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A Student's Guide to Symplectic Spaces, Grassmannians

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A Student's Guide to Symplectic Spaces, Grassmannians A Student’s Guide to Symplectic Spaces, Grassmannians and Maslov Index Paolo Piccione Daniel Victor Tausk DEPARTAMENTO DE MATEMATICA´ INSTITUTO DE MATEMATICA´ E ESTAT´ISTICA UNIVERSIDADE DE SAO˜ PAULO Contents Preface v Introduction vii Chapter 1. Symplectic Spaces 1 1.1. A short review of Linear Algebra 1 1.2. Complex structures 5 1.3. Complexification and real forms 8 1.3.1. Complex structures and complexifications 13 1.4. Symplectic forms 16 1.4.1. Isotropic and Lagrangian subspaces. 20 1.4.2. Lagrangian decompositions of a symplectic space 24 1.5. Index of a symmetric bilinear form 28 Exercises for Chapter 1 35 Chapter 2. The Geometry of Grassmannians 39 2.1. Differentiable manifolds and Lie groups 39 2.1.1. Classical Lie groups and Lie algebras 41 2.1.2. Actions of Lie groups and homogeneous manifolds 44 2.1.3. Linearization of the action of a Lie group on a manifold 48 2.2. Grassmannians and their differentiable structure 50 2.3. The tangent space to a Grassmannian 53 2.4. The Grassmannian as a homogeneous space 55 2.5. The Lagrangian Grassmannian 59 k 2.5.1. The submanifolds Λ (L0) 64 Exercises for Chapter 2 68 Chapter 3. Topics of Algebraic Topology 71 3.1. The fundamental groupoid and the fundamental group 71 3.1.1. The Seifert–van Kampen theorem for the fundamental groupoid 75 3.1.2. Stability of the homotopy class of a curve 77 3.2. The homotopy exact sequence of a fibration 80 3.2.1. Applications to the theory of classical Lie groups 92 3.3. Singular homology groups 99 3.3.1. The Hurewicz’s homomorphism 109 Exercises for Chapter 3 113 Chapter 4. Curves of Symmetric Bilinear Forms 116 4.1. A few preliminary results 116 4.2. Evolution of the index: a simple case 117 4.3. Partial signatures and another “evolution of the index” theorem 120 Exercises for Chapter 4 127 iii iv CONTENTS Chapter 5. The Maslov Index 129 5.1. A definition of Maslov index using relative homology 129 5.2. A definition of Maslov index using the fundamental groupoid 138 5.3. Isotropic reduction and Maslov index 141 5.4. Maslov index for pairs of Lagrangian curves 145 5.5. Computation of the Maslov index via partial signatures 149 5.6. The Conley–Zehnder index 152 Exercises for Chapter 5 153 Appendix A. Kato selection theorem 157 A.1. Algebraic preliminaries 157 A.2. Polynomials, roots and covering maps 160 A.3. Multiplicity of roots as line integrals 163 A.4. Eigenprojections and line integrals 165 A.5. One-parameter holomorphic families of linear maps 167 A.6. Regular and singular points 170 Exercises for Appendix A 175 Appendix B. Generalized Jordan Chains 181 Exercises for Appendix B 184 Appendix C. Answers and Hints to the exercises 185 C.1. From Chapter 1 185 C.2. From Chapter 2 189 C.3. From Chapter 3 192 C.4. From Chapter 4 194 C.5. From Chapter 5 194 C.6. From Appendix A 196 C.7. From Appendix B 200 Bibliography 201 Index 202 Preface This is a revised edition of a booklet originally published with the title “On the geometry of grassmannians and the symplectic group: the Maslov index and its applications”. The original text was used as a textbook for a short course given by the authors at the XI School of Differential Geometry, held at the Universidade Federal Fluminense, Niteroi, Rio de Janeiro, Brazil, in 2000. This new edition was written between November 2007 and September 2008 at the University of Sao˜ Paulo. Several changes and additions have been made to the original text. The first two chapters have basically remained in their original form, while in Chapter 3 a section on the Seifert–van Kampen theorem for the fundamental groupoid of a topological space has been added. Former Chapter 4 has been divided in two parts that have become Chapters 4 and 5 in the present edition. This is where most of the changes appear. In Chapter 4 we have added material on the notion of partial signatures at a singularity of a smooth path of symmetric bilinear forms. The partial signatures are used to compute the jump of the index for a path of real analytic forms in terms of higher order derivatives. In Chapter 5, we have added a new and more general definition of Maslov index for continuous curves with arbitrary endpoints in the Lagrangian Grassmannian of a finite dimensional symplectic space. In the original edition, we discussed only the homological definition, which applies only to curves with “nondegenerate” endpoints. The presentation of this new definition employs the Seifert–van Kampen theorem for the fundamental groupoid of a topological space that was added to Chapter 3. We also discuss the notion of Maslov index for pairs of Lagrangian paths, and related topics, like the notion of Conley–Zehnder index for symplectic paths. Given an isotropic subspace of a symplectic space, there is a natural construc- tion of a new symplectic space called an isotropic reduction of the symplectic space (see Example 1.4.17). In this new edition of the book we have also added a section in Chapter 5 containing some material concerning the computation of the Maslov index of a continuous path of Lagrangians that contain a fixed isotropic subspace. This is reduced to the computation of the Maslov index in an isotropic reduction of the symplectic space. Finally, two appendices have been added. Appendix A contains a detailed proof of the celebrated Kato’s selection theorem, in the finite dimensional case. Kato’s theorem gives the existence of a real analytic path of orthonormal bases of eigenvectors for a given real analytic path of symmetric operators. The proof of Kato’s theorem presented in Appendix A is accessible to students with some back- ground in Differential Geometry, including basic notions of covering spaces and analytic functions of one complex variable. Kato’s theorem is needed for the proof of the formula giving the Maslov index of a real analytic path of Lagrangians in v vi PREFACE terms of partial signatures. Appendix B contains an algebraic theory of general- ized Jordan chains. Generalized Jordan chains are related to the notion of partial signature discussed in Chapter 4. Former Chapter 5, which contained material on some recent applications of the notion of Maslov index in the context of linear Hamiltonian systems, has been removed from the present version. Incidentally, this book will be published 150 years after the publication of C. R. Darwin’s On the origin of species, in 1859. Both authors are convinced evolution- ists, and they wish to give a tribute to Darwin’s scientific work with two quotations of the scientist at the beginning and at the end of the book. The authors grate- fully acknowledge the partial financial support provided by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnologico´ (CNPq), Brazil, and by Fundac¸ao˜ de Amparo a Pesquisa do Estado de Sao˜ Paulo (Fapesp), Sao˜ Paulo, Brazil. Dedication. This book is dedicated to Prof. Elon Lages Lima and to Prof. Man- fredo Perdigao˜ do Carmo on occasion of their 80th anniversary. Elon and Manfredo are both authors of great math books from which the authors have learned and still learn a lot. Sao˜ Paulo, October 2008 Introduction The goal of this book is to describe the algebraic, the topological and the geo- metrical issues that are related to the notion of Maslov index. The authors’ inten- tion was to provide a self-contained text accessible to students with a reasonable background in Linear Algebra, Topology and some basic Calculus on differential manifolds. The new title of the book reflects this objective. The notion of symplectic forms appears naturally in the context of Hamiltonian mechanics (see [1]). Unlike inner products, symplectic forms are anti-symmetric and may vanish when restricted to a subspace. The subspaces on which the sym- plectic form vanishes are called isotropic and the maximal isotropic subspaces are called Lagrangian. Hamiltonian systems naturally give rise to curves of symplecto- morphisms, i.e., linear isomorphisms that preserve a symplectic form. In such con- text, subspaces of the space where the symplectic form is defined may be thought of as spaces of initial conditions for the Hamiltonian system. Lagrangian initial conditions appear in many situations such as the problem of conjugate points in Riemannian and semi-Riemannian geometry. Lagrangian initial conditions give rise, by the curve of symplectomorphisms, to curves of Lagrangian subspaces. The Maslov index is a (semi-)integer invariant associated to such curves. In many ap- plications it has an interesting geometric meaning; for instance, in Riemannian geometry the Maslov index of the curve of Lagrangians associated to a geodesic is (up to an additive constant) equal to the sum of the multiplicities of conjugate (or focal) points along the geodesic (see [6, 12]). Applications of Malsov index (and other related indexes) can be found, for instance, in [2, 3, 4, 5, 10, 14, 15, 16, 18]. Chapter 1 deals with the linear algebraic part of the theory. We introduce the basic notion of symplectic space, the morphisms of such spaces and the notion of isotropic and Lagrangian subspaces of a symplectic space. Symplectic structures are intimately related to inner products and complex structures, which are also discussed in the chapter. The last part of the chapter deals with the notion of index of a symmetric bilinear form and its main properties. Chapter 2 deals with the geometrical framework of the theory. We describe the differential structure of Grassmannians, that are compact manifolds whose el- ements are subspaces of a given vector space.
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