , Hg ] is a Lie algebra homomorphism. If f is in the kernel of this mapping, then we have Hf = 0 ⇐⇒ σ (Hf , −) = 0 ⇐⇒ iHf σ = 0 ⇐⇒ −df = 0 ⇐⇒ f is constant .

.

⨆ ⨅

3.6 Darboux’s Theorem Let .(M, σ ) be a 2n-dimensional symplectic manifold. The Darboux’s theorem ∑ states that locally .σ can be considered as the standard symplectic form .σst = ni=1 dξi ∧

3.6 Darboux’s Theorem

99

dxi on .R2n . We studied this theorem in Chap. 2, Sect. 2.8, using local theorems. Here we explain the original proof given by Darboux. Theorem 3.2 (Darboux [11]) For every .m0 ∈ M there exists an open neighborhood U of .m0 in M and a diffeomorphism .Ф from U onto an open subset V of n n ∗ .R × R such that .Ф σst = σ or Ф∗ (

n ∑

.

dξi ∧ dxi ) = σ on U.

i=1

Proof Let .ϕ be a smooth function in a neighborhood U of .m0 such that .ϕ(m0 ) = 0 and .dϕm0 /= 0. We choose any co-dimension one smooth submanifold S of M containing .m0 such that .Hϕ (m0 ) ∈ / Tm0 S. Then there is a locally unique smooth function f on a neighborhood of .m0 such that .{ϕ, f } = LHϕ f = 1 and .f = 0 on .S. (Because if g is another smooth function with the properties of f , then we have .σ (Hϕ , Hf ) = σ (Hϕ , Hg ) = 1, since .σ is isomorphism, we get .Hf = Hg or .iHf σ = iHg σ , which locally gives that .−df = −dg, so .f = g + c. Since .f = g = 0 on S, so we have the constant c is zero, and therefore .f = g.) So we have .σ (Hϕ , Hf ) = 1. Therefore, at every point, the two vectors .Hϕ and .Hf are linearly independent, and so .dϕ and df are linearly independent at every point. Therefore, N := {m ∈ U | f (m) = ϕ(m) = 0}

.

is a smooth co-dimension two submanifold of the open neighborhood U of .m0 on which .ϕ and f are defined. In other words, for .x ∈ N, we have Tx N = ker(dϕx ) ∩ ker(dfx )

.

or the symplectic orthogonal complement of .Tx N is generated by the vectors .Hϕ (x) and .Hf (x), i.e., (Tx N)σ = ,

.

because .ker(dϕ) = ker(iHϕ ) = ker(σ (Hϕ , −)) = {v ∈ Tx N | σ (Hϕ , v) = 0}, which means that .Hϕ is in .(Tx N)σ . Similarly, .Hf is in .(Tx N)σ . Hence, the restriction .σN = σ |N of .σ to N is a symplectic form. We have .iH{ϕ,f }=1 σ = −d{ϕ, f } = 0 or .σ (H{ϕ,f } , −) = 0. Since .σ is isomorphism, we get .H{ϕ,f }=1 = 0. From .[Hϕ , Hf ] = H{ϕ,f }=1 = 0, we conclude that .Hϕ and .Hf commute. So their flows commute. For .x ∈ N and .t, τ ∈ R, we define the mapping Ф : N × R × R −→ U ⊂ M

.

Ф(x, t, τ ) := etHϕ ◦ e−τ Hf (x) = e−τ Hf ◦ etHϕ (x).

.

100

3 Hamiltonian Systems

Taking derivative of .Ф with respect to t and .τ , we get .

∂Ф(x, t, τ ) ∂Ф(x, t, τ ) = Hϕ (Ф(x, t, τ )) and = −Hf (Ф(x, t, τ )). (3.1) ∂t ∂τ

The pullback of the symplectic form .σ by the mapping .Ф over the pair of vectors (x, ˙ t˙, τ˙ ), (x˙' , t˙' , τ˙' ) ∈ T(x,t,τ ) (N × R × R) is obtained as follows:

.

.

(Ф∗ σ )(x,t,τ ) ((x, ˙ t˙, τ˙ ), (x˙' , t˙' , τ˙' )) = σФ(x,t,τ ) ((T(x,t,τ ) (etHϕ ◦ e−τ Hf )(x, ˙ t˙, τ˙ ), T(x,t,τ ) (etHϕ ◦ e−τ Hf )(x˙' , t˙' , τ˙' )) = σФ(x,t,τ ) (((Tx (etHϕ ◦ e−τ Hf )x) ˙ + t˙Hϕ − τ˙ Hf ), (Tx (etHϕ ◦ e−τ Hf ))x˙' ) + t˙' Hϕ − τ˙' Hf ) = σФ(x,t,τ ) (Tx (etHϕ ◦ e−τ Hf )x, ˙ Tx (etHϕ ◦ e−τ Hf )x˙' ) +σФ(x,t,τ ) (Tx (etHϕ ◦ e−τ Hf )x, ˙ t˙' Hϕ − τ˙' Hf ) +σФ(x,t,τ ) (t˙Hϕ − τ˙ Hf , Tx (etHϕ ◦ e−τ Hf ))x˙' ) +σФ(x,t,τ ) (t˙Hϕ − τ˙ Hf , t˙' Hϕ − τ˙' Hf ) = σx (x, ˙ x˙' ) + 0 + 0 + τ˙ t˙' − τ˙' t˙.

Here in the second equality, we used (3.1) in the third equality, and we used bilinearity of .σ. In the last equality, in the first part, we used the fact that Hamiltonian flows preserve the symplectic form, i.e., .(etHϕ ◦ e−τ Hf )∗ σ = σ , or equivalently, σФ(x,t,τ ) (Tx (etHϕ ◦ e−τ Hf )x, ˙ Tx (etHϕ ◦ e−τ Hf )x˙' ) = σx (x, ˙ x˙' ).

.

In the second and third parts of the last equality, we used the fact that .Hϕ and .Hf are in the symplectic orthogonal complement of .Tx N, and in the last part we used .σ (Hϕ , Hf ) = 1 and anti-symmetric property of .σ. Therefore we have obtained (Ф∗ σ )(x,t,τ ) = σx (x, ˙ x˙ ' ) + τ˙ t˙' − τ˙' t˙.

.

This shows .Ф∗ σ is equal to the direct sum of .σN and .σst (the standard symplectic form in .R2 ). The proof now follows by induction on .n, i.e., for .n = 1, we have twodimensional case and the same as above argument we find two linear independent vectors to obtain standard symplectic form on .R × R. Then we assume the above argument for the step .n − 1, i.e., for .2n − 2 dimension. Then we get .Nn−1 × R × R × · · · × R, where by .Nn−1 we mean a two-dimensional submanifold, and it is with the product of .2n − 2 copies of .R. Using the first step of the argument for .Nn−1 , we get standard symplectic form for .Nn−1 and therefore for the whole .U. ⨆ ⨅ Remark 3.14 The above proof is the classical proof given by Darboux in [11] (see also [3], p.230 and [50]) using induction on the dimension. There is another proof

3.7 Poisson Structures

101

of the Darboux Theorem given by Weinstein [52] (see also [25, 37]) that is based on deformation argument that has been introduced in normal form theory by Moser [42]. We studied this proof in Sect. 2.8. Exercise 3.4 Let .(M, σ ) be a 2n-dimensional symplectic manifold and .Ф : M −→ smooth mapping with coordinate functions .x1 , . . . , xn , ξ1 , . . . , ξn . Show R2n be a∑ that .Ф∗ ( ni=1 dξi ∧ dxi ) = σ, if and only if .{xi , xj } = 0, {ξi , xj } = δij , and .{ξi , ξj } = 0.

3.7 Poisson Structures Below, using symplectic form and Poisson brackets defined by it, we explain how to generalize Poisson structures as bi-vectors on a manifold. The general concept of Poisson structures was invented by Lichnerowicz [35]. Consider the symplectic form .σ on .M. Then we have the isomorphisms σm : Tm M −→ (Tm M)∗

.

σm−1 : (Tm M)∗ −→ (Tm M) ∼ = ((Tm M)∗ )∗ . Let f and g be smooth functions on .M. Then we can write their Poisson brackets in terms of .σ −1 as follows: {f, g}(m) = LHf g(m)

.

= dg(Hf )(m) = dg(−σm−1 (df ))(m) = −(σm−1 (df ))(dg)(m) = −σm−1 (df (m), dg(m)). Here .πm := σm−1 : (Tm M)∗ −→ Tm M is regarded as an anti-symmetric bi-linear form on .(Tm M)∗ or as an element of .Λ2 Tm M, which is also called a bi-vector in .Tm M. Moreover, in the third equality, we used .iHf σ = −df so we have −1 (df ). In the fourth equality .dg ∈ T ∗ M and .σ (Hf , −) = −df or .Hf = −σ −1 .σ (df ) ∈ T M, and in the fifth equality we consider .σ −1 (df ) ∈ (T ∗ M)∗ . Definition 3.19 Let M be a smooth manifold. A Poisson structure on M is defined as a smooth bi-vector field .πm ∈ Λ2 Tm M, m ∈ M, such that the corresponding Poisson brackets .{f, g}, defined by {f, g}(m) := πm (df (m), dg(m)), m ∈ M,

.

satisfy the Jacobi identity.

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If we consider .πm as a linear mapping from .(Tm M)∗ to .Tm M, then we can define the Hamiltonian vector field .Hf of the function f by .Hf (m) := πm df (m). So, we have {f, g}(m) = πm (df (m), dg(m))

.

= πm (df (m))(dg(m)) = LHf g. Remark 3.15 (1) When .πm is surjective, we conclude that it is bijective, since .Tm M and .(Tm M)∗ have the same dimension. Then, for a symplectic form on M, we have .πm = −σm−1 . This shows that we only obtain new examples of Poisson structures when .πm is not surjective. (2) Consider .Hm = πm ((Tm M)∗ ). When the rank of .πm , i.e., the dimension of .Hm , is constant (as a function of .m ∈ M), .Hm , m ∈ M, define a smooth vector subbundle .H of .T M. Let .πm df (m) and .πm dg(m) be vectors in .Hm = πm ((Tm M)∗ ). Then we obtain [πm df (m), πm dg(m)] = [Hf (m), Hg (m)]

.

= H{f,g} (m) = πm d{f, g}(m) ∈ Hm , which means that .H is integrable. (3) If I is an integral manifold of .H, then restricting .{f, g} to I only depends on the restrictions .f |I and .g |I . So we obtain Poisson structure on .I, which is actually defined by a symplectic structure on .I. Hence, our Poisson manifold .(M, π ) can be considered as a manifold that is foliated by symplectic leaves I ’s, in such a way that the Poisson brackets are defined by the Poisson brackets of the functions restricted to the symplectic leaves.

3.8 Hamiltonian Group Actions In this section we consider the case when a Lie group G acts on a symplectic manifold. In particular, we consider the infinitesimal action and explain Hamiltonian action that is defined using momentum mapping. As an example, we study coadjoint action. Let G be a Lie group acting on the smooth manifold M as G × M −→ M

.

(g, m) ‫׀‬−→ g.m.

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103

For each .g ∈ G, we consider the diffeomorphism .gM : M −→ M; m − ‫ → ׀‬g.m. Let .Te G = g be the Lie algebra of the Lie group .G, where e denotes the identity element of .G. Then the exponential map is the mapping .exp : g −→ G. Recall that an action of Lie group G on M is a group homomorphism ϕ : G −→ Diff (M)

.

g− ‫ → ׀‬ϕg , where usually we use the evaluation mapping G × M −→ M

.

(g, m) ‫׀‬−→ ϕg (m) := g.m. If this mapping is smooth, then the action is smooth. For .G = R, we call .{ϕt } a 1parameter group of diffeomorphism, and there is .1 − 1 correspondence as follows: { complete vector fields on M } ←→ { smooth actions of R on M}

.

Xp :=

d |t=0 ϕt (p) ←− ϕ dt X −→ exp(tX).

Definition 3.20 Let .(M, ω) be a symplectic manifold. An action .ϕ : G −→ Diff (M) is a symplectic action if ϕ : G −→ Symp(M) ⊆ Diff (M),

.

where .Symp(M) is the subgroup of diffeomorphisms preserving .ω. ∑ Example 3.2 Let .M = R2n with .σst = ni=1 dxi ∧ dyi . Consider the action of .R on M by translation in .y1 direction, i.e., ϕ : R × M −→ M

.

(t, (x1 , y1 , x2 , y2 , . . . , xn , yn )) − ‫( → ׀‬x1 , y1 + t, x2 , y2 , . . . , xn , yn ). It is easy to check that for any .t ∈ R, we have .ϕt∗ σst = σst , that is, we have a symplectic action. To construct symplectic group actions, we can use functions. Consider function f ∈ C ∞ (M), and the one-form .df ∈ Ω1 (M). In Sect. 3.2 we saw that (by the non-degeneracy of the symplectic form) there is a unique vector field .Xf (called Hamiltonian vector field) such that .iXf ω = −df. Also, we saw that the symplectic form is invariant under the flow of this vector field. In other words if we integrate .Xf to a 1-parameter family .{ρt } of diffeomorphism .ρt : M −→ M such that .ρ0 = idM

.

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d and . dt |t=0 ρt (p) = Xf (p), then we have .ρt∗ ω = ω for all .t ∈ R. Therefore, .{ρt } is a symplectic .R-action. Above, we obtained a symplectic .R-action using a Hamiltonian function. Now, considering the case when G is not equal to .R, we are going to see a generalization of the above idea. This is actually a motivation to define Hamiltonian actions. In fact the action of a group G on .(M, ω) is called Hamiltonian if it is symplectic, and there exists a momentum mapping .μ : M −→ g∗ for it. The momentum mapping plays the role of Hamiltonian function f in above.

Definition 3.21 For each element X in the Lie algebra .g of G, the infinitesimal action of X on M is defined by XM :=

.

d (exp(tX))M |t=0 , dt

which is a smooth vector field on .M. Lemma 3.12 If .etXM denotes the flow of the vector field .XM , then .(exp(tX))M = etXM for every .t ∈ R. Proof Set .exp(tX).m := γ (t). For the curve .γ (t), we have: • .γ (0) = exp(0).m = e.m therefore γ (0) = m. d |t=0 exp(tX).m = XM (m), and • .γ ' (0) = dt

. .

d exp(tX).m dt d = |s=0 exp((t + s)X).m ds d = |s=0 exp(sX).(exp(tX).m) ds = XM (γ (t)).

γ ' (t) =

.

This shows that .γ (t) is an integral curve of .XM with the initial point .m, so etXM = (exp(tX))M , for every .t ∈ R. ⨆ ⨅

.

Lemma 3.13 We have .[X, Y ]M = −[XM , YM ], where .X, Y ∈ g. Proof We start from the left-hand side. d (exp(t[X, Y ]))M |t=0 dt d d = (exp(t ( (esX )∗ Y |s=0 )))M |t=0 dt ds

[X, Y ]M =

.

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105

d sX ∗ d (e ) Y |s=0 )(exp(t ( (esX )∗ Y |s=0 ))M |t=0 ds ds d = ( (esX )∗ Y |s=0 .e)M ds d d −sX rY = ◦ e ◦ esX |r=0 )M |s=0 ( e ds dr d d = ( (−exp(sX)exp(rY )exp(sX) |r=0 )M |s=0 ds dr d d = ( (−(exp(sX))M ◦ (exp(rY ))M ◦ (exp(sX))M |r=0 ) |s=0 ds dr d d = − ( esXM ◦ erYM ◦ etXM |r=0 ) |s=0 ds dr = −[XM , YM ]. = ((

Here, in the first equality, we used definition of infinitesimal action of .[X, Y ] on M. In the second equality we used definition of the Lie bracket, then in the third equality we took derivative with respect to t, and in the fourth equality we evaluate at .t = 0. In the fifth equality we used definition of pullback of vector .Y. In the eighth equality we used previous Lemma. In the last equality we used definition of Lie brackets (and pullback). ⨆ ⨅

.

Corollary 3.1 The mapping .g −→ χ (M); X ‫׀‬−→ XM from the Lie algebra .g to the Lie algebra .χ (M) is an anti-homomorphism. Definition 3.22 Let .σ be a symplectic form on .M. The action of the Lie group G on M is called Hamiltonian action with respect to .σ, if for every .X ∈ g there exists a smooth function .μ(X) = : M −→ R such that: (1) .XM = H . (2) . depends linearly on .X ∈ g. (3) .{, } = − for .X, Y ∈ g. In other words .μ : g −→ F(M); X ‫׀‬−→ is an anti-homomorphism of Lie algebras from .g to the Poisson Lie algebra .F(M). Remark 3.16 The condition (1) of the above definition means that the infinitesimal actions are Hamiltonian that implies that the one-parameter subgroups, i.e., .etXM , preserve the symplectic form .σ for all .t. Hence, if G is connected, it follows that the G-action leaves the symplectic form invariant. In other words the action is a homomorphism from G to the group of canonical transformations in .(M, σ ). Definition 3.23 For every .m ∈ M, the mapping .μ(m) : g −→ R; μ(m) : X ‫׀‬−→ (m) is a linear form on .g and defines a smooth mapping μ : M −→ g∗ ; m ‫׀‬−→ μ(m),

.

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which is called the momentum mapping of the Hamiltonian action of G on .M, [1]. Remark 3.17 Note that this notation leads to . = (m) for every m ∈ M.

.

In the following we discuss the co-adjoint action as an example of Hamiltonian action. Consider left translation and right translation on the Lie group G defined by the mapping cx = lx ◦ rx−1 : G −→ G

.

y ‫׀‬−→ xyx −1 . Then, the mapping Ad is defined by .Ad(x) := Te cx : Te G −→ Te G. Hence, Ad(x) ∈ GL(Te G), and we have the following mapping:

.

Ad : G −→ GL(Te G)

.

x ‫׀‬−→ Ad(x). The adjoint action of G on .g is defined as .

G × g −→ g (g, X) ‫׀‬−→ g.X = (Ad(g))(X).

From .Ad(g) : g −→ g, we have .Ad ∗ (g) : g∗ −→ g∗ . The co-adjoint action of G on .g∗ , the dual of the Lie algebra .g, is defined by G × g∗ −→ g∗

.

(g, ξ ) ‫׀‬−→ g.ξ = ((Ad(g))∗ )−1 (ξ ) ξ ∈ g∗ . Lemma 3.14 For the co-adjoint action of G on .g∗ , and for every .X ∈ g, the infinitesimal co-adjoint action of .X ∈ g is given by the linear mapping Xg∗ = −(adX)∗ : g∗ −→ g∗ .

.

Proof Consider the mapping .Ad(g) : Te G −→ Te G, so we have .Ad : G −→ GL(Te G). Consider .ad := Te (Ad) : Te G −→ End(Te G). For .X ∈ g and .ξ ∈ g∗ , we obtain d (exp(tX))g∗ |t=0 dt d = exp(tX).ξ |t=0 dt

Xg∗ =

.

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107

d (Ad ∗ (exp(tX)))−1 (ξ ) |t=0 dt = −(adX)∗ . =

Here, in the third equality, we used the definition of co-adjoint action of G on .g∗ . The last equality is obtained by chain rule and definition of .ad. ⨅ ⨆ Lemma 3.15 For .ξ ∈ g∗ and .X, Y ∈ g, we have . = −. Proof = (Xg∗ (ξ ))(Y )

.

= ((−(adX)∗ )(ξ ))(Y ) = −ξ((ad(X))Y ) = −ξ([X, Y ]) = −, where in the third equality we used the definition of pullback, and in the fourth equality, we used the fact that .(ad(X))Y = [X, Y ] from Lie theory. ⨆ ⨅ Lemma 3.16 If in previous lemma we substitute .ξ = μ(m), then .LXM μ = Xg∗ μ. Proof Remember that we have the mappings μ : M −→ g∗

.

m ‫׀‬−→ μ(m) and μ(m) : g −→ R

.

X ‫׀‬−→ (m). By previous lemma, we have = −

.

= −(m) = {, }(m) = LH (m) = LXM (m).

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Here in the second equality, we used remark, and in the third equality, we used part (3) of the definition of Hamiltonian action. In the fourth equality, we used the definition of Poisson brackets. In the last equality, we used part (1) of the definition of Hamiltonian action. Therefore, we showed that . = LXM . So .(Xg∗ μ)(Y ) = LXM μ(Y ). Thus .Xg∗ μ = LXM μ. ⨆ ⨅ Remark 3.18 (1) The above lemma, i.e., .LXM μ = Xg∗ μ, shows that the momentum mapping .μ : M −→ g∗ intertwines the infinitesimal action of .g on M with the infinitesimal co-adjoint action of .g on .g∗ . Therefore, we have the following commutative diagram:

(2) If G is connected, then this implies that the momentum mapping intertwines ∗ μ = the action of G on M with co-adjoint action of G on .g∗ , i.e., .gM ∗ −1 ((Ad(g)) ) μ, g ∈ G, because ((Ad(g))∗ )−1 (μ(m)) = ((Ad(g))−1 )∗ (μ(m))

.

= μ(g.m) = μ(gM (m)) ∗ = gM μ(m).

In other words, the following diagram commutes.

(3) As an example, consider the Poisson structure in .g∗ defined by πξ (X, Y ) = −,

.

ξ ∈ g∗ , X, Y ∈ (g∗ )∗ = g.

The symplectic leaves are the co-adjoint orbits in .g∗ . Part (3) of the definition of Hamiltonian action shows that for a Hamiltonian action of G on the symplectic manifold .(M, σ ), the momentum mapping .μ

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109

intertwines the Poisson structure on .(M, σ ) with the Poisson structure on the dual of the Lie algebra of .G. Exercise 3.5 Show that the translation action of .R on .(R2n , σst ) is a symplectic action. Exercise 3.6 Let .M = C with the symplectic form .ω = 2i dz ∧ d z¯ . Consider the group G as the circle .G = S 1 = {λ ∈ C; | λ |= 1} acting on M by complex multiplication, i.e., ϕ : S 1 × C −→ C

.

(λ, z) ‫׀‬−→ (λ.z = λz). Show that this is a symplectic action, and in other words show that .ϕλ∗ ω = ω.

3.9 Symplectic Quotients The original idea of what comes below is in Sect. 2.2.2 where we explained reduction, a technique to construct reduced symplectic manifolds having a fibration. In fact, the construction below is reduction when we have a G-principal fiber bundle. Let .ϕ : G −→ Diff (M) be any action. The orbit of G through .P ∈ M is Op := {ϕg (p) | g ∈ G}.

.

The stabilizer of .p ∈ M is the subgroup Gp := {g ∈ G | ϕg (p) = p}.

.

Recall that the action .ϕ is called transitive if there is just one orbit and is called free if all stabilizer groups are trivial, i.e., .{e}. Consider a group G acting smoothly over a manifold .M. The group action defines an equivalence relation on .M, and in other words for .p, q ∈ M we say that .p ∼ q if there exists an element .g ∈ G such that .g.p = q. The orbit space .M/G = M/ ∼ is the space of all equivalence classes, and it is equipped with quotient topology with respect to which the projection π : M −→ M/G

.

p ‫׀‬−→ Op is continuous. In other words, .U ⊆ M/G is open if and only if .π −1 (U ) is open in .M.

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Definition 3.24 Let G be a Lie group and B be a manifold. A principal G-bundle over B is a manifold P with a smooth mapping .π : P −→ B satisfying: (1) The group G acts freely on .P . (2) The manifold B is the orbit space for the action of G on .P , and .π is the orbit projection. (3) There is an open covering of B such that for all U open in the covering, there corresponds a mapping ϕU : P ⊇ π −1 (U ) −→ U × G

.

such that for all p ∈ π −1 (U ), ϕU (p) = (π(p), sU (p)),

.

and .sU (p) is G-equivariant, i.e., .sU (g.p) = g.sU (p). The G-valued mappings .sU are determined by the corresponding .ϕU . The condition (3) of the definition is called the property of being locally trivial. Let P be a principal G-bundle with mapping .π : P −→ B over the manifold .B. Then B is called the base, and the manifold P is called the total space, the Lie group G is called the structure group, and the mapping .π is called the projection. The principal bundle is represented by

Example 3.3 Assume that P is the 3-sphere, the set of unit vectors in .C2 , i.e., P = S 3 = {(z1 , z2 ) ∈ C2 ; | z1 |2 + | z2 |2 = 1}.

.

Also, assume that the group G is the circle group .S 1 with elements .eit ∈ S 1 . Consider that the group .S 1 acts on .S 3 by complex multiplication, i.e., S 1 × S 3 −→ S 3

.

eit .(z1 , z2 ) ‫׀‬−→ (eit z1 , eit z2 ). Then we obtain a principal .S 1 -bundle, where the base B is the quotient space .S 3 /S 1 that is the complex projective space .CP1 , that is, .S 2 . The diagram below represents this principal .S 1 -bundle that is known as the Hopf fibration.

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111

Theorem 3.3 If a compact Lie group G acts freely on a manifold .M, then the orbit space .M/G is a manifold and .π : M −→ M/G is a principal G-bundle. The proof of this theorem and the next theorem can be found in any standard book on Lie group theory or in [7]. In fact this theorem is a consequence of the slice theorem. Let .Op denote the orbit passing through point .p ∈ M. In fact, one can show that even if the action is not free, the orbit .Op is a compact embedded submanifold of M and the orbit is diffeomorphic to the quotient of G by stabilizer group of .p, i.e., .Op  G/Gp . A transverse section to the orbit .Op is called slice, and we denote it by .S. We choose coordinate system .x1 , . . . , xn around point p such that for .Op  G we have .x1 = · · · = xk = 0 and for S we have .xk+1 = · · · = xn = 0. Now, consider n n .Sɛ = S ∩ Bɛ (0, R ), where .Bɛ (0, R ) is the ball of radius .ɛ with the center at 0 n in .R . Also, consider the mapping .η : G × S −→ M, where .η(g, s) = g.s. The following theorem is the equivariant version of tubular neighborhood theorem. Theorem 3.4 (Slice Theorem) Let G be a compact Lie group acting on a manifold M. Assume that G acts freely at .p ∈ M. For sufficiently small .ɛ, the mapping .η : G×Sɛ −→ M maps .G×Sɛ diffeomorphically onto a G-invariant neighborhood U of the G-orbit of .p. .

Now we have all ingredients to rephrase the symplectic quotient theorem. Theorem 3.5 (Marsden–Weinstein–Meyer [32, 40, 41]) Let .(M, ω) be a symplectic manifold with a Hamiltonian action of a compact connected Lie group G on it. For a momentum mapping .μ, consider the inclusion mapping .i : μ−1 (0) ᶜ→ M. Moreover, assume that G acts freely on .μ−1 (0). Then: • The orbit space .μ−1 (0)/G is a manifold. −1 (0) −→ μ−1 (0)/G is a principal G-bundle. .• The mapping .π : μ −1 (0)/G such that .i ∗ ω = π ∗ ω .• There is a symplectic form .ωred on .Mred := μ red −1 on .μ (0). .

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Proof Let .p ∈ M and .Gp the stabilizer of .p. We denote the Lie algebra of .Gp by gp . Consider the mapping

.

dμp : Tp M −→ Tμ(p) g∗ ∼ = g∗ .

.

Using the equation . = ωp (Xp , v), where .Xp denotes a vector field on M, for all .X ∈ g, and .v ∈ Tp M, we obtain that

.

Ker(dμp ) = (Tp (G.p))ωp

.

I m(dμp ) = annihilator of gp in g∗ := {ξ ∈ g∗ | = 0, ∀X ∈ gp }. Since G acts freely on .μ−1 (0), we obtain that .dp μ is surjective for all .p ∈ μ−1 (0), where .gp is the Lie algebra of the stabilizer .Gp , which in this case is trivial, and hence .I m(dp μ) = g∗ , since .gp = 0. Therefore, 0 is a regular value of .μ; hence, −1 (0) is a closed submanifold of .M, and .dim(μ−1 (0)) = dimM − dimG, i.e., .μ of codimension .dim(G). So, the first and second part of the theorem is just an application of Theorem 3.3 to the free action of G on .μ−1 (0). Also we conclude that .Ker(dμp ) = Tp μ−1 (0). Note that the G-orbit through p is isotropic, i.e., .ωp |Tp (Op ) = 0, because G acts on .μ−1 (0), and we have Tp (Op ) ⊆ Ker(dμp ) = Tp μ−1 (0) = Tp (Op )ωp .

.

Remember in Chap. 1, Sect. 1.5, we defined induced symplectic structure, and in exercise 10 we saw that if .(V , ω) is a symplectic vector space and .U ⊆ V is isotropic subspace, then .ω induces a symplectic structure on .U ω /U. Since a point .[p] ∈ μ−1 (0)/G has tangent space T[p] (μ−1 (0)/G) = Ker(dμp )/Tp (Op ) = Tp (Op )ω /Tp (Op ),

.

ω induces a canonical symplectic structure .ωred on .T[p] (μ−1 (0)/G), which is welldefined because .ω is G-invariant. Thus we have .π ∗ ωred = i ∗ ω. Moreover, .ωred is closed because

.

π ∗ dωred = dπ ∗ ωred = di ∗ ω = i ∗ dω = i ∗ 0 = 0

.

and .π is injective (because .π is the projection of a fiber bundle). Therefore, (Mred , ωred ) is a symplectic manifold of dimension .dim(M) − 2dim(G). ⨆ ⨅

.

Definition 3.25 The symplectic manifold .(Mred , ωred ) is the reduction of .(M, ω) with respect to the Lie group G and momentum mapping .μ, and is called the symplectic quotient, or the Marsden–Weinstein–Meyer quotient.

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113

Example 3.4 In Sect. 2.2.3 using reduction, we equipped complex projective manifolds with symplectic structure. Here in a special case when .E = Cn , we show the obtained reduced symplectic manifold .CPn−1 as a symplectic quotient. Let ω=

.

∑ ∑ i ∑ dzj ∧ d z¯ j = dxj ∧ dyj = rj drj ∧ dθj 2 n

n

n

j =1

j =1

j =1

be the standard symplectic form on .Cn . Consider the .S 1 -action on .(Cn , ω) as multiplication by .eit , that is, ϕ : S 1 × Cn −→ Cn

.

(t, z) ‫׀‬−→ ϕt (z) = eit z. We leave as an exercise to check that .ϕ is a Hamiltonian action with the momentum mapping μ : Cn −→ R

.

z− ‫→ ׀‬−

| z |2 + constant. 2

For the constant equal to . 12 , we obtain .μ−1 (0) = S 2n−1 , the unit sphere. Then by Theorem 3.5 the orbit space μ−1 (0)/S 1 = S 2n−1 /S 1 = CPn−1

.

is a symplectic quotient space. Exercise 3.7 Show that the action defined in Example 3.4 is Hamiltonian with the momentum mapping .μ defined there. Exercise 3.8 Consider .S 2 as the unit sphere in .R3 equipped with symplectic form 3 2 .ω as we described in Example 2.4. The standard action of .SO(3) on .R maps .S 2 onto itself; therefore, it induces an action on .S . Compute the momentum mapping of this action of .SO(3) on .(S 2 , ω).

Chapter 4

Contact Manifolds

In the study of geometric aspects of classical mechanics, besides symplectic manifolds that are even dimensional, there are contact manifolds that are odd dimensional. Contact manifolds are used in relation to mechanical systems dependent on time and in the study of energy surfaces. Contact geometry is the study of contact manifolds, and it is in fact the odddimensional counterpart of symplectic geometry. In other words, most of the facts in symplectic geometry have analogs in contact geometry. Moreover, these two areas of geometry have close interactions. The same as symplectic geometry, contact geometry has origins in classical mechanics. An even-dimensional phase space of a mechanical system is a symplectic manifold. A constant-energy hyperspace, which is of co-dimension one, has odd dimension and is a contact manifold. Contact geometry also has applications in geometric optic and thermodynamics. In this chapter we explain the definition of a contact manifold, examples and constructions of contact manifolds, and their relation to symplectic manifolds. For instance, we explain how to obtain a contact manifold as an energy surface, or as a hypersurface in a symplectic manifold. We explain contactomorphisms, Legendrian submanifolds, and Legendrian knots. In particular, we study the characteristic foliation of an embedded surface in a three-dimensional contact manifold, induced by its contact structure. We study Darboux’s theorem for contact manifolds, which states that locally, (2n+1)-dimensional contact manifolds are the same. Hence, the same as symplectic manifolds, there is no local invariant to distinguish contact manifolds. Moreover, we study geometric optic as a motivation for contact geometry. We explain contact surgery as a way to construct contact manifolds and, briefly, we talk about contact invariants.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. Eslami Rad, Symplectic and Contact Geometry, Latin American Mathematics Series, https://doi.org/10.1007/978-3-031-56225-9_4

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4.1 Definition of Contact Manifold Let M be a (2n−1)-dimensional manifold and ξ ⊂ T M be a hyperplane field on M. Recall that a submanifold N of a manifold M is said to be an integral submanifold if Tq N = ξq for any point q ∈ N. Definition 4.1 Let M be a (2n − 1)-dimensional manifold. A contact structure ξ ⊂ T M on M is a hyperplane field that is maximally non-integrable. The maximally non-integrable condition in the definition means that (at least locally) there is no n-dimensional integral submanifold, and at most we have up to (n − 1)-dimensional integral submanifolds. More precisely, first note that the following lemma indicates that we can write locally any hyperplane distribution as the kernel of a differential one-form. Lemma 4.1 Locally any hyperplane distribution ξ can be written as the kernel of a differential one-form α. Proof Consider a Riemannian metric g on M. We define the line bundle ξ ⊥ as the orthogonal complement of ξ in T M with respect to the metric g. Hence, we can write T M ∼ = ξ ⊕ ξ ⊥ , and therefore we have T M/ξ ∼ = ξ ⊥ . So around each point x ∈ M, we consider a neighborhood Ux such that the line bundle ξ ⊥ is trivial over Ux . Hence, it has a nowhere vanishing section. We denote a non-zero section of ξ ⊥ |Ux by X, and we define a one-form αUx on Ux by αUx := g(X, −). Then clearly, ξ ⊥ |Ux = ker(αUx ). ⨆ ⨅ Now, let α be the one-form on M such that ker(α) = ξ (locally). On the other hand, the Frobenius theorem gives integrability condition. Theorem 4.1 (Frobenius) A (2n − 2)-dimensional integral submanifold exists if and only if α ∧ dα = 0. As a corollary of the Frobenius theorem, an n-dimensional integral submanifold exists if and only if α ∧ (dα)n−1 = 0. Therefore, the maximally non-integrable condition in above definition is equivalent to say that α ∧ (dα)n−1 /= 0.

.

Note that the condition α ∧ dα /= 0 means that ξ is not integrable; however, the condition α ∧ (dα)n−1 /= 0 means that ξ is maximally non-integrable. Up to now, using the above lemma, we have assumed that the one-form α is locally defined. If the hyperplane distribution is co-orientable, then it admits a global defining one-form α. Lemma 4.2 A hyperplane field ξ is co-orientable if and only if there exists a global defined one-form α such that ξ = ker(α).

4.1 Definition of Contact Manifold

117

Proof If ξ is a co-orientable hyperplane field, then ξ ⊥ the orthogonal complement of ξ in T M with respect to an arbitrary Riemannian metric g on M, is orientable. Since ξ ⊥ is line bundle (T M = ξ ⊕ ξ ⊥ ), it is trivial. So there exists a non-zero section X, and thus there exists a one-form α globally. Conversely, if ξ = ker(α) with a globally defined one-form α, then one can define a global section of ξ ⊥ by g(X, X) = 1 and α(X) > 0; hence, ξ is co⨆ ⨅ orientable. Example 4.1 To understand the notion of co-orientability of a hyperplane field, here we bring an example (from [20]) of a hyperplane field that is “not” coorientable. Let M = Rn+1 × RP n . We denote Cartesian coordinates on Rn+1 by (x0 , . . .∑ , xn ) and homogeneous coordinates on RP n by (y0 : · · · : yn ). Then ξ := ker( nj =0 yj dxj ) is a well-defined hyperplane field on M because the oneform in the right-hand side is well-defined up to scaling by a non-zero constant. If n is even, then M is not orientable. So there can be no global constant form defining ξ, i.e., ξ is not co-orientable. From now on we assume that the hyperplane field is co-orientable, i.e., it admits a global defined one-form α. Definition 4.2 Such a global defined one-form α is called a contact form, and (M, ξ ) is called a contact manifold. Any other choice of a contact form for ξ is given by f α, where f is a nonvanishing function on M. The conformal class of the symplectic form dα on ξ is independent of f, since d(f α) restricts to f dα on ξ. For a global defined form α the condition α ∧ (dα)n−1 /= 0 means that α ∧ (dα)n−1 is a volume form for the manifold M and so M is an orientable manifold. So, the maximally non-integrabililty of a hyperplane field ξ means that dα |ξ has maximal rank. In other words, the two-form dα |ξ is a skew-symmetric matrix (through basis), so it is invertible. This means that dα |ξ is non-degenerate, so it defines a symplectic form on the hyperplane distribution ξ = kerα. In other words (ξ, dα) is a symplectic vector bundle, i.e., at any point p ∈ M the space (ξp , dα |ξp ) is a symplectic vector space. Definition 4.3 Let α be a contact form. The Reeb vector field Rα is the vector field uniquely defined by the equations dα(Rα , −) = 0 and α(Rα ) = 1. This means that the Reeb vector field is always transverse to the contact plane, as shown in Fig. 4.1. In other words, we have ξp⊥ = p or p ⊕ ξp = Tp M.

.

Note that different contact forms defining the same contact structure may have Reeb vector fields with different dynamics, i.e., the dynamic properties of Rα strongly depend on the choice of the contact form α for ξ.

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4 Contact Manifolds

Fig. 4.1 Reeb vector field

Lemma 4.3 The flow of the Reeb vector field Rα preserves the contact form α. Proof We want to show that (etRα )∗ α = α. This is equivalent to show that LRα α = 0. By Cartan formula we have LRα α = iRα dα + d(α(Rα ))

.

= 0 + d(1) = 0. Therefore the flow of Rα preserves the contact form α. Exercise 4.1 Consider the one-form

R2n+1

⨆ ⨅

with coordinates (x1 , y1 , . . . , xn , yn , z). Show that

α = dz +

.

n ∑ (xi dyi − yi dxi ) i=1

is a contact form on R2n+1 . Exercise 4.2 Let (x1 , y1 , . . . , xn+1 , yn+1 ) be Cartesian coordinates on R2n+2 . Then the one-form α :=

.

n+1 ∑ (xj dyj − yj dxj ) j =1

is a contact form the unit sphere S 2n+1 in R2n+2 where the standard form on ∑on n+1 2n+2 R is ωst = j =1 dxj ∧ dyj .

4.2 Examples and Constructions In this section we explain some examples and constructions of contact manifolds. The first and the most important example is the standard contact structure.

4.2 Examples and Constructions

119

Fig. 4.2 The standard contact structure ξst = ker(dz − ydx)

Example 4.2 (Standard Contact Structure) Let M = R2n+1 with coordinates (x1 , . . . , xn , y1 , . . . yn , z). Consider the one-form α := dz −

n ∑

.

yi dxi

i=1

on R2n+1 . Then ξ = ker(α) is a contact structure because dα = and

∑n

i=1 dxi

∧ dyi

α ∧ (dα)n = n!dz ∧ dx1 ∧ · · · ∧ dxn ∧ dy1 ∧ · · · ∧ dyn /= 0.

.

Hence, α ∧ (dα)n is a volume form on R2n+1 . Later we will see that by Darboux’s theorem any contact structure is locally the same as the above contact structure ξ ; for this reason we call such a contact structure as the standard contact structure, and we denote it by ξst . In particular case, when n = 1, we have M = R3 with coordinates (x, y, z). Then the standard contact structure on R3 (shown in Fig. 4.2) is equal to ξst = ker(dz − ydx) =
. ∂y ∂x ∂z

Note that the Reeb vector field Rαst of the standard contact structure ξst = ∂ ker(αst ) on R2n+1 is equal to ∂z . In fact, one can consider more examples of contact forms on R2n+1 . Example 4.3 Consider R2n+1 with coordinates (x1 , y1 , . . . , xn , yn , z). Then the following one-form α = dz +

.

n ∑ (xi dyi − yi dxi ) i=1

is a contact form. We leave as an exercise to check that α ∧ (dα)n /= 0. Therefore, ξ = ker(α) is a contact structure on R2n+1 .

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4 Contact Manifolds

In the following example, we equip the odd-dimensional sphere S 2n+1 with a contact structure. Example 4.4 Let (x1 , y1 , . . . , xn+1 , yn+1 ) be Cartesian coordinates on R2n+2 . Then a contact structure on the unit sphere S 2n+1 in R2n+2 is given by the contact form α :=

n+1 ∑

.

(xj dyj − yj dxj ),

j =1

∑ where the standard form on R2n+2 is ωst = n+1 j =1 dxj ∧dyj . We leave as an exercise to check that α satisfies the contact condition. In fact, the idea of defining such a contact form on sphere comes from the next example and the theorem below of it. Example 4.5 Consider R2n with the usual coordinates (x1 , . . . , x2n ). Assume that M is a submanifold of the form x2n = c for c ∈ R. We claim that the restriction of the one-form ∑ .α = dx2n−1 + xi dxm−1+i i≤n−1

to M is a contact form. We have α ∧ (dα)n−1 = adx1 ∧ · · · ∧ dx2n−1 , where a /= 0. On the other hand, since the vector field ∂x∂2n is orthogonal to M, it follows that i

.

∂ ∂x2n

(dx1 ∧ · · · ∧ dx2n )

is a volume form on M. This volume form is equal to −dx1 ∧ · · · ∧ dx2n−1 . By comparing with α ∧(dα)n−1 , our claim is obtained. Below, we will see an important generalization of this example in the study of energy surfaces. Recall from theorem of regular value that if c is a regular value of a smooth mapping H : M −→ R, then H −1 (c) is a submanifold of M with co-dimension equal to 1. Definition 4.4 Each connected component of H −1 (c) whose all points are regular is called a regular energy level. Theorem 4.2 Let (M, σ ) be a symplectic manifold, c be a regular value for the smooth mapping H : M −→ R, and XH be Hamiltonian vector field associated to H. Assume that: (i) The symplectic form σ is exact, i.e., there exists a one-form θ such that σ = dθ . (ii) Over H −1 (c) we have θ (XH ) /= 0.

4.2 Examples and Constructions

121

Consider the inclusion mapping j : H −1 (c) ᶜ→ M. Then (H −1 (c), j ∗ θ ) is a contact manifold. Proof We should prove that at each point x ∈ H −1 (c) we have that (j ∗ θ ) ∧ (dj ∗ θ )n−1 is a volume form. We consider (2n − 2) tangent vectors X1 , . . . , X2n−2 on H −1 (c) so that the vectors XH , X1 , . . . , X2n−2 are linear independent. Then we have (j ∗ θ ) ∧ (dj ∗ θ )n−1 (XH , X1 , . . . , X2n−2 ) = θ ∧ (dθ )n−1 (XH , X1 , . . . , X2n−2 )

.

= iXH (θ ∧ (dθ )n−1 )(X1 , . . . , X2n−2 ) = (θ (XH )(dθ )n−1 − θ ∧ iXH (dθ )n−1 ) ×(X1 , . . . , X2n−2 ) = θ (XH )(dθ )n−1 (X1 , . . . , X2n−2 ), where the second equality follows from definition of interior product and the third equality follows from properties of interior product. We obtain the fourth equality because we know that dH = iXH (dθ ) and Tx (H −1 (c)) = ker(dHx ). We can choose the vectors X1 , . . . , X2n−2 in such a way that (dθ )n−1 (X1 , . . . , X2n−2 ) /= 0. Hence by assumption (ii) of the theorem, i.e., θ (XH ) /= 0 over H −1 (c), we conclude that (j ∗ θ ) ∧ (dj ∗ θ )n−1 /= 0. ⨆ ⨅ Example 4.6 Let M be an n-dimensional smooth manifold with the Riemannian metric g. Consider mapping K : T M −→ R defined by K(u) =

.

1 g(u, u) 2

and mapping K ∗ : T ∗ M −→ R defined by K ∗ (α) =

.

1 ∗ g (α, α). 2

Assume that V : M −→ R is a smooth mapping. Remember from previous chapter that Hamiltonian vector field XH on (T ∗ M, σ ) for which H = K∗ + V ◦ τ ∗

.

(τ ∗ : T ∗ M −→ M is the natural projection) is called a classical mechanical system on T ∗ M (with the Hamiltonian function H ). The mappings V and K ∗ are called potential energy and kinetic energy, receptively. In this case, we know that the symplectic form σ is exact, i.e., σ = dλcan , so the assumption (i) of the theorem holds. For the energy function H = K ∗ + V ◦ τ ∗ , all energy levels are regular except those which consist of critical points of H. Let N be a regular energy level. Then locally we can write

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4 Contact Manifolds

∑ ∑ ∂H ∂ ∑ ∂H ∂ θ (XH ) = ( pi dqi )( − ) ∂pi ∂qi ∂qi ∂pi ∑ ∂H . = pi ∂pi

.

∗

∂K ∗ Since V does not depend on pi ’s, we have ∂H ∂pi = ∂pi . Since K with respect to pi ’s is homogeneous of second degree, it follows that θ (XH ) = 2K ∗ . Therefore, if the energy level does not intersect with the zero section of T ∗ M, the assumption (ii) of the theorem holds. In fact, if N does not intersect the zero section, then N is a regular energy level, since all critical points of H are on the zero section (x ∈ M is a critical point for the potential energy if and only if its image under the zero section is a singular point for the classical mechanical system). Therefore, the assumptions (i) and (ii) of the theorem hold; hence, (N, θ | N) is a contact manifold.

Remark 4.1 Recall from the second chapter that the only even-dimensional sphere that accepts symplectic form is S 2 (by cohomological reason). However, using the above theorem, we can see that any odd-dimensional sphere has a contact form. We leave this as an exercise. Exercise 4.3 Show that any odd-dimensional sphere has a contact form. (Hint: Consider S 2n−1 as a submanifold of R2n and use Theorem 4.2.) Exercise 4.4 Show that each of the following manifolds has a contact form. T 3 , T 2k × S2k−1 , RP 3 ∼ = SO(3).

.

(Hint: For three-torus T 3 , consider the four-dimensional phase space or cotangent bundle of T 2 , i.e., T ∗ (T 2 ). Then use Example 4.6. Similarly, for the next two manifolds consider a suitable phase space and use Example 4.6.)

4.2.1 Hypersurfaces in Symplectic Manifolds Contact manifolds can be obtained as hypersurfaces in symplectic manifolds. Definition 4.5 Let (W 2n , ω) be a symplectic manifold and X ∈ X(W ) be a Liouville vector field, i.e., LX ω = ω. Also let ∑ ⊂ W be a hypersurface such that X is transverse to ∑, i.e., RX ⊕ T ∑ = T W. Then we say that ∑ is a hypersurface of contact type. Proposition 4.1 If ∑ is a hypersurface of contact type in the symplectic manifold (W, ω) corresponding to the Liouville vector field X, then (∑, ξ = ker(iX ω) |∑ ) is a contact manifold. Proof Let α = iX ω. Then for the Liouville vector field X, by Cartan’s formula, we have

4.2 Examples and Constructions

123

dα = d(iX ω) = −iX dω + LX ω = ω.

.

So (dα)n = ωn is a volume form on W and iX (dα)n is a volume form on ∑. Hence, iX (dα)n = nα ∧ (dα)n−1 implies that α ∧ (dα)n−1 /= 0, which means that α ∧ (dα)n−1 is a volume form on ∑, i.e., kerα |∑ is a contact structure. ⨆ ⨅ Example 4.7 Consider M = R2n with the standard symplectic form ωst = ∑n i=1 dxi ∧ dyi . Define ∂ ∂ 1∑ .X := xi + yi . 2 ∂xi ∂yi n

i=1

The vector field X is Liouville, because iX ω = Cartan’s formula we have

1 2

∑n

i=1 xi dyi

− yi dxi , and by

LX ω = d(iX ω)

.

1∑ xi dyi − yi dxi ) 2 n

= d(

i=1

=

1 2

n ∑

dxi ∧ dyi − dyi ∧ dxi

i=1

1∑ 2dxi ∧ dyi 2 n

=

i=1

= ω. Now, consider the hypersurface ∑ in R2n as the unit sphere ∑ = S 2n−1 = {(x, y) ∈ R2n |

n ∑

.

xi2 + yi2 = 1}.

i=1

Then by above proposition, the one-form 1∑ xi dyi − yi dxi 2 n

α := iX ω =

.

i=1

is a contact form on S 2n−1 . Inspired by the above example and using the previous proposition, we can find a contact structure on the unit cotangent bundle. Example 4.8 (The Unit Cotangent Bundle) Consider an n-dimensional manifold L with the local coordinates q = (q1 , . . . , qn ), and denote the corresponding dual

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4 Contact Manifolds

coordinates in fibers of the cotangent bundle T ∗ L by p = (p1 , . . . , pn ). Remember from Chap. 2 that the canonical one-form is n ∑

λcan =

.

pi dqi .

i=1 ∗ ∗ The ∑n cotangent bundle T L with this canonical one-form, i.e., (T L, dλcan = i=1 dpi ∧ dqi ) is a symplectic manifold. Moreover,

X :=

n ∑

.

pi

i=1

∂ ∈ X(T ∗ L) ∂pi

is a Liouville vector field. Since n ∑

iX (dλcan ) = i∑n

.

∂ i=1 pi ∂pi

=

n ∑

=

n ∑

j =1

pi (dpj (

i,j =1

dpj ∧ dqj

∂ ∂ ) ∧ dqj − dpj ∧ dqj ( ) ∂pi ∂pi

pi dqi

i=1

= λcan , and by Cartan’s formula, we have LX dλcan = iX ddλcan + diX dλcan

.

= dλcan . A Riemannian metric g on L defines a bundle isomorphism Ψ from the tangent bundle T L to the cotangent bundle T ∗ L, which is fiber-wise given by Ψl : Tl L −→ Tl∗ L

.

V ‫׀‬−→ gl (V , −). This induces a bundle metric g ∗ on T ∗ L, defined by gl∗ (u1 , u2 ) = gl (Ψl−1 (u1 ), Ψl−1 (u2 ))

.

for u1 , u2 ∈ Tl∗ L. The unit cotangent bundle ST ∗ L is defined fiber-wise by STl∗ L = {u ∈ Tl∗ L | gl∗ (u, u) = 1},

.

4.2 Examples and Constructions

125

Fig. 4.3 Unit cotangent bundle

see Fig. 4.3. The Liouville vector field X is transverse to the unit cotangent bundle S ∗ T L, because: The integral curves of X are tangent to the fibers of T ∗ L and can be written as u(t) = (l, et u0 ) ∈ Tl∗ L, t ∈ R.

.

d ∗ So gl∗ (u(t), u(t)) = e2t gl∗ (u0 , u0 ). For u0 /= 0, we obtain dt gl (u(t), u(t)) > 0, which means that X is transverse to the positive level sets of gl∗ . Along the zero section of T ∗ L, the vector field X vanishes identically. Therefore, by the above proposition, (S ∗ T L, kerλcan ) is a contact manifold.

4.2.2 Contactization Contactization is a way to obtain a contact manifold from a symplectic manifold. Before we explain contactization, first we explain symplectization. In Chap. 2, we saw different ways to construct symplectic manifolds. Another way to obtain a symplectic manifold is symplectization, i.e., obtaining a symplectic manifold from a contact manifold. Proposition 4.2 Let (M, ξ = kerα) be a contact manifold. Then (R × M, ω = d(et α)), where t ∈ R is a symplectic manifold. Proof From ω = d(et α) = et dt ∧ α + et dα, we obtain ωn = nent dt ∧ α ∧ (dα)n−1 .

.

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4 Contact Manifolds

Since dt is transverse to M and α ∧ (dα)n−1 is a volume form on M, we conclude that ω is a volume form on W. So ω is a symplectic form on W. ⨆ ⨅ Definition 4.6 The symplectization of the contact manifold (M, ξ = kerα) is the symplectic manifold (R × M, ω = d(et α)). In fact, this symplectization depends only on ξ. Any other choice of α leads to a symplectomorphic manifold. Definition 4.7 A symplectic manifold (W, ω) is exact if ω = dβ for some oneform β ∈ Ω1 (W ). The contactization of (W, dβ) is the contact manifold (M = R × W, ξ ) with the contact structure ξ = kerα, where α = dt + β. Note that dα = dβ, and therefore α ∧ (dα)n = α ∧ (dβ)n

.

= (dt + β) ∧ (dβ)n = dt ∧ (dβ)n + β ∧ (dβ)n = dt ∧ (dβ)n + 0 = dt ∧ (dβ)n , which is a volume form on M. Now, the question is if the one-form α = dt + β is the only possibility to define α. The answer to this question is positive and is given in the following lemma. Lemma 4.4 Any form α ∈ Ωp (R × M), where p ≥ 1, is uniquely written as α = β + dt ∧ α¯

.

such that i ∂ α¯ = 0, i ∂ β = 0. ∂t

∂t

∂ ∂t

Proof Here, is the unit vector field in the direction of the first component. In d other words, if dt is the unit vector field on R, then for (t, x) ∈ [0, 1] × M we have ∂ d ∂t (t, x) = ( dt (t), 0). To prove the uniqueness, we assume that α can be written in two ways, i.e., β + dt ∧ α¯ = β ' + dt ∧ α¯ ' . So we have β − β ' = dt ∧ (α¯ ' − α). ¯ If at the point (t, x) we have α¯ /= α¯ ' , then there exist vector fields X2 , . . . , Xp in the subspace {0} × Tx M such that (α¯ ' − α)(X ¯ 2 , . . . , Xp ) /= 0. Now, if we apply both sides on ( ∂t∂ , X2 , . . . , Xp ), we get contradiction. Therefore, α¯ ' = α¯ and so β ' = β. To show the existence, for a given α, we define β = α − dt ∧ α¯ and α¯ = i ∂ α. It ∂t ⨆ ⨅ follows that i ∂ α¯ = 0 and i ∂ β = 0. ∂t

∂t

Example 4.9 Again, remember that the cotangent bundle with the canonical form λcan is the symplectic manifold (W, ω) = (T ∗ L, dλcan ), which is an exact

4.2 Examples and Constructions

127

symplectic manifold. So the contactization of the cotangent bundle gives rise to the contact manifold (R × T ∗ L, ker(dt + λcan )).

.

Example 4.10 (The Jet Bundle) Let f ∈ C ∞ (L, R) be a smooth function. We consider a manifold such that f is a section of that manifold. This manifold is called the 0-jet space of L, and we denote it by J 0 (L). We denote the 0-jet of f by j 0 (f ). In other words if we consider the mapping π0 : L × R −→ L,

.

then we have L × R = J 0 (L) and f = j 0 (f ). Now, we consider a manifold such that (f, df ) is a section of that manifold. This manifold is called the 1-jet space of L, and we denote it by J 1 (L). We denote the 1-jet of (f, df ) by j 1 (f ). In other words, if we consider the mapping π1 : T ∗ L × R −→ L,

.

then we have T ∗ L × R = J 1 (L) and (f, df ) = j 1 (f ). Let s be a section of J 1 (L) = T ∗ L × R. Consider the coordinates on L as x1 , . . . , xn , on T ∗ L as x1 , . . . , xn , y1 , . . . , yn and on R as z. Now the question is when s is obtained as (f, df ) = j 1 (f ) for some f ∈ C ∞ (L, R), or, if exists f ∈ C ∞ (L, R) such that s = j 1 (f ) = (f, df ). To answer this question, consider s(p) = (yi ◦ s(p), z ◦ s(p)) ∈ T ∗ L × R,

.

p ∈ L.

We set f = z ◦ s ∈ C ∞ (L, R). Therefore we have df =

.

n ∑ ∂f dxi dxi i=1

=

n ∑ (yi ◦ s)dxi . i=1

Therefore the answer is equivalent to dz −

.

∑

yi dxi | graph of s = 0.

Hence, in general, we shall consider the jet bundle (J 1 (L), ker(dz − λcan )) as a contact manifold. In summary, let L be an n-dimensional manifold and f ∈ C ∞ (L, R). Then the 1-jet space of L, denoted by J 1 (L), is a bundle over L whose sections are 1-jet of

128

4 Contact Manifolds

f i.e., j 1 (f ) = (f, df ). So we can identify J 1 (L) with the space R × T ∗ L by mapping jx1 f to (f (x), dfx ). Let λcan be the canonical form on T ∗ L that is given in local coordinates q on L and dual coordinates p by λcan = pdq. Writing z for the R-coordinate in R × T ∗ L, we have a natural contact structure on the 1-jet space given by ξj et = ker(dz − λcan ).

.

4.3 Contactomorphisms Up to now, we have studied several examples and constructions of contact manifolds. Now the question is which of these contact manifolds are the same. To answer this question, we need to define a mapping between two contact manifolds that preserves their contact structure. Definition 4.8 Two contact manifolds (M1 , ξ1 ) and (M2 , ξ2 ) are contactomorphic if there is a diffeomorphism f : M1 → M2 such that Tf (ξ1 ) = ξ2 , where Tf : T M1 → T M2 denotes the differential of f. Such a diffeomorphism f is called a contactomorphism. Example 4.11 Let M1 , M2 = R2n+1 with coordinates (x1 , . . . , xn , y1 , . . . yn , z). Consider the one-forms α1 := dz +

n ∑

.

xi dyi

i=1

and α2 = dz +

n ∑

.

xi dyi − yi dxi .

i=1

Two contact manifolds (R2n+1 , ξ1 = ker(α1 )) and (R2n+1 , ξ2 = ker(α2 )) are contactomorphic. One can define a contactomorphism f by f (x, y, z) = (

.

xy (x + y) (y − x) , ,z + ), 2 2 2

where x denotes∑coordinates (x1 , . . . , xn ) and y denotes coordinates (y1 , . . . , yn ) and xy denotes i xi yi . Moreover, both these contact∑ structures are contactomorphic to the standard contact structure ξst = ker(dz − i yi dxi ) in Example 4.2.

4.3 Contactomorphisms

129

Note that any of contact structures in the above example can be considered as the standard contact structure on R2n+1 ∑ (since they are all contactomorphic). Here, usually we consider ξst = ker(dz − i yi dxi ); otherwise, we specify the standard contact structure. The above examples of contact structures on R3 are all contactomorphic to the standard contact structure. In the next example we see a contact structure that is not contactomorphic to them. Example 4.12 (Overtwisted Contact Structure on R3 ) In cylindrical coordinates (r, θ, z), consider α = f1 (r)dz + f2 (r)dθ as a contact form in which f1 , f2 are smooth functions such that f1 (0) = 1, f2 (0) = 0, and f1 = cos(2π r), f2 = sin(2π r).

.

The overtwisted contact structure is defined as ξot = ker(α) = ker(f1 (r)dz + f2 (r)dθ )

.

on R3 . Bennequin in [4] proved that this contact structure (R3 , ξot ) is not contactomorphic to previous examples of standard contact structure (R3 , ξst ). Below, the contact planes in overtwisted contact structure are illustrated. Note that the standard contact structure ξst on R3 in Fig. 4.2 and the overtwisted contact structure ξot in Fig. 4.4 are both horizontal along the z-axis, i.e., when r = 0 all the rays are perpendicular to the z-axis (with z and θ constant) and are tangent to both ξst and ξot . When r /= 0, the contact structure ξot is spanned as ξot =
. ∂r ∂θ ∂z

Thus ξot turns counterclockwise as we move outward from the z-axis along any ray. Note that in ξst the contact planes never become vertical (approaches but never Fig. 4.4 Overtwisted contact structure ξot on R3

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4 Contact Manifolds

reaches the vertical direction). However, the contact planes of ξot make infinitely many complete turns as we move in radial direction. The following example shows that a contact manifold can be considered as a hypersurface in its symplectization. Then the contact manifold is contactomorphic with that contact hypersurface. Example 4.13 Consider contact manifold (M, ξ = ker(α)). Using symplectization, we obtain symplectic manifold (R × M, d(et α)). Now, using Proposition 4.1, we can construct a contact manifold in our symplectic manifold (R × M, d(et α)). The vector X=

.

∂ ∈ χ (R × M) ∂t

is a Liouville vector field because by Cartan formula we have LX d(et α) = iX dd(et α) + dix (det α)

.

= 0 + d(et α) = d(et α). Here we have used the following calculation: iX d(et α) = i ∂ (et dt ∧ α + et dα)

.

∂t

= et i ∂ (dt ∧ α) + et i ∂ dα ∂t

∂t

∂ ∂ = et (dt ( ) ∧ α − dtα( )) + et i ∂ dα ∂t ∂t ∂t = et α. We can choose ∑ = {0}×M, t = 0 as a hypersurface; therefore, by Proposition 4.1, the one-form iX d(et α) |∑ is a contact form. So we started by a contact manifold, then got a symplectic manifold, and again made a contact manifold. In fact these two contact manifolds are contactomorphic (globally). The next example shows that a symplectic manifold is symplectomorphic to symplectization of its contact hypersurface. Example 4.14 If we start with a symplectic manifold, then we obtain a contact manifold as a hypersuface in the symplectic manifold, and again using symplectization, we obtain a symplectic manifold. More precisely, if ∑ is a hypersurface of contact type in the symplectic manifold (W, ω) corresponding to the Liouville vector field X, then (∑, ξ = ker(iX ω) |∑ ) is a contact manifold. Using symplectization we obtain a new symplectic manifold

4.3 Contactomorphisms

131

Fig. 4.5 Symplectomorphism between W and ∑ × R

(∑ × R, d(et (iX ω) |∑ )). In fact these two symplectic manifolds, i.e., W and ∑ × R, are symplectomorphic in a neighborhood of ∑ and ∑ × {0}, as in Fig. 4.5. In general we cannot extend this symplectomorphism to the whole manifold, since maybe in the outside of the neighborhood we have another symplectic form. Exercise 4.5 Show that the mapping f defined in Example 4.11 is a contactomorphism between contact manifolds (R2n+1 , ξ1 = ker(α1 )) and (R2n+1 , ξ2 = ker(α2 )). Exercise 4.6 Show both contact structures in the previous exercise are contacto∑ morphic to ξst = ker(dz − i yi dxi ). Exercise 4.7 Show that the two contact manifolds (M, ξ = ker(α)) and (∑, iX d(et α) |∑ ), described in Example 4.13, are contactomorphic.

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4 Contact Manifolds

Exercise 4.8 Show that the two symplectic manifolds (W, ω) and (∑ × R, d(et (iX ω) |∑ )), described in Example 4.14, are symplectomorphic.

4.4 Complex Structure Remember from the definition of the contact structure ξ ⊂ T M that for any (local) one-form α with ξ = kerα we have that (dα)n−1 /= 0, i.e., (ξp , dα |ξp ) is a symplectic vector space for all points p ∈ M. This provides (ξ, dα) as a symplectic vector bundle, i.e., a vector bundle with a symplectic form on every fiber, varying smoothly with the base point. In the following, we will see how to equip this vector bundle with a complex structure. Definition 4.9 A complex structure on a real vector space V is an automorphism J :V →V

.

satisfying J 2 = −idV . If (V , ω) is a symplectic vector space, then a complex structure J on V is called ω-compatible if it satisfies: 1. ω(J u, J v) = ω(u, v) for all u, v ∈ V . 2. ω(v, J v) > 0 for all non-zero v ∈ V . In fact if J is a complex structure on a vector space V , we can define a complex scalar √multiplication by (a + ib)v := av + bJ v for a, b ∈ R and v ∈ V , where i = −1. This turns V into a vector space over C, and the real dimension of V is even. Definition 4.10 An inner product g on V is called J -compatible if g(J u, J v) = g(u, v)

.

for all u, v ∈ V . Therefore if J is a complex structure compatible with a symplectic form ω, then gJ (u, v) := ω(u, J v)

.

defines a J -compatible inner product on V . Proposition 4.3 The space J(ω) of ω-compatible complex structures on the symplectic vector space (V , ω) is non-empty and contractible. Proof Let g be an inner product on V ; then there is a unique linear mapping A ∈ End(V ) such that g(., A) = ω(., .). Thus for v, w ∈ V we have g(v, Aw) = ω(v, w) = −ω(w, v) = −g(w, Av) = −g(Av, w).

.

4.4 Complex Structure

133

So we obtain A = −A∗ , and A∗ A = −A2 , which is symmetric and positive definite. This means that there is an orthogonal matrix P such that A∗ A = −A2 = P DP −1 with ⎛ ⎞ λ1 0 ⎜ .. ⎟ .D = ⎝ ⎠ . 0

λ2n

1

1

and λi > 0. We can define (A∗ A) 2 = P D 2 P −1 with ⎛√ ⎞ λ1 0 1 ⎜ ⎟ .. .D 2 = ⎝ ⎠. . √ λ2n 0 1

1

Hence, (A∗ A) 2 is still symmetric and positive definite. Define B := (A∗ A)− 2 A. Then we have 1

1

1

1

B 2 = (A∗ A)− 2 A(A∗ A)− 2 A = (−A2 )− 2 A(−A2 )− 2 A =

.

1 −1 1 −1 A A A A = −I. i i

Therefore, B is a complex structure. The complex structure B is ω-compatible, since we have 1

1

g(Bv, Bw) = g((A∗ A)− 2 Av, (A∗ A)− 2 Aw)

.

= g((A∗ A)−1 Av, Aw) = g(A∗ A(A∗ A)−1 v, w) = g(v, w). Therefore, ω(Bv, Bw) = g(Bv, B 2 w) = g(v, Bw), by compatibility of g. So we get ω(Bv, Bw) = ω(v, w). Hence, J(ω) is non-empty. To show that the space J(ω) is contractible, we consider the space G of inner products on V . The topology on G can be defined by choosing a basis for V and identifying G with the space of symmetric and positive-definite matrices. The above construction gives a smooth mapping j : G −→ J(ω)

.

g ‫׀‬−→ J = j (g). Let J ∈ J(ω) and the J -compatible inner product gJ be defined by gJ (v, w) = ω(v, J w), so ω(v, w) = gJ (J v, w). Using the above construction for gJ , we obtain j (gJ ) = J.

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4 Contact Manifolds

Now, we fix an element J0 ∈ J(ω), and we define a family of continuous mappings ft : J(ω) −→ J(ω), t ∈ [0, 1],

.

by ft (J ) = j ((1 − t)gJ0 + tgJ ).

.

This gives a homotopy between the constant mapping f0 ≡ J0 and the identity f1 = idJ(ω) , which shows that the space J(ω) is contractible. ⨆ ⨅ Extending the above notion from vector spaces to vector bundles, we have the following definition. Definition 4.11 A complex structure on a symplectic vector bundle (E, ω) → B is called ω-compatible if Jb is ωb - compatible on Eb for each b ∈ B. A complex bundle structure on the tangent bundle T M of a manifold M is called an almost complex structure on M. Definition 4.12 Given a contact manifold (M, ξ = kerα), a complex structure J on ξ is called ξ -compatible if Jp : ξp → ξp is a dα-compatible complex structure on ξp , i.e., dα(Jp ., Jp .) = dα(., .)

.

and dα(., Jp .) > 0,

.

for each p ∈ M. By Proposition 4.3, the set of compatible complex structures on ξ is non-empty and contractible; hence, a contact structure defines a complex vector bundle up to homotopy.

4.5 Darboux’s Theorem Just as for symplectic manifolds, contact manifolds have no local invariants. This is because of Darboux’s theorem for contact manifolds that indicates that any (2n−1)dimensional contact manifold locally looks like R2n−1 with the standard contact structure. The same as symplectic manifolds, the proof of Darboux’s theorem for contact manifolds relies on the Moser trick.

4.5 Darboux’s Theorem

135

Lemma 4.5 (Moser Trick) Let αs , s ∈ [0, 1] be a 1-parameter family of contact forms on M. Then there exist a 1-parameter family of vector fields Xs , s ∈ [0, 1] and a 1-parameter family of smooth functions gs , s ∈ [0, 1], satisfying .

dαs + iXs dαs = gs .αs ds

(∗).

Proof The equation (∗) reminds us the similarity to ξp ⊕p = Tp M for p ∈ M. If we project the equation (∗) on the natural subspace ξs = kerαs of Tp M, then we obtain iXs dαs = −

.

dαs . ds

Since dαs is a symplectic form, we get the unique solution Xs . On the other hand, if we project equation (∗) to the line spand by the Reeb vector field Rαs , then we have .

dαs (Rαs ) = gs .αs (Rα ) ds = gs .

Here by definition of the Reeb vector field Rαs , the second term in (∗) is zero, so we obtain the first equality. In the second equality, we used the fact that αs (Rαs ) = 1. ⨆ ⨅ Therefore we get the unique gs . Theorem 4.3 (Darboux’s Theorem) (i) Every contact (2n − 1)-manifold (M, ξ ) locally looks like (R2n−1 , ξst ), i.e., for every p ∈ M, there exists an open neighborhood U of p and a contactomorphism ψ : (U, ξ ) → (V , ξst ), with V as an open neighborhood of ψ(p) in R2n−1 . (ii) Let (M, ξ ) be a contact manifold with contact form α. For every p ∈ M, there exists an open neighborhood U of p and a diffeomorphism ψ : U → V ⊂ R2n−1 with coordinates x1 , . . . , xn−1 , y1 , . . . , yn−1 , z such that α = ψ ∗ (dz − ydx) = ψ ∗ (dz −

n−1 ∑

.

yi dxi ),

i=1

or equivalently ξ = ker(dz −

∑n−1 i=1

yi dxi ).

Proof (i) Let α0 = α be the contact form as∑ξ = kerα, and α1 = αst be the standard contact form as ξst = ker(dz − ni=1 yi dxi ). We can always find a linear

136

4 Contact Manifolds

coordinate transformation so that ξp = ker(dz), locally. Also, we can always find a linear coordinate transformation∑of (x1 , . . . , xn−1 , y1 , . . . , yn−1 ) so that n we get the symplectic form dαp = i=1 dxi ∧ dyi , by Darboux’s theorem for symplectic manifolds. Therefore at the point p we have (α1 )p = (α0 )p and (dα1 )p = (dα0 )p . Now we want to verify this on a neighborhood of p. We define αs = sα1 + (1 − s)α0 ,

.

s ∈ [0, 1].

Note that not only α1 and α0 are contact forms but also αs ’s are, because at the point p we have α1 = α0 therefore αs = sα0 + α0 − sα0 = α0 . Hence, αs ∧ (dαs )∧n = α0 ∧ (dα0 )∧n

.

/= 0, which by continuity we can extend it to a neighborhood of p. Now, we are going to verify fs α0 = ψs∗ αs for some 1-parameter family of diffeomorphism ψs and 1-parameter family of functions fs . Taking derivative from both sides, we obtain d d αs + ψs∗ LXs αs = fs .α0 , ds ds

ψs∗

.

where Xs (q) =

d ds ψs (q).

ψs∗ (

.

If we denote gs :=

This is equivalent to

dαs 1 dfs + LXs αs ) = fs α0 ds fs ds ( d(logfs ) −1 ) = ψs∗ .ψs .αs . ds

d(logfs ) −1 ds .ψs ,

.

we want

dαs + LXs αs = gs .αs . ds

Equivalently, by Cartan formula we want .

dαs + iXs dαs + d(αs (Xs )) = gs .αs ds

or .

dαs + iXs dαs + 0 = gs .αs ds

4.5 Darboux’s Theorem

137

This is exactly Moser equation, and by Moser trick we know that it has unique solution for Xs and gs . Therefore, we find a local diffeomorphism ψ such that f1 α0 = ψ ∗ α1 . (ii) In order to prove Darboux’s theorem for contact forms, we are going to use the Moser trick to find an isotopy ψs of a neighborhood of p such that ψs∗ αs = α0 . As we see below, locally this equation has always solution. So locally the Moser trick can be applied to the equation ψs∗ αs = α0 . We take derivative from both sides (assuming that ψs is the flow of some time-dependent vector field Xs ), and we obtain ψs∗ (

.

d αs + LXs αs ) = 0, ds

so by Cartan formula Xs should satisfy .

d αs + d(αs (Xs )) + iXs dαs = 0. ds

Replacing Xs = fs Rs + Ys (where Rs is the Reeb vector field of αs and Ys ∈ kerαs and fs is a smooth family of functions) in this equation, we get .

d α(Rs ) + dfs (Rs ) = 0. ds

A smooth family of functions fs satisfying the last equation can always be found by integration on a small neighborhood of p so that none of the Rt has d any closed orbits there. As ds α is zero at p, we take fs (p) = 0 and dfs |p = 0 for all s ∈ [0, 1]. Once we choose fs , then Ys is defined uniquely by .

d α + dfs + iYs dαs = 0. ds

Note that with the assumptions on fs , we have Xs (p) = p for all s. Now consider ψs as the local flow of Xs . This local flow fixes p, and it is defined for all s ∈ [0, 1]. Since the domain of definition of a local flow on a manifold M in R × M is always open, we can say that ψs is defined for all s ∈ [0, 1] on a sufficiently small neighborhood of p. ⨆ ⨅ By Gray’s theorem, contact structures are stable under isotopy. Theorem 4.4 (Gray Stability Theorem) Let {ξs }s∈[0,1] be a smooth family of contact structures on a closed manifold M. Then there exists a smooth path {ψs }s∈[0,1] of diffeomorphisms of M such that ψ0 = id and T ψs (ξ0 ) = ξs for all s ∈ [0, 1].

138

4 Contact Manifolds

Proof Let αs be contact forms for ξs , s ∈ [0, 1]. We show that ψs∗ αs = fs α0 , for all s ∈ [0, 1]. Using Moser trick we obtain Xs for this equation. Since M is compact, we can integrate Xs . ⨆ ⨅ We have seen that we can find some contact structure on sphere S 2n−1 and unit cotangent bundle ST ∗ L, using transverse Liouville vector fields. The Gray stability theorem says that we always get diffeomorphic contact structure independent of choice of the shape of hypersurfaces.

4.6 Legendrian Submanifolds Let (M 2n−1 , ξ ) be a contact manifold with contact form α, i.e., ξ = kerα. In Sect. 4.1, we saw that the maximally non-integrability of a hyperplane field ξ means that dα |ξ has maximal rank. In other words, the two-form dα |ξ is nondegenerate, so it defines a symplectic form on the hyperplane distribution ξ = kerα. Hence, at any point p ∈ M the space (ξp , dα |ξp ) is a symplectic vector space of 2n − 2 dimension. Therefore, the maximal dimension of an isotropic subspace of (ξp , dα |ξp ) is n − 1. Definition 4.13 Let (M 2n−1 , ξ ) be a contact manifold. A submanifold L ⊂ M is called isotropic if it is tangent to ξ, i.e., Tp L ⊂ ξp for all p ∈ L. If L has dimension n − 1, then it is called a Legendrian submanifold. Let (T L)⊥ ⊂ ξ |L be the subbundle of ξ |L that is symplectically orthogonal to T L with respect to the symplectic bundle structure dα |ξ , i.e., (T L)⊥ := {v ∈ ξ : dα(u, v) = 0 for all u ∈ T L}.

.

Then, if L is isotropic, it follows that T L ⊂ (T L)⊥ . Example 4.15 Any curve in the xy-plane in the contact manifold (R3 , ξst ) is isotropic, and in fact it is Legendrian submanifold of (R3 , ξst ), since it is tangent to the contact structure and is of dimension 1. Later we will study about Legendrian knots that are examples of Legendrian submanifolds. Definition 4.14 Let (W, ω = dβ) be an exact symplectic manifold. A Lagrangian submanifold L of W is exact if β |L is exact, i.e., there exists f ∈ C ∞ (L, R) such that β |L = df. Proposition 4.4 Let L be an exact Lagrangian submanifold of (W, dβ), an exact symplectic manifold. Then there exists Legendrian submanifold Λ of the contact manifold (R × W, ker(dt + β)) such that Λ projects on L under the natural projection. Proof Define Λ = {(g(p), p) ∈ R × W | p ∈ L} for some g ∈ C ∞ (L, R). We show that dgp + βp |L = 0 for all p ∈ L, and equivalently Λ is Legendrian, i.e.,

4.6 Legendrian Submanifolds

139

Tp Λ ⊆ ξp = ker(αp ). We can choose g = −f + c for any c ∈ R. Note that L is an exact Lagrangian submanifold, so there exists f ∈ C ∞ (L, R) such that β |L = df. Therefore we have −dfp + βp |L = 0 or dgp + βp |L = 0. ⨆ ⨅ Example 4.16 Consider w = T ∗ L the cotangent bundle of L. Let β ∈ Ω1 (L), i.e., β is a section of π : T ∗ L −→ L. Let L˜ β be the image of the section corresponding to β, i.e., L˜ β = {(p, βp ) ∈ T ∗ L | p ∈ L} ∼ = L.

.

In fact L˜ β is diffeomorphic to L via natural projection. L˜ β is Lagrangian when ω |L˜ β = 0. On the other hand λcan |L˜ β = π ∗ β |L˜ β , equivalently ω |L˜ β = π ∗ dβ |L˜ β . Therefore, equivalently we have π ∗ dβ |L˜ β = 0, and this is equivalent to dβ = 0. Hence, we showed that L˜ β being Lagrangian is equivalent to dβ = 0. L˜ β is an exact Lagrangian when equivalently β is exact on L. (Note that π is a diffeomorphism, so the exactness on L˜ β is equivalent to the exactness on L.) So equivalently β = df with f ∈ C ∞ (L, R). We write in this case L˜ β = L˜ f = {(p, dfp ) ∈ T ∗ L | p ∈ L}.

.

Now we use the contactization to get the contact manifold (M = J 1 (L) = T ∗ LR, ξ = ker(dz − λcan )).

.

Above of L˜ f , we have a Legendrian submanifold Λf = {(p, dfp , f (p)) ∈ J 1 (L) = T ∗ × R | p ∈ L} = image of j 1 (f ).

.

Definition 4.15 In particular case, when L = Rn with coordinates x1 , . . . , xn , we have T ∗ L with the coordinates x1 , . . . , xn , y1 , . . . , yn and J 1 (L) ∑ with coordinates x1 , . . . , xn , y1 , . . . , yn , z. The contact form on J 1 (Rn ) is dz − ni=1 yi dxi . We can project our Legendrian submanifolds by the front projection σ : J 1 (L) = T ∗ L × R −→ L × R,

.

and we have σ (Λf ) = σ (p, dfp , f (p))

.

= (p, f (p)) = graph of f = j 0 (f ).

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4 Contact Manifolds

Fig. 4.6 Front projection of the curve c onto the curve c' in the xz-plane

In low dimension when n = 1, we have J 1 (R) = R3 and α = dz − ydx. Therefore, the front projection is σ : R3 −→ R2

.

(x, y, z) ‫׀‬−→ (x, z), which is projecting to the xz-plane. Note that in the front projection we never get vertical tangent direction on diagrams in xz-plane, but we have cusp points, as it is shown in Fig. 4.6. The reason is that for the contact form α we have ξ = ker(α), and in other words, α is zero on the standard contact plane ξ. Hence, for a Legendrian submanifold that is tangent to dz ξ , we have the condition dz − ydx = 0, so y = dx . If we have a vertical direction dz in xz-plane, it means that dx → +∞, which means that y → +∞, and this is in contradiction for embedded Legendrian submanifolds. Definition 4.16 The other projection is the Lagrangian projection that is projecting to the xy-plane as it is shown in Fig. 4.7. π : R3 −→ R2

.

(x, y, z) ‫׀‬−→ (x, y).

4.6 Legendrian Submanifolds

141

Fig. 4.7 Lagrangian projection of the curve c onto the curve c' in the xy-plane

Theorem 4.5 (Legendrian Neighborhood Theorem) Let L be a Legendrian submanifold of (M, ξ ). Then there exist contactomorphic neighborhoods U of L and V of the 0-section of the 1-jet space of L. Proof Let α be a contact form for ξ. For all p ∈ L we have Tp L ⊂ ξp . Let Vp = ξp /Tp L, which is n-dimensional. The mapping Tp L −→ Vp∗

.

v− ‫ → ׀‬iv dαp is an isomorphism. Hence, Vp ∼ = Tp∗ L. Therefore, ξp ∼ = Tp L ⊕ Tp∗ L, and naturally, ∗ ∼ we have Tp M = R ⊕ Tp L ⊕ Tp L. Here the first component, i.e., R, represents the normal direction that is globally defined. Therefore we have T M |L ∼ = R ⊕ T L ⊕ T ∗ L.

.

Therefore, α and dα are naturally identified along L ⊂ (M, ξ ) and 0-section in J 1 (L). So there exists a local diffeomorphism ψ : V ⊂ J 1 (L) −→ U ⊂ M, where V is a neighborhood of the 0-section of J 1 (L) with the contact form on J 1 (L) as α = dz − ydx and U is a neighborhood of L with the contact form α0 for ξ. Consider contact forms α0 and α1 on U such that α0 = α1 and dα0 = dα1 along L. Consider a 1-parameter family of one forms αs = sα1 + (1 − s)α0 ,

.

s ∈ [0, 1].

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4 Contact Manifolds

Note that on L we have αs = α0 and dαs = dα0 . So αs ∧ (dαs )∧n /= 0 along L. By continuity we can extend it to a neighborhood of L, so near L all one-forms αS are contact forms. We want to find a 1-parameter family of diffeomorphisms ψ˜ s of U such that ∗ ˜ ψs αs = fs α0 . Using Moser trick, we obtain Xs and gs for this equation. Note that Xs = 0 along L. So ψ˜ s is defined up to time 1 near L (for example on U ). Now we look at the composition ψ˜ 1 ◦ ψ : V −→ U.

.

This is a local contactomorphism.

⨆ ⨅

From Legendrian neighborhood theorem we conclude that we cannot restrict to a neighborhood of L in order to find a Legendrian invariant. In other words, if we have another Legendrian L' in the same neighborhood U of L, then the neighborhoods V ⊂ J 1 (L) and V ' ⊂ J 1 (L' ) are contactomorphic and therefore L ∼ = L' . Definition 4.17 A Legendrian isotopy is a smooth mapping j : L × [0, 1] → M such that: • jt = j |L×{t} is an embedding for all t ∈ [0, 1]. • jt (L) = Lt is Legendrian in (M, ξ ) for all t ∈ [0, 1]. Definition 4.18 A contact isotopy of (M, ξ ) is a 1-parameter family of diffeomorphisms ψs of M, s ∈ [0, 1] such that (ψs )∗ ξ = ξ for all s ∈ [0, 1]. This means that if we deform our manifold, our contact structure does not change. Remark 4.2 Given such ψs , diffeomorphisms of M, and L ᶜ→ (M, ξ ) Legendrian submanifold, we obtain a Legendrian isotopy ψs (L), i = ψs |L : L × [0, 1] −→ (M, ξ ). So if we have contact isotopy, then we have Legendrian isotopy. The converse is described in the following theorem. Theorem 4.6 (Isotopy Extension Theorem) If j : L × [0, 1] −→ (M, ξ ) is a Legendrian isotopy, then there exists contact isotopy ψt of (M, ξ ) such that jt = ψt ◦ j0 , for all t ∈ [0, 1]. Proof Given Legendrian embedding j0 : L −→ (M, ξ ), we can find extension j˜0 : V0 ⊂ J 1 (L) −→ U ⊂ (M, ξ )

.

(by neighborhood theorem), which is contactomorphism, i.e., it preserves contact structures. Starting with jt , we get the extension j˜t : Vt × [0, 1] −→ (M, ψ). We define vector fields Xt by d Xt ◦ j˜t = j˜t , dt

.

which is only defined near L. We can always extend Xt to M (by cut off and Moser trick). We can integrate Xt , because Xt = 0; therefore ψt = id outside of our

4.7 Geometric Optic

143

neighborhood. Near L we can integrate Xt to j˜t . So we have a 1-parameter family of diffeomorphisms ψt of M such that ψt = j˜t near L. Let (ψt )∗ ξ /= ξ in the neighborhood of L. Consider ξt = (ψt )∗ ξ as a 1-parameter family of contact structures on M. We apply Gray stability theorem (integrate vector field near L) to get a 1-parameter family ϕt of diffeomorphisms of M so that ϕt ◦ ψt are contactomorphisms. ⨆ ⨅

4.7 Geometric Optic In the previous chapter, we talked about classical mechanics of a physical system, in particular, Hamiltonian formalism that is the origin of symplectic geometry. Here we are going to explain geometric optic as a motivation for contact geometry. Recall that in symplectic geometry, we consider a physical system with configuration space Q (in local coordinates we consider q1 , . . . , qn ) and with phase space T ∗ Q, (in local coordinates we consider q1 , . . . , qn , p1 , . . . , pn ). The phase space T ∗ Q is equipped with the symplectic form ω = dλcan =

n ∑

.

dqi ∧ dpi .

i=1

Moreover, we saw that to a Hamiltonian function H ∈ C ∞ (T ∗ Q, R) we associate a Hamiltonian vector field XH ∈ χ (T ∗ (Q)) determined by iXH ω = −dH. In coordinates this equation is written as XH =

.

n ∑ ∂H ∂ ∂H ∂ − . ∂pi ∂qi ∂qi ∂pi i=1

Hence the integral curves qi' =

.

∂H ∂pi

pi' = −

∂H ∂qi

form Hamiltonian equations. On the other hand we saw that the physical evolution, i.e., flow of XH , preserves ω if and only if LXH ω = 0. Remember from the previous chapter where we talked about geodesic flow. Definition 4.19 Let M be a manifold with a Riemannian metric g. The unique vector field G on the tangent bundle T G with trajectories of the form t ‫ →׀‬γ ' (t) ∈ Tγ (t) M ⊂ T M (where γ is a geodesic on M, not necessarily of unit speed) is called geodesic field, and its (local) flow is called the geodesic flow.

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4 Contact Manifolds

If (M, g) is a complete Riemannian manifold, the geodesic flow can be defined globally, i.e., for all times. Using the Riemannian metric g on M, one can define a bundle isomorphism Ψ from the tangent bundle T M to the cotangent bundle T ∗ M, which is fiber-wise given by Ψp : Tp M −→ Tp∗ M

.

X ‫׀‬−→ gp (X, −) for p ∈ M. This mapping induces a bundle metric g ∗ on T ∗ M such that gp∗ (u1 , u2 ) = gp (Ψp−1 (u1 ), Ψp−1 (u2 )),

.

where u1 , u2 ∈ Tp∗ M. Definition 4.20 The unit tangent bundle ST M fiber-wise is defined by STP M = {X ∈ Tp M, gp (X, X) = 1}.

.

Similarly, the unit cotangent bundle ST ∗ B is defined as ˜ X) ˜ = 1}. STP∗ M = {X˜ ∈ Tp∗ M, gp∗ (X,

.

Given a geodesic γ on M, the length of its tangent vector γ ' is constant, because = 0. Therefore, the flow of the geodesic vector field G preserves the unit tangent bundle ST M, and along ST M the vector field G is in fact tangent to ST M. Hence, we can consider the geodesic flow on ST M. d ' ' dt g(γ , γ )

Theorem 4.7 ([20]) Let (M, g) be a Riemannian manifold. (i) The Liouville form λ on the cotangent bundle T ∗ M induces a contact form on the unit cotangent bundle ST ∗ M. The Reeb vector field Rλ of this contact form is dual to the geodesic vector field G, i.e., T Ψ (G) = Rλ . (ii) Let H : T ∗ M −→ R be the Hamiltonian function defined by H (u) =

.

1 ∗ g (u, u). 2

Then along ST ∗ M = H −1 (2), the Reeb vector field Rλ is equal to the Hamiltonian vector field XH with respect to the symplectic form ω = dλ on T ∗ M. Definition 4.21 The flow of T Ψ (G) on ST ∗ M is called the cogeodesic flow. By theorem, the cogeodesic flow is equivalent to the Reeb flow of Rλ and is equivalent to the Hamiltonian flow of H on ST ∗ M, i.e., the flow of XH . Let the speed of light in a given medium be equal to nc where n is refraction index greater or equal to 1. We consider the contact manifold as the unit cotangent bundle

4.7 Geometric Optic

145

ST ∗ R3 . The configuration space is ST ∗ R3 , which is a hypersurface in T ∗ R3 , so (x, y) ∈ ST ∗ R3 with x ∈ R3 , y ∈ S 2 ⊂ R3 . Consider the identification ψ : ST R3 −→ ST ∗ R3

.

(x, y) ‫׀‬−→ (x, R3 ). By linearity it follows that ψ is a diffeomorphism and in fact contactomorphic. Since ST R3 and ST ∗ R3 are contactomorphic, so we have the contact form α := ψ ∗ λcan

.

∑ on ST R3 , where λcan = 3i=1 yi dxi in the canonical contact form on ST ∗ R3 . Now, a question one can ask is that “which path will follow a light ray?.” To answer this question we need to compute the travel time between two points, minimizing the time to travel from point A to point B (according to Fermat’s principal of least time). The curve γ : I ⊂ R −→ R3

.

t ‫׀‬−→ x(t) induces the curve .

γ˜ : I ⊂ R −→ ST R3 t ‫׀‬−→ (x(t),

x ' (t) ). ║ x ' (t) ║

The length of γ is equal to .

║ x ' (t) ║ dt =

║ γ ║=

γ˜

γ

λcan

because λcan (γ˜ ' ) =

3 ∑

.

i=1

xi' (t) · x ' (t) =║ x ' (t) ║ . ║ x ' (t) ║ i

Now, with the speed of light equal to

c n

the elapsed time is equal to

t=

.

γ˜

n λcan c

because from x = vt we have t = xv ; therefore t =

λcan c n

.

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4 Contact Manifolds

Set c = 1 and α := nλcan , where n ∈ C ∞ (ST R3 , R) is the index of refraction. So the elapsed time t is the action functional of the curve γ˜ defined by S : C ∞ (I, ST R3 ) −→ R α. S(γ˜ ) :=

.

γ˜

Consider a 1-parameter family γ˜s such that γ˜0 = γ˜ . We want to find the minimum for S that is time; hence we set .

d S(γ˜s ) |s=0 = 0. ds

d We have vector field X = ds γ˜s . So we have the flow ϕX which we can extend it on entire manifold or a neighborhood. Note that approximately we have γ˜s ≈ (ϕX )s ◦ γ˜ at the first order and s = 0.

.

d d S(γ˜s ) |s=0 = |s=0 ds ds d |s=0 = ds LX α = = = =



γ˜s =(ϕX )s ◦γ˜

γ˜

α

ϕs∗ α

γ˜

γ˜

γ˜

γ˜

iX dα + d(iX α) iX dα + 0 iX dα.

Consider fixed endpoints A and B for γ˜s ; therefore X = 0 at A and B. d Therefore ds S(γ˜s ) |s=0 = 0 for all γ˜s is equivalent to iX dα = 0 or dα(X, γ˜ ' ) = 0 d at any point for all X. Hence, equivalently for X = ds γ˜s = γ˜ ' , we obtain i(γ˜ )' dα = 0

.

at any point. On the other hand, α(γ˜ ' ) = nλcan (γ˜ ' )

.

= n ║ γ˜ ║ = 1,

4.7 Geometric Optic

147

since ║ γ˜ ║= nc and c = 1. Therefore, γ˜ ' is exactly the Reeb vector field of α. Remember that the flow of Rα preserves the contact form α. In fact, we interpreted the Riemannian (M, g) as a model for an optical medium, so that geodesics with respect to the metric g correspond to light rays. Given a metric g on M, Fermat principle says light travels along geodesics. Then using g to identify the geodesic flow with the Reeb flow, we saw above that light will travel along trajectories of the Reeb vector field. Definition 4.22 The wave front Fp0 (t) of a point p0 ∈ M at time t is the set of points p ∈ M such that there exists a unit speed geodesic γ in M with γ (0) = p0 and γ (t) = p. The wave front Fp0 (t) is a smooth hypersurface in M, for | t | small. If | t | increases, then Fp0 (t), in general, develops singularities. Consider ST M with its natural contact structure, and write π : ST M −→ M for the bundle projection that is called (wave) front projection. Every fiber π −1 (p), p ∈ M is a Legendrian submanifold of ST M. The Legendrian fiber π −1 (p) of the bundle π : ST M −→ M consists of the pairs (p, v), where v is a unit tangent vector at p. If we denote the geodesic flow on ST M by Фt , the front Fp (t) can be written as Fp (t) = π(Фt (π −1 (b))).

.

The submanifold Фt (π −1 (p)) ⊂ ST M is a Legendrian submanifold, since Фt is a contactomorphism of ST M for each t. Let the source of light be at point x ∈ R3 , which is switched on at the time t = 0, see Fig. 4.8. Now the question is that “what is the shape for the surface of the light front?” In ST R3 , the light front at t = 0 is given by STx R3 ∼ = S 2 ⊂ ST R3 ; in other words, at t = 0, the initial surface L0 is the fiber at x. Note that the dimension of T R3 is equal to 6, so the hypersurface ST R3 has dimension equal to 5, and L0 is of dimension 2.

Fig. 4.8 Smooth wave front and wave front with singularity

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4 Contact Manifolds

If this surface starts moving, what is the equation of motion? At time t > 0, the light front is given by ϕtRα (L0 ) = Lt .

.

By mapping π : ST R3 −→ R3 , we get the light front π(Lt ) in R3 . Note that λcan |L0 = 0, i.e., L0 is Legendrian submanifold, because λcan is a contact form and R L0 ∼ = S 2 ⊂ ξ. Since ϕt α preserves the contact form, hence Lt ’s are Legendrian submanifolds for all t ≥ 0.

4.8 Characteristic Foliation Let (M, ξ ) be a contact manifold with dimension equal to 3. Let ∑ be a compact surface (which may have boundary) embedded in M. In the following we are going to see how does ξ behave near ∑. Definition 4.23 A distribution Δ (with singularities) on the surface ∑ is defined, at each point p ∈ M, by Δp =

.

⎛ Tp ∑ ∩ ξ p

when this intersection has dimension 1 if Tp ∑ = ξp

0

.

Recall that a one-dimensional distribution is integrable, so we can always integrate Δ to a singular foliation in which the dimensions of the leaves are one or zero, and we denote this foliation by ξ ∑. Definition 4.24 The characteristic foliation ξ ∑ of a surface ∑ induced by the contact structure ξ in (M, ξ ) is the singular one-dimensional foliation of ∑ defined by the distribution Δ. Example 4.17 Let ∑ be the round sphere in R3 around origin. Consider the singular foliation for ∑ as it is shown in the left side of Fig. 4.9. This foliation has two degenerate singularities in the south pole and north pole. By perturbation we get a singular foliation depicted in the right side of Fig. 4.9. This foliation has two non-degenerate singularities in the south pole and north pole. This foliation represents the characteristic foliation of ∑ induced by the standard contact structure ξ = ker(dz + 12 (xdy − ydx) in R3 . The two singular points are (0, 0, −1) and (0, 0, +1). The characteristic foliation is spanned by (xz − y)

.

∂ ∂ ∂ + (yz + x) − (x 2 + y 2 ) . ∂x ∂y ∂z

4.8 Characteristic Foliation

149

Fig. 4.9 Characteristic foliation of sphere

Theorem 4.8 (Giroux, [22]) Let ∑ be an embedded compact surface in M. Let ξ0 and ξ1 be contact structures on M, such that ξ0 ∑ = ξ1 ∑ (up to diffeomorphism, i.e., for a diffeomorphism ψ on ∑, we have ψ(ξ0 ∑) = ξ1 ∑). Then there exists a 1-parameter family of diffeomorphisms ψs of M such that ψ0 = id, (ψ1 )∗ ξ0 = ξ1 , and ψs (∑) = ∑, ψs (∂∑) = ∂∑. Moreover, (ψs )∗ ξ0 all have the same characteristic foliation. Proof A neighborhood of ∑ has the form ∑ × (−ε, ε) ⊂ ∑ × R. We can use local coordinates (x, y) on ∑ and t on R. On ∑ × R a contact form looks like α = ut dt + βt with ut ∈ C ∞ (∑), βt ∈ Ω1 (∑). Hence, we consider the contact form for ξ0 as α0 = ut,0 dt + βt,0

.

and the contact form for ξ1 as α1 = ut,1 dt + βt,1 .

.

Note that on ∑, i.e., at t = 0, we have α0 = β0,0 and α1 = β0,1 . By assumption the characteristic foliation of ξ0 and ξ1 are the same, i.e., ξ0 ∑ = ξ1 ∑. Therefore ker(β0,0 ) = ker(β0,1 ), i.e., ker(α0 ) = ker(α1 ). Up to a multiplication of α1 (by a smooth function f ), we get β0,0 = β0,1 . Now, set αs = sα1 + (1 − s)α0 ,

.

ut,s = sut,1 + (1 − s)ut,0 , βt,s = sβt,1 + (1 − s)βt,0 .

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4 Contact Manifolds

Therefore, we have αs ∧ dαs = (ut,s dt + βt,s ) ∧ (dut,s ∧ dt + dβt,s + dt ∧

.

dβt,s ). dt

So, on ∑, i.e., when t = 0, we have dβ0,s ) dt dβ0,s = u0,s dt ∧ dβ0,0 + β0,0 ∧ dut,s ∧ dt + β0,0 ∧ dt ∧ dt = s(α1 ∧ dα1 ) + (1 − s)(α0 ∧ dα0 ) /= 0.

αs ∧ dαs = (u0,s dt + β0,0 ) ∧ (dut,s ∧ dt + dβ0,0 + dt ∧

.

Thus, αs is a contact form near ∑. Now, to obtain ψs , we use Moser trick to find Xs and gs . Note that Xs is obtained by αs (Xs ) = 0

.

iXs dαs =

−dαs ds

on ξ.

s Let v ∈ ξs ∩ T ∑ = Δ; then dαs (X, v) = −dα ds (v) = 0. Since dαs is symplectic form and it is non-degenerate, Xs is linearly dependent on v, i.e., Xs ∈ Δ (because v ∈ Δ). In particular, Xs is tangent to ∑ (Xs ∈ Δ = ξs ∩T ∑). So for the compact ∑, we can integrate Xs globally, so flow of Xs on ∑ is defined for all s ∈ [0, 1]. So we can integrate to ψs and ψs (∑) = ∑. Moreover, ψs preserves Δ, i.e., (ψs )∗ Δ = Δ, which is equivalent to the fact that ξs ∑ is independent of s. Note that we are in R3 and ∂∑ is one-dimensional, and we know that every curve is Legendrian, i.e., T ∂∑ ⊂ ξ. If ∂∑ is Legendrian, then ∂∑ is a leaf of ξ ∑ or union of leaves. Hence, ψs (∂∑) = ∂∑. ⨆ ⨅

If ∂∑ /= ∅, then ∂∑ is Legendrian curve. Therefore, ∂∑ is the union of leaves (and singularities) of the characteristic foliation ξ ∑. In particular, if X (vector field) is tangent to ξ ∑, the flow of X never leaves ∑. Example 4.18 Let ∑ = D 2 be the embedded disc in (R3 , ξ0 = ker(dz − ydx)), such that its boundary is Legendrian curve. In Lagrangian projection, i.e., in xyplane, the disc with the characteristic foliation induced by standard contact structure is depicted as in the left side of Fig. 4.10. In R3 , we obtain the embedded disc shown in the right side of Fig. 4.10. Then the description of the characteristic foliation of ξ ∑ is as in the right side of Fig. 4.10. Note that the boundary has two leaves and two singular points. These two singular points coincide on the origin in the Lagrangian projection.

4.8 Characteristic Foliation

151

Fig. 4.10 Characteristic foliation of disc induced by standard contact structure Fig. 4.11 Overtwisted disc-characteristic foliation of disc induced by overtwisted contact structure

Fig. 4.12 A leaf inside of overtwisted disc

Definition 4.25 An overtwisted disc D in (M, ξ ) is an embedded disc in M such that its characteristic foliation ξ D looks like the disc in Fig. 4.11. Note that the overtwisted disc, as it is shown in Fig. 4.11, has only one singularity at the center and there is no singularity on the boundary. In fact, any (non-singular) leaf inside of the overtwisted disc in Fig. 4.11 is a non-compact leaf as it is shown in Fig. 4.12. Thus, the embedded overtwisted disc in Lagrangian projection has the picture as in Fig. 4.13. We know that by Darboux’s theorem all contact manifolds are locally the same, so locally the characteristic foliation is induced by the standard contact structure. In Fig. 4.13 one can see this fact that in a very small neighborhood near the center of the disc, the characteristic

152

4 Contact Manifolds

Fig. 4.13 Overtwisted disc in Lagrangian projection

foliation is as the left side of Fig. 4.10 that shows the characteristic foliation induced by the standard contact structure. Example 4.19 Remember from Example 4.12 the overtwisted contact structure on R3 with the contact form α = cos(2π r)dz + sin(2π r)dθ

.

in cylindrical coordinates (r, θ, z). Now consider disc D 2 in (R3 , ξot = ker(αot )). For any p ∈ ∂D, we have Tp ∂D = ξp . Therefore there is no singularity on the boundary of the disc. The characteristic foliation ξot D induced by ξot is shown as in Fig. 4.11.

4.9 Legendrian Knots in Contact 3-Manifolds Before talking about Legendrian knots, let us recall some basic definitions of knots in R3 . Legendrian knots and (topological) knots are defined in higher dimensions as well; however, in this section, we focus on three-dimension, which gives us images of knots; hence it will be easier to generalize the facts to higher dimensions. Definition 4.26 A (topological) knot in R3 is an embedding K of the circle in R3 , i.e., K : S 1 −→ R3 .

.

Definition 4.27 Two knots are equivalent if there exits an isotopy, i.e., a smooth mapping F : S 1 × [0, 1] −→ R3

.

such that F |S 1 ×{0} = K0 , F |S 1 ×{1} = K1 and F |S 1 ×{t} is an embedding. To visualize a knot K on paper, i.e., in two-dimension, we project K to R2 along a generic projection, so that the projection is an immersion with at most double points. A knot diagram is equipped with crossing information such as under-crossing or over-crossing, see Fig. 4.14.

4.9 Legendrian Knots in Contact 3-Manifolds

153

Fig. 4.14 Crossings of a knot

Fig. 4.15 Equivalent diagrams of unknot

Fig. 4.16 A disc and a torus with a disc removed as two different Seifert surfaces of the unknot

An equivalence class of a knot gives rise to many different diagrams. For instance for the unknot, i.e., {x 2 + y 2 = 1, z = 0} ⊂ R3 , we have different equivalent diagrams as in Fig. 4.15. Table of knots is a list of diagrams by increasing the number of crossings, and we choose minimum for a given equivalent class of knots. Definition 4.28 A Seifert surface for a knot K is an orientable compact surface ∑ embedded in R3 such that the boundary of ∑ is K, i.e., ∂∑ = K. Theorem 4.9 (Frankl and Pontryagin, [19]) Any knot K admits a Seifert surface. In fact a Seifert surface associated to a knot K is not unique. As a simple observation, see Fig. 4.16, which shows two Seifert surfaces for the unknot. Definition 4.29 The genus of a knot K is the minimal genus for a Seifert surface of K. For instance, the genus of unknot is zero, since unknot can be considered as the boundary of disc. Conversely, if the genus of a knot is zero, we conclude that it is in fact an unknot. As another example, the genus of a knot as trefoil is equal to one. Let M = R3 be equipped with the standard contact structure ξst := ker(dz − ydx). The closed Legendrian submanifolds in R3 are knots or links. Definition 4.30 A Legendrian knot is a topological knot such that it is tangent to the contact structure.

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4 Contact Manifolds

Lemma 4.6 The algebraic area surrounded by a closed Legendrian (immersed) curve is equal to zero. Proof Note that in Lagrangian projection, the area of each region enclosed by a Legendrian curve is with a sign induced by the orientation of the curve. By algebraic area we mean the algebraic sum of such areas with their sign. Consider R2 with the symplectic form ω = −dλcan = dx ∧ dy, where the canonical form is λcan = ydx. Any curve in R2 is Lagrangian, since dimension of a curve is one so it is isotropic, i.e., dx ∧ dy | curve = 0. Moreover, a closed curve in R2 is an exact Lagrangian, since we have α = dz − ydx = 0; therefore λcan | curve = ydx | curve = dz | curve . Hence, the one-form ydx is exact on

a closed 1-manifold. From De Rham cohomology, we remember that we have c ydx = 0, where c is a closed curve. Now we have 0=

ydx

.

c



=−

dx ∧ dy D

= −area(D), where D is a domain bounded by c. We conclude that, in general, immersed curves enclosing zero algebraic area. ⨆ ⨅ Example 4.20 By the fact indicated in the above lemma, the simplest diagram to consider for a Legendrian unknot is depicted in the following diagram, in Fig. 4.17. As we see in the figure, there are two domains bounded by the (immersed) unknot, which have the same area but with different signs according to the orientation of the unknot. The following theorem indicates that we can always approximate a smooth curve with a Legendrian curve in R3 with the standard contact structure. Fig. 4.17 Simple Legendrian unknot

4.9 Legendrian Knots in Contact 3-Manifolds

155

Fig. 4.18 Legendrian C 0 -approximation in front projection

Theorem 4.10 (Legendrian Approximation Theorem) Let γ1 : [0, 1] → (R3 , ξst ) be a smooth curve in the contact manifold (R3 , ξst ); then there exists a Legendrian curve γ2 : [0, 1] → (R3 , ξst ), which is C 0 -close to γ1 , i.e., ║ γ1 (t) − γ2 (t) ║< ε for all t ∈ [0, 1] and some ε > 0. Proof Let γ be a curve in standard R3 . We consider the front projection σ (γ ) that is a regular curve without vertical tangencies and with isolated cusps, see Fig. 4.18. Then Legendrian lift of this front is a C 0 -approximation of γ . In the case that γ is defined on a closed interval, we can choose the slope of the front at the endpoints so that the Legendrian approximation of γ has the same endpoints. If γ itself is Legendrian near its endpoints, then we assume that the Legendrian approximation coincides with γ near the endpoints. Hence, its Legendrian lift coincides with γ near endpoints. Note that our Legendrian approximating curve does not have any selfintersection. We approximate by a front with only transverse self-intersections. Since we assume that the curve γ is embedded, its Legendrian C 0 -approximation does not have any self-intersection neither. Moreover, during the above procedure, we avoid any non-trivial knotting so that the Legendrian C 0 -approximation is topologically isotopic to the original curve. Using Lagrangian projection, one gives another proof. Similarly, we consider γ in standard R3 . To find the Legendrian C 0 approximation of γ , we approximate its Lagrangian projection by an immersed curve whose area integral (the area under the curve) is close to the area integral of the original curve. We can obtain such a curve by considering small loops oriented positively or negatively, see Fig. 4.19. If the Lagrangian projection has self-intersection, we can choose this approximating curve so that along loops the area integral is non-zero, in such a way that we do not have any self-intersection in the Legendrian approximation of γ . ⨆ ⨅

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4 Contact Manifolds

Fig. 4.19 Legendrian C 0 -approximation in Lagrangian projection

Thanks to the Legendrian approximation theorem, any topological knot can be represented by a Legendrian knot. In fact, in [20] it is mentioned that there is a similar statement in higher dimension. Definition 4.31 Two Legendrian knots are equivalent if they are isotopic through Legendrian knots. A given topological knot gives rise to several (in fact, infinitely many) equivalence classes of Legendrian knots. Hence, classifying Legendrian knots modulo Legendrian isotopy (deformation of a Legendrian knot through Legendrian knots) can be seen as a refinement of the classification for topological knots, i.e., a topological knot type splits into many Legendrian knot types. The Thurston–Bennequin invariant and the rotation number are the two simplest Legendrian invariants for Legendrian knots in dimension 3. To define Thurston–Bennequin invariant, for a knot K, fix a Seifert surface that is an orientable surface ∑ embedded in (R3 , ξst ) such that ∂∑ = K. Definition 4.32 The Seifert framing of K is the trivialization of the normal bundle N K corresponding to the parallel curve (a longitude on the boundary of a tubular neighborhood of K) obtained by pushing K along ∑. The contact framing of K is the trivialization of the normal bundle N K corresponding to the parallel curve obtained by pushing K in a direction transverse to ξst . Definition 4.33 The twisting of the contact framing of a Legendrian knot K with respect to the Seifert framing in a contact 3-manifold (R3 , ξst ) is called the Thurston–Bennequin invariant, with right-handed twists being counted positively. We denote this invariant by tb(K). In fact if we choose a vector field along K transverse to ξst and define a parallel knot K ' by pushing K along this vector field, then tb(K) equals the linking number of K and K ' that is the number (with sign) of times that K intersects the Seifert surface of K ' . This number is independent of the choice of ∑, see [23].

4.9 Legendrian Knots in Contact 3-Manifolds

157

Fig. 4.20 Computing the Thurston–Bennequin invariant of the simple Legendrian unknot

Example 4.21 Let K be a Legendrian unknot, with the Siefert surface ∑ as twisted disc. As it is shown in Fig. 4.20, the orientation of ∑ on the left side is counterclockwise and the orientation on the right side is clockwise; moreover, the ∂ ∂ ∂ ∂ orientation on K ' is + ∂z . Therefore we have T K ' ⊕ T ∑ = < ∂z , ∂y , ∂x >. Reversing the orientation, we obtain T K' ⊕ T ∑ =
. ∂x ∂y ∂z

Therefore tb(K) = −1. Remember that the standard contact structure lies in xy-plane and the z-axis is the direction transverse to it. Since K is Legendrian, it is tangent to the contact structure, and it is also in the xy-plane. In other words, K lies in this page, and to visualize K ' , look at this page vertically from above. So you can see that the linking number of K and K ' is −1. Or you can see that the Siefert surface of K ' intersects the Siefert surface of K once. Definition 4.34 Let γ : S 1 → K ⊂ M be a (regular) parametrization of K compatible with its orientation. Given a trivialization ξst |∑ = ∑ × R2 (as oriented bundle), this induces a map γ ' : S 1 → R2 \ {0}. The rotation number denoted by rot (L) is the degree of the map γ ' . In other words, rot (L) counts the number of rotations of the (positive) tangent vector to K relative to the trivialization of ξst |∑ as we go once around K. The Thurston–Bennequin number is independent of the choice of Seifert surface and the orientation of K. The rotation number is also independent of the choice of Seifert surface; however, it depends on the choice of orientation of K. Also, the sign of the rotation number depends on the choice of orientation for ξ, for more details, see [23]. Moreover, the rotation number rot (K) does not depend on the choice of trivialization of ξ |∑ , see [lemma 3.5.14, [20]].

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4 Contact Manifolds

Fig. 4.21 Crossing signs

The above definitions of the Thurston–Bennequin number and rotation number are generalized to any contact 3-manifold, see [20]. There is a recipe for computing the Thurston–Bennequin number and rotation number of a Legendrian knot in standard R3 from its Lagrangian projection or front projection. Definition 4.35 The writhe of a knot diagram is the sum of signs associated to the crossings of the diagram as it is shown in Fig. 4.21. Hence, the writhe of an oriented knot diagram is the signed number of selfcrossings of the diagram. Remark 4.3 Note that writhe is not a knot invariant. We leave it as an exercise to find examples of knots that have the same writhe but are not equivalent. Proposition 4.5 The Thurston–Bennequin invariant tb(K) of a Legendrian knot K in (R3 , ξst ) is equal to the writhe of its Lagrangian projection, i.e., tb(K) = writhe(π(K)).

.

Proof We consider the parallel copy K ' of K obtained by pushing K in a direction tangent to xist and transverse to K. Using Lagrangian projection, ξst projects isomorphically onto the (x, y)-plane. Therefore, K ' can be represented by a diagram in Lagrangian projection parallel to the diagram of K in Lagrangian projection in (x, y)-plane. Now the result follows from calculating the linking number of K with K ' . ⨆ ⨅ Example 4.22 Consider the following diagrams of two Legendrian unknots in Lagrangian projection, in Fig. 4.22. According to above proposition, the Thurston– Bennequin invariant of them is equal to the writhe of their Lagrangian projection, i.e., the sum of the signs of the crossings in diagrams. Therefore we obtain tb(K1 ) = writhe(π(K1 )) = −1

.

tb(K2 ) = writhe(π(K2 )) = −2.

.

Note that these are two equivalent topological unknots but are not Legendrian isotopic, since their Thurston–Bennequin invariant is different.

4.9 Legendrian Knots in Contact 3-Manifolds

159

Fig. 4.22 Two non-isotopic Legendrian unknots in Lagrangian projection Fig. 4.23 Simple Legendrian Unknot in front projection

Below, we see a formula to compute Thurston–Bennequin invariant in front projection. Proposition 4.6 Let K be a Legendrian knot in (R3 , ξst ). Then the Thurston– Bennequin invariant of K is given by 1 tb(K) = writhe(σ (K)) − ♯cusps(σ (K)), 2

.

where σ (K) is the front projection of K. Proof We fix an orientation for K. We construct K ' in (x, z)-plane by pushing K ∂ in the ∂z direction. Then we compute the linking number of K and K ' , i.e., we count the crossings of K ' with K. with sign. Note that a self-crossing of K will give a crossing of K ' under K of the same sign. Moreover, a cusp on the right gives a negative crossing of K ' under K, and a cusp on the left gives a crossing of K ' over K. Since the number of cusps on the left is the same as the number of the cusps on the right, we obtain the formula. ⨆ ⨅ Example 4.23 (1) Let K be the simple Legendrian unknot in front projection, as shown in Fig. 4.23. The Thurston–Bennequin invariant of K using the formula in above proposition is equal to 1 tb(K) = 0 − (2) = −1. 2

.

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4 Contact Manifolds

Fig. 4.24 A Legendrian unknot in front projection

Fig. 4.25 A Legendrian unknot in front projection

Here, note that there is no crossings, so the first term is zero. Moreover, there are two cusps. (2) Consider K1 as a Legendrian unknot shown in Fig. 4.24, in front projection. The Thurston–Bennequin invariant of K1 is equal to 1 tb(K) = 0 − (4) = −2. 2

.

Here, again we do not have any crossing, so the first terms are zero, and we have four cusps. (3) Let K2 be the Legendrian unknot in front projection, shown in Fig. 4.25. Here, there is no crossing and we have four cusps; therefore the Thurston–Bennequin invariant of K2 is equal to 1 tb(K) = 0 − (4) = −2. 2

.

As we saw, the Legendrian unknot K1 is not Legendrian isotopic to Legendrian unknots K1 and K2 . Although Legendrian unknots K1 and K2 have the same Thurston– Bennequin invariant, they are not Legendrian isotopic. Because they have different rotation numbers which we compute in below. In order to compute the rotation number of Legendrian knots in Lagrangian projection and front projection, we can use two formulas given in the two following propositions. For the proof of these propositions, see [20]. First we bring the statement in Lagrangian projection. Proposition 4.7 Let K be a Legendrian knot in (R3 , ξst ). Then the rotation number of K is equal to the rotation number of its diagram in Lagrangian projection, i.e.:

4.9 Legendrian Knots in Contact 3-Manifolds

161

Fig. 4.26 A Legendrian unknot in Lagrangian projection

Example 4.24 Consider the Legendrian unknot K shown in Fig. 4.26. The rotation number of this unknot, using the formula in the above proposition, is equal to rot (K) =

.

1 [2 + 2 − 0 − 0] = 2. 2

The formula to compute the rotation number in front projection is given as follows. Proposition 4.8 Let K be an oriented Legendrian knot in (R3 , ξst ). Then the rotation number of K is given by rot (K) =

.

1 (cd − cu ), 2

where cd is the number of cusps going down, and cu is the number of cusps going up in front projection. Example 4.25 (1) Let K be simple Legendrian unknot in Fig. 4.27. The rotation number of K using the formula in front projection is equal to rot (K) =

.

1 (1 − 1) = 0, 2

since in the diagram we have one cusp going down and one cusp going up. (2) Consider K1 as the Legendrian unknot shown in Fig. 4.28. The rotation number of K1 is equal to rot (K1 ) =

.

1 (3 − 1) = 1. 2

Here, we have three cusps going down and one cusp going up. (3) Now consider the Legendrian unknot K2 shown in Fig. 4.29, in front projection. The rotation number of K2 is equal to rot (K2 ) =

.

1 (1 − 3) = −1. 2

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4 Contact Manifolds

Fig. 4.27 Simple Legendrian unknot in front projection

Fig. 4.28 A Legendrian unknot in front projection

Fig. 4.29 A Legendrian unknot in front projection

Here, we have one cusp going down and three cusps going up. As we see in these examples, the simple unknot is not Legendrian isotopic to Legendrian unknots K1 and K2 . Since the rotation number of K is not the same as the rotation number of K1 and k2 . Moreover, we see that the rotation numbers of K1 and K2 are not the same, so K1 and K2 are not Legendrian isotopic, although they have the same Thurston– Bennequin invariant (as we saw before). As we saw above through some examples, classical invariants, i.e., Thurston– Bennequin invariant and rotation number, can distinguish different Legendrian unknots that are topologically the same. In particular, up to a Legendrian isotopy, there is an unique Legendrian knot L that is smoothly the unknot with (rot (K) = 0, tb(K) = −1). The following theorem shows that the Thurston–Bennequin invariant together with rotation number is a complete set of Legendrian invariants for unknots. Theorem 4.11 (Eliashberg-Fraser, [15]) Let K and K ' be two topologically trivial Legendrian knots in (R3 , ξst ). Then K and K ' are Legendrian isotopic if and only if their Thurston–Bennequin invariant and rotation number are equal, i.e., tb(K) = tb(K ' ) and rot (K) = rot (K ' ). The following Fig. 4.30 gives the image of the above theorem, in other words the complete classification of Legendrian unknots. Here, the horizontal axis represents rotation number and the vertical axis represents Thurston–Bennequin. Hence, as it is shown in the figure any Legendrian unknot is represented by a pair (rot, tb). Let us remark that Eliashberg and Fraser proved this theorem for topological trivial Legendrian knots in any “tight” contact 3-manifold.

4.9 Legendrian Knots in Contact 3-Manifolds

163

Fig. 4.30 Complete classification of Legendrian unknots

Definition 4.36 An embedded disc D ⊂ (M, ξ ) is an overtwisted disc in the contact 3-manifold (M, ξ ) if ∂D = K is a Legendrian knot with tb(K) = 0, i.e., if the contact framing of K coincides with the Seifert framing given by the disc D. The contact manifold (M, ξ ) is overtwisted if it contains an overtwisted disc; otherwise it is called tight. Up to now, we have been talking about Legendrian unknots. We can use classical invariants to classify Legendrian knots. However, classical invariants do not give complete classification for Legendrian knots. In other words, there are Legendrian knots with the same classical invariants, but which are not Legendrian isotopic. An example of this case was given by Chekanov and Eliashberg, as in Fig. 4.31. These are the first two Legendrian knots with the same Thurston–Bennequin invariant and rotation number and knot type that are not Legendrian isotopic [14]. In this direction, Chekanov has defined a new Legendrian invariant by constructing a differential graded algebra corresponding to the knot diagram, [8, 9]. This invariant distinguishes some Legendrian knots and their mirror versions. The Chekanov invariant is in fact a combinatorial version of a more general invariant, called Legendrian contact homology, defined using holomorphic curves. For a discussion of how the theory described by Chekanov fits into this more general invariant, see [18]. Exercise 4.9 Find examples of knots that have the same writhe but are not equivalent to conclude that writhe is not a knot invariant.

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4 Contact Manifolds

Fig. 4.31 Non-isotopic Legendrian knots with the same classical invariants

Exercise 4.10 Compute Thurston–Bennequin invariant and the rotation number of the Legendrian knots in Fig. 4.31.

4.10 Contact Surgery The construction of contact manifolds has been a basic problem in contact geometry. One can obtain a contact manifold by performing a contact surgery along an isotropic submanifold of a given contact manifold. Here we first describe topological surgery, and then we equip this description with contact data. Let M be an n-dimensional smooth manifold and S k ⊂ M be an embedded k-sphere with trivial normal bundle. This implies that we can find an embedded copy of S k × D n−k in M. The boundary of S k × D n−k in M can be regarded as S k ×S n−k−1 . On the other hand S k ×S n−k−1 can be seen as the boundary of D k+1 × S n−k−1 . In the following definition, we move out S k × D n−k from M and fill in its place by D k+1 × S n−k−1 . Definition 4.37 Given S k × D n−k ⊂ M, we can form a new manifold M ' := (M \ S k × I nt (D n−k )) ∪S k ×S n−k−1 (D k+1 × S n−k−1 )

.

by making the identification along S k × S n−k−1 = ∂(M \ S k × I nt (D n−k )) = ∂(D k+1 × S n−k−1 ).

.

The procedure of constructing M ' from M is called a surgery along S k ⊂ M. The fact that M ' is a manifold is a consequence of the collaring theorem for manifolds with boundary, see [20].

4.10 Contact Surgery

165

In the following we see another description of surgery in terms of handle attaching. Consider the cylinder [−1, 1] × M over M. Given an embedded copy of S k × D n−k ⊂ M ≡ {1} × M,

.

one can form W = ([−1, 1] × M) ∪S k ×D n−k (D k+1 × D n−k ).

.

Definition 4.38 We say that W is obtained from [−1, 1]×M by attaching a (k +1)handle (or handle of index k + 1) D k+1 × D n−k−1 to the boundary component {1} × M. In fact, W is not a manifold, and one needs to smooth the corners at S k ×S n−k−1 . The boundary of W is the disjoint union of M ≡ {−1} × M and the result M ' of performing surgery on M defined by S k × D n−k ⊂ M. In other words, W is a cobordism between M and M ' . Let H be the locus of points (x, y) ∈ Rk+1 × Rn−k satisfying the inequalities .

− 1 ║ y ║2 − ║ x ║2  1

and .

║ x ║ . ║ y ║< sinh 1. cosh 1.

This manifold H is not a handle in the strict sense, i.e., it is not diffeomorphic to D k+1 × D n−k , but it is diffeomorphic to a copy of D k+1 × D n−k with the corners cut off. We call the subset of H described by .

║ y ║2 − ║ x ║2 = −1

the lower boundary of H , and we denote it by ∂ − H (the green lines shown in Fig. 4.32). We call the subset described by .

║ y ║2 − ║ x ║2 = 1

the upper boundary of H , and we denote it by ∂ + H (the red lines shown in Fig. 4.32). When k  1, the lower boundary is connected, and when k = 0, it has two components, as it is shown in Fig. 4.32. Similarly, this statement holds for the k be the k-sphere along which we want to perform surgery. upper boundary. Let SH This k-sphere is shown in Fig. 4.32 in the lower boundary, visualized by the two points on the x-axis.

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4 Contact Manifolds

Fig. 4.32 The model handle H

Definition 4.39 The (k + 1)-disc D k+1 × {0} ⊂ H is called the core of the handle; the (n − k)-disc {0} × D n−k ⊂ H, the belt disc or cocore; the (n − k − 1)-sphere {0} × ∂D n−k ⊂ ∂H, the belt sphere. The handle H is used to describe the cobordism corresponding to a surgery on the n-manifold M, see [20]. Now we describe contact surgery using the attaching symplectic handles constructed by Weinstein, [55]. We perform surgery along an isotropic (k − 1)-sphere L with trivial conformal symplectic normal bundle (which we define next) in a given (2n − 1)-dimensional contact manifold M, where 1  k  n. The resulting manifold carries again a contact structure that coincides with the old one outside the neighborhood where surgery takes place. This surgery is called contact surgery by constructing the corresponding symplectic cobordism. Definition 4.40 Let L be an isotropic submanifold of the contact manifold (M, α). The quotient bundle CSN (M, L) := (T L)⊥ /T L with the conformal symplectic structure induced by dα is called the conformal symplectic normal bundle of L in M. So the normal bundle N(M, L) := (T M |L )/T L of L in M can be split as N (M, L) ∼ = (T M |L )/(ξ |L ) ⊕ (ξ |L )/(T L)⊥ ⊕ CSN (M, L).

.

The second bundle is naturally isomorphic to the cotangent bundle T ∗ L via the bundle isomorphism ψ : (ξ |L )/(T L)⊥ → T ∗ L

.

[Y ] ‫ →׀‬iY dα |T L ,

.

4.10 Contact Surgery

167

since, by the definition of (T L)⊥ , the bundle homomorphism ψ is well-defined and injective. As both bundles have rank, k − 1, ψ is an isomorphism. Therefore, if we have a trivialization of CSN (M, L) as a conformally symplectic vector bundle and a trivialization of (T M |L )/(ξ |L ) ⊕ T ∗ L, we get (up to a homotopy) a trivialization of N(M, L), also called a framing of L in M. If L is diffeomorphic to the standard sphere S k−1 ⊂ Rk , then the stabilized tangent bundle T S k−1 ⊕ ɛ has a natural trivialization, corresponding to the identification of the trivial line bundle ɛ with the normal bundle of S k−1 ⊂ Rk . This induces the natural trivialization of (T M |L )/(ξ |L ) ⊕ T ∗ L, and so we get a framing by specifying a trivialization of CSN (M, L). We call this framing of S k−1 the natural framing determined by the chosen trivialization of CSN (M, S k−1 ). Using this framing, we obtain a new manifold from M by surgery along L. The rank of CSN (M, S k−1 ) is 2(n − k) (for dimM = 2n − 1). Hence, contact surgery is always possible along a Legendrian sphere S n−1 ⊂ M, and the framing for the contact surgery is completely determined by the embedding of that sphere. The bundle (T M |L )/(ξ |L ) is a trivial line bundle because ξ is co-oriented. The Reeb vector field Rα induces a nowhere vanishing section of this quotient bundle, so it is convenient to identify (T M |L )/(ξ |L ) with the line subbundle ⊂ T M spanned by Rα . Moreover, it is proved in [20, Proposition 2.5.5] that the bundle (ξ |L )/(T L)⊥ is isomorphic to J (T L), where J is a complex bundle structure on ξ compatible with the symplectic bundle structure given by dα. Therefore, we can write the normal bundle N(M, L) as N(M, L) ∼ = ⊕ J (T L) ⊕ CSN (M, L).

.

Theorem 4.12 Let (Mi , ξi ), i = 0, 1 be contact manifolds with closed isotropic submanifolds Li . Suppose there is an isomorphism of conformal symplectic normal bundles Ф : CSN (M0 , L0 ) → CSN (M1 , L1 )

.

that covers a diffeomorphism φ : L0 → L1 .

.

Then this diffeomorphism φ extends to a contactomorphism ψ : N(L0 ) → N(L1 ) of suitable neighborhoods N(Li ) of Li such that the bundle maps T ψ |CSN (M0 ,L0 ) and Ф are bundle homotopic (as conformal symplectic bundle isomorphism). For the proof of this theorem see [20]. By this neighborhood theorem for isotropic submanifolds in a contact manifold, we can identify (by a contactomorphism) a neighborhood of an isotropic sphere in a given contact manifold with a neighborhood of an isotropic sphere in a model handle. The isotropic spheres in the model handle will have trivial conformal symplectic normal bundle, so the same

168

4 Contact Manifolds

condition will have to be imposed on any sphere along which we want to perform contact surgery. The symplectic cobordism between manifolds M and M ' can be written as the union of M × I (I is an interval), and a standard handle that is embedded in R2n and has the standard symplectic structure. M × I has a symplectic structure as part of the symplectization of M. Using a normal form for neighborhoods of isotropic submanifolds in contact manifolds, these two symplectic structures can be glued together on a neighborhood of the sphere where surgery takes place. Moreover, the boundary of the standard handle can be chosen transversal to a conformally symplectic vector field, so that M ' inherits a contact structure from the symplectic structure on the cobordism. In the following, we show the construction of the standard handle that will be glued to M × I to produce the cobordism. In the standard symplectic space R2n with coordinates (q1 , . . . , qk , qk+1 , . . . , qn , p1 , . . . , pn ) ∈ R2n = Rk × R2n−k

.

and the standard symplectic form ωst = X is given by X :=

.

∑n

j =1 dpj

∧ dqj , a Liouville vector field

n k ∑ 1 ∑ (−qj ∂qj + 2pj ∂pj ) + (qj ∂qj + pj ∂pj ), 2 j =k+1

j =1

which is the gradient vector field, with respect to the standard Euclidean metric on R2n , of the Morse function g : (q, p) ‫→׀‬

.

k n ∑ 1 1 ∑ 2 (− qj2 + pj2 ) + (qj + pj2 ). 2 4 j =1

j =k+1

∼ S k−1 × I nt (D 2n−k ) be an open neighborhood in the hypersurface Let NH = −1 g (−1) ⊂ R2n of the (k − 1)-sphere k−1 SH := {

k ∑

.

qj2 = 2, qk+1 = · · · = qn = p1 = · · · = pn = 0}.

j =1

This NH plays the role of the lower boundary. Definition 4.41 We define the symplectic handle H as the locus of points (q, p) ∈ (R2n , ωst ) satisfying the inequality −1  g(q, p)  1 and lying on a gradient flow line of g through a point of NH .

4.10 Contact Surgery

169

The Liouville vector field X is transverse to the level sets of g, so the one-form λst := iX ωst =

.

k n ∑ 1 ∑ (qj dpj + 2pj dqj ) + (−qj dpj + pj dqj ) 2 j =k+1

j =1

induces a contact form on the lower and upper boundaries of H. With respect to k−1 this contact form, SH is an isotropic sphere in the lower boundary. Its symplectic k−1 normal bundle SN∂H (SH ) is trivialized by the vector fields ∂qj , ∂pj , j = k + 1, . . . , n. Using the following lemma [20], we glue this model handle H symplectomorphically to M ×I so as to obtain a symplectic structure on the cobordism corresponding to the surgery. Lemma 4.7 Let M0 and M1 be hypersurfaces in the symplectic manifolds (W0 , ω0 ) and (W1 , ω1 ) transverse to Liouville vector fields X0 and X1 . Given a contactomorphism φ : (M0 , α0 ) → (M1 , α1 ),

.

extend it to a diffeomorphism φ˜ of a cylindrical neighborhood of M0 in W0 to a corresponding neighborhood of M1 in W1 by sending the flow lines of X0 to those of X1 . Then φ˜ is a symplectomorphism, i.e., φ˜ ∗ ω1 = ω0 . Performing a surgery along the isotopic sphere S k−1 ⊂ M by attaching the handle H corresponds to the framing given by the natural trivialization of ⊕ J (T S k−1 ) and an arbitrary choice of the trivialization of CSN (M, S k−1 ), which k−1 ). then determines a bundle isomorphism φ : CSN (M, S k−1 ) → CSN (∂H, SH The above discussion gives rise to the following theorem, see [20]. Theorem 4.13 (Contact Surgery Theorem) Let S k−1 be an isotropic sphere in a contact manifold (M, ξ = kerα) with a trivialization of the conformal symplectic normal bundle CSN (M, S k−1 ). Then there is a symplectic cobordism from (M, ξ ) to the manifold M ' obtained from M by surgery along S k−1 using the natural framing. In particular, the surged manifold M ' carries a contact structure that coincides with the one on M away from the surgery region. In comparison with the notation for topological surgery, here we attach a khandle to the symplectic manifold M 2n−1 × I along the isotropic sphere S k−1 . And the handle is diffeomorphic to D (k−1)+1 × D (2n−1)−(k−1) . Definition 4.42 For k = n, the contact surgery along the Legendrian sphere S n−1 is called Legendrian surgery. The handle is diffeomorphic to D n × D n , and the boundary of this handle is D n × S n−1 , which is diffeomorphic to the unit cotangent bundle of the n-disc, i.e., ST ∗ D n .

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4 Contact Manifolds

In other words, by Legendrian surgery we replace a tubular neighborhood of the Legendrian sphere by the unit cotangent bundle of the n-disc.

4.11 Contact Invariants The construction and classification of contact manifolds has been a fundamental problem in differential topology. As we saw in the previous section, contact surgery (see also [46, 55]) is a way to construct contact manifolds. For instance, in [12] it is shown that any closed connected contact 3-manifold can be obtained by contact surgery along a Legendrian link in (S 3 , ξst ). By Darboux’s theorem, there is no local invariant besides the dimension for contact manifolds. Therefore to classify contact manifolds, we should look for global invariants. Contact homology of contact manifolds, as a J -holomorphic invariant, is a part of Symplectic Field Theory [16] to classify contact manifolds. Moreover, Legendrian knots or links in contact manifolds and their classification enrich this theory. What comes in the following is a quick review of contact homology and its refined versions and then Legendrian contact homology for Legendrian knots and links. Consider a contact manifold as a (2n−1)-dimensional manifold Y equipped with a maximally non-integrable field of tangent hyperplanes ξ. Let λ be a one-form with ξ = ker(λ). The Reeb vector field Rλ of λ is the unique vector field that satisfies λ(Rλ ) = 1 and iRλ dλ = 0. For a generic λ, periodic Reeb orbits form a discrete set, see [5]. The algebra A(Y ) of Y is the super-commutative differential graded algebra (DGA) over the group ring R = Q[H2 (Y ; Z)] freely generated by the good Reeb orbits (see [5]). The grading of a Reeb orbit is obtained by its Conley–Zehnder index (see [47]), and the differential is defined by counting holomorphic curves in the symplectization of Y (see [5]). The latter one is the manifold R × Y equipped with the exact symplectic form d(et λ), t ∈ R. Definition 4.43 The homology of the differential graded algebra A(Y ) is the contact homology of Y denoted by HC(Y, ξ ). By the filtration of the algebra A(Y ) by the length of its words, we look at A1 l 1 in A = ⊕∞ l=0 A , i.e., we consider the submodule A of A/R linearly generated by words of length 1. A DGA homomorphism ε : A → R where R is equipped with the trivial differential is called an augmentation. An augmentation ε together with the differential d on A(Y ) induces a differential d ε on A(Y ), which respects the filtration. Definition 4.44 The linearized contact homology HC ε (Y, ξ ) is the homology of A1 with the induced differential on A1 . As we saw before, a Legendrian submanifold of Y is an (n−1)-submanifold Λ ⊂ Y that is tangent to ξ everywhere. Consider contact manifold Y with a Legendrian

4.12 Legendrian Surgery Exact Sequences

171

sphere Λ ⊂ Y. Performing contact surgery on Y along Λ, we obtain a new contact manifold YΛ and a symplectic cobordism X from Y to YΛ . Definition 4.45 Let Λ be a Legendrian submanifold. A Reeb chord on Λ is a flow line of Rλ that begins and ends on Λ. For a generic λ, Reeb chords are isolated, and the two endpoints of any Reeb chords are distinct. The algebra A(Y, Λ) of Λ ⊂ Y is the non-commutative differential graded algebra generated by Reeb orbits and Reeb chords over the group ring Q[H2 (Y, Λ; Z)]. A Reeb chord is graded by its Conley–Zehnder index. For more details, see, e.g., [13]. The differential is defined by counting holomorphic curves in R × Y with Lagrangian boundary condition R × Λ (see [38]). Definition 4.46 The homology of the differential graded algebra A(Y, Λ) is the Legendrian contact homology LHC(Λ) of Λ. Definition 4.47 If we restrict the Legendrian contact homology on Reeb chords, one corresponds a Legendrian homology algebra to the Legendrian submanifold. The corresponding algebra, LH A(Λ), carries the differential dLH A : LH A(Λ) → LH A(Λ), which satisfies the graded Leibniz rule d(ab) = (da)b + (−1)|a| a(db),

.

for arbitrary generators a and b.

4.12 Legendrian Surgery Exact Sequences An interesting part of the contact homology is the computation of the invariant for different examples. In [6], it is explained a cyclic version of Legendrian contact homology and the following theorem. 2 = 0 and the cyclic Legendrian homology, Theorem 4.14 ([6]) We have that dcyc cyc cyc LH (Λ) = H∗ (LH (Λ), dcyc ), is independent of all choices and is a Legendrian isotopy invariant of Λ.

Using Legendrian surgery, linearized contact homology, and cyclic Legendrian homology, the following surgery exact sequence is given. Theorem 4.15 (Bourgeois et al. [6]) There exists an exact sequence .

cyc

∗

¯ε Ф

Ψ¯ ε

cyc

· · · → LHk (Λ) → HCkФ ε (YΛ , ξΛ ) → HCkε (Y, ξ ) → LHk−1 (Λ) → . . . , −→

where Ф is the chain map induced by the cobordism (Y Y Λ , ω) and Ψ is a chain map induced by the symplectization (R × Y, d(et λ)) with its Lagrangian submanifold R × Λ.

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4 Contact Manifolds

Therefore, knowing the cyclic Legendrian homology of the Legendrian knot or link Λ and linearized contact homology of a contact manifold Y, using this exact sequence, we can compute the linearized contact homology of the constructed contact manifold YΛ by Legendrian surgery. Let Λ = Λ1 ⨆ Λ2 be a Legendrian link in the contact manifold (Y0 , ξ0 ). In [17], it is explained precisely that one can associate a “cyclic version” to the differential graded algebra (LH A(Λ), dLH A ). We perform Legendrian surgery along the link Λ to obtain a new contact manifold (Y2 , ξ2 ). Then, using the above surgery exact sequence, we will be able to compute the linearized contact homology of the new contact manifold Y2 . To this end, we need to know the cyclic Legendrian homology of Λ denoted by LHcyc (Λ). In general, computations to obtain LHcyc (Λ) are hard and long. In order to compute this invariant, we carry out the Legendrian surgery in two steps. First along Λ1 in Y0 and we call the obtained manifold Y1 , then along the ˜ 2 in Y1 . In [17] we showed that there exists the following second component called Λ long exact sequence for Legendrian links. Theorem 4.16 ([17]) Let Λ = Λ1 ⨆ Λ2 be a Legendrian link. Then there exists the following exact sequence in terms of cyclic Legendrian homology of the Legendrian link and its components. .

˜ 2 ) −→ LH (Λ) −→ LH (Λ1 ) −→ LH (Λ ˜ 2 ) −→ . . . . . . . −→ LHk (Λ k k k−1 cyc

cyc

cyc

cyc

Hence, computing the cyclic Legendrian homology of Λ will be reduced to ˜ 2 . In [17] we described computing the cyclic Legendrian homologies of Λ1 and Λ the cyclic Legendrian homology deduced from Legendrian contact homology, and we presented some examples of computing this invariant. There, we used the surgery exact sequence given in Theorem 4.15 to compute the linearized contact homology of (RP 3 , ξst ) explicitly. In fact, we used the classification theorem for tight contact structures on lens spaces given by Honda in [29]. Also, we computed the cyclic Legendrian homology for the double unknot in three-dimension and in higher dimension and for the Legendrian link of simple Legendrian unknots. Then we computed the linearized contact homology of several lens spaces.

References

1. R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings Publishing Company, 1978. 2. V.I. Arnol’d, Characteristic class entering in quantization condition, Func. Anal. Appl. 1 (1967) 1–13. 3. V. I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, 60. Springer–Verlag, New York 1989. 4. D. Bennequin, Entrelacements et equations de Pfaff, in: IIIe Rencontre de Geometrie du Schnepfenried, Vol, 1, Asterisque 107–108(1983), 87–161. 5. F. Bourgeois, A survey of contact homology, New Perspectives and Challenges in Symplectic Field Theory, CRM Proceedings and Lecture Notes 49, (2009), 45–72. 6. F. Bourgeois, T. Ekholm, and Y. Eliashberg, Effect of Legendrian surgery, Geometry and Topology 16, pages 301–389, (2012). 7. A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. 8. Yu. Chekanov, Differential algebra of Legendrian links, Invent. Math. 150, pages 441–483, (2002). 9. Yu. Chekanov, Invariants of Legendrian knots, Proceedings of the International Congress of Mathematicians, Vol. II, Higher Education Press, Beijing, pages 385–394, (2002). 10. E.A. Coddington and N. Levinson, Theory of ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, 1955. 11. Darboux, Gaston , Sur le problème de Pfaff., Bull. Sci. Math. 6 (1882) 14–36, 49–68. 12. F. Ding and H. Geiges, A Legendrian surgery presentation of contact 3–manifolds, Math. Proc. Cambridge Philos. Soc. 136, pages 583–598, (2004). 13. T.Ekholm, J.Etnyre, M.Sullivan,The contact homology of Legendrian submanifolds in .R2n+1 , J. Differential Geom., 71(2005) No.2, 177–305. 14. Y. Eliashberg, Invariants in contact topology, Proceedings of the International Congress on Mathematics, Vol. II, Document on Mathematics, pages 327–338, (1998). 15. Y. Eliashberg and M. Fraser, Classification of topologically trivial Legendrain knots. Geometry, Topology and Dynamics (Montreal 1995), CRM Proc. Lecture Notes 15, American Mathematical Society, Providence, pages 17–51, (1998). 16. Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume, Part II, pages 560–673, (2000). 17. A. Eslami Rad, An exact sequence for Legendrian links, Algebraic and Geometric Topology 15-1 (2015), 191–230.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. Eslami Rad, Symplectic and Contact Geometry, Latin American Mathematics Series, https://doi.org/10.1007/978-3-031-56225-9

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174

References

18. J. Etnyre, L. Ng, and J. Sabloff, Invariants of Legendrian Knots and Coherent Orientations, J. Symplectic Geom. 1, no. 2, pages 321–367, (2002). 19. F. Frankl and L. Pontrjagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Annalen (in German). 102 (1): 785–789, (1930). 20. H. Geiges, An introduction to contact topology. Cambridge Studies in Advanced Mathematics, 109. Cambridge University Press, Cambridge, (2008). 21. Georges de Rham, Sur l’Analyse Situs des Varietes a n Dimensions, Doctoral Thesis, Paris, 1931. 22. E. Giroux, Convexite en topologie de contact, Comment. Math. Helv. 66 (1991), 637–677. 23. M.Gotay, On coisotropic imbeddings of presymplectic manifolds, Proc. Amer. Math. Soc. 84 (1982), 111–114. 24. M. L. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicae 82 (1985), 307–347. 25. Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. 26. Victor Guillemin and Shlomo Sternberg, Geometric Asymptotics, Vol. 14. Math. Surveys and Monographs. Amer. Math. Soc., 1977. 27. Brian C. Hall, Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, 2015. 28. W.R. Hamilton, On a general method in dynamics, Phil. Trans. Royal Soc. London (1834) 247–308, (1835) 95-144. 29. K. Honda, On the classification of tight contact structures I. Geom. Topol. 4, pages 309–368, (2000). 30. L. Hormander: An Introduction to Complex Analysis in Several Variables. North-Holland Publ. Co. Amsterdam, 1973. 31. C.G.J. Jacobi, Nova methodus, equations differentiales partiales primi orins inter numerum variabilium quemcuque propositas integrandi, Crelle journal fur die reine und angewandte Mathematik 60 (1862) 1–181, Also in: Gesammelte Werke, V. Band, 1–189., Berlin 1890. Reprint Chelsea Publ. Cy. New York, 1969. 32. P. Kronheimer, T. Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997), 209– 255. 33. J.-L. Lagrange, Mecanique Analytique, First print 1788 in Paris, Reprint by Blanchard, Paris, 1965. 34. J. M., Lee, Introduction to smooth manifolds, Springer Graduate Texts in Mathematics, 218, 2003. 35. A. Lichnerowicz, Les varietes de Poisson et leurs algebres de Lie associees. J. Differential Geometry 12 (1977) 253–300. 36. S. Lie, Theorie der Transformationsgruppen I–III, Unter Mitwirkung von Prof.dr. F. Engel. Teubner, Leipzig, 1888, 1890, 1893. 37. Dusa MacDuff and Dietmar Salamon, Introduction to Symplectic Topology, Second Edition, Oxford Mathematical Monographs, Oxford, 2005. 38. D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, (2004). 39. B. Malgrange, Sur l’integrabilite des structures presques-complexes. pp. 289–296 in: Symposia Mathematica II, INDAM, Rome, 1968. 40. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Reports on Math. Phys. 5 (1974), 121–130. 41. J. Martinet, Formes de contact sur les varietes de dimension 3, Proceedings of Liverpool Singularities Symposium II (1969/1970), 142–163, Lecture Notes in Math. 209, Springer, Berlin, 1971. 42. J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc, 120 (1965) 286–294. 43. A. Newlander and L. Nirenberg, Complex analytic coordinates in an almost complex manifold, Annals of Math, 65 (1957) 391–404.

References

175

44. Isaac Newton, The Principia, Mathematical Principles of Natural Philosophy, A New Translation by I.B. Cohen and A. Whitman, with A Gide To Newton’s Principia by I.B. Cohen, University of California Press, 1999. 45. Emmy Noether, Invariante Variations probleme, Gottingen Nachrichten (1918), 235–257. 46. B. Ozbagci and A. Stipsicz, Surgery on contact 3–manifolds and Stein surfaces. Bolyai Society Mathematical Studies, 13, Springer-Verlag, Berlin, (2004). 47. J. Robbin, D. Salamon, The Maslov index for paths, Topology, 32(1993) No. 4, 827–844. 48. W. Rossmann, Lie Groups–An Introduction through linear groups, Oxford Graduate Texts in Mathematics, Oxford Science Publication, 2002. 49. M. Spivak, Calculus on manifolds, Addison-Wesley, 1965. 50. S.Sternberg, Lectures on Differential Geometry, Second edition, Chelsea, 1983. 51. Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott-Foresman and Company, 1971. 52. A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Mathematics 6 (1971) 329–346. 53. A. Weinstein, Lectures on Symplectic Manifolds, Vol. 29. Regional Conference Series in Mathematics. Amer. Math. Soc., Providence, 1977. 54. A. Weinstein, Neighborhood classification of isotropic embeddings, J. Differ. Geom. 16 (1981), 125–128. 55. A. Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J., 20(2), pages 241–251, (1991). 56. H. Whitney, Analytic extensions of functions defined in closed sets,Trans. Amer. Math. Soc. 36.1 (1934) 63–89.

Index

A Almost complex structure, 39, 54–57, 134

C Characteristic foliation, 148–152 Classical mechanics, vii, viii, 39, 77, 91–94, 97, 115, 121, 143 Complex projective manifold, viii, 39, 51–54 Complex structure, 1, 27, 30–32, 35, 39, 51, 55–57, 132–134 Contact manifold, vii, viii, 115–172 Contact surgery, 164–170

D Darboux’s theorem, vii, viii, 1, 14, 39, 67, 71, 74, 77, 98–101, 115, 119, 134–138, 170 E Euler-Lagrange equation, vii, 77, 87, 89–92

G Geodesic flow, viii, 65–67, 143, 144, 147 Geometric optic, viii, 115, 143–148

H Hamiltonian group action, viii, 77, 102–109 Hamiltonian system, vii, viii, 67, 77–113 Hermitian form, viii, 1, 27–35, 39, 51, 54, 56

I Isotropic subspace, 1, 6, 9–12, 61, 138

L Lagrangian Grassmannian, viii, 15–20, 22, 33, 34 Lagrangian submanifold, viii, 39, 61–67, 72, 73, 138, 139, 171 Lagrangian subspace, 1, 10, 11, 15, 17, 18, 27, 61, 62, 73 Legendre transform, viii, 77, 87–94 Legendrian knot, viii, 115, 138, 152–164, 170, 172 Legendrian submanifold, viii, 138–143, 147, 148, 153, 170, 171

M Momentum mapping, viii, 77, 88, 89, 91, 93, 94, 102, 104, 106, 108, 111–113 Moser trick, viii, 60, 67, 70, 134, 135, 137, 138, 142, 150

O Overtwisted contact structure, 129, 151, 152

P Poisson brackets, 77, 94–98, 101, 102, 108

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. Eslami Rad, Symplectic and Contact Geometry, Latin American Mathematics Series, https://doi.org/10.1007/978-3-031-56225-9

177

178 R Reduced symplectic manifold, viii, 47–51, 109, 113 S Standard contact structure, 118, 119, 123, 129, 148, 150, 151, 153, 154, 157 Standard symplectic form, vii, 12, 13, 27, 31, 36, 40, 46, 59, 62, 67, 98, 100, 113, 122, 168 Symplectic form, vii, viii, 1–3, 6, 9–15, 19, 20, 24, 27, 28, 30, 31, 35–37, 39, 40, 42, 46–48, 50, 51, 53–62, 65–67, 72–74,

Index 77, 84, 86, 87, 94, 95, 98–103, 105, 109, 111, 113, 117, 120–122, 126, 131, 132, 135, 136, 138, 143, 144, 154, 168, 170 Symplectic linear group, viii, 1, 20–27, 32, 33 Symplectic manifold, vii, viii, 39–75, 84, 86, 94, 97, 98, 101, 103, 108, 109, 112, 113, 115, 120, 122, 124–127, 130–132, 134, 136, 138, 169 Symplectic quotient, viii, 51, 77, 109–113 Symplectic vector space, 1, 2, 9–12, 14, 15, 17, 20, 27, 31, 40, 54, 112, 117, 132, 138 Symplectomorphism, viii, 1, 21, 27, 32, 39, 46, 59–61, 64, 66, 67, 71, 74, 131, 169