**Mathematical Surveys and Monographs**

Volume: 14;
1977;
480 pp;
Softcover

MSC: Primary 20;
Secondary 22; 35; 44; 49

Print ISBN: 978-0-8218-1633-2

Product Code: SURV/14

List Price: $96.00

Individual Member Price: $76.80

**Electronic ISBN: 978-0-8218-3208-0
Product Code: SURV/14.E**

List Price: $96.00

Individual Member Price: $76.80

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# Geometric Asymptotics

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*V Guillemin; S Sternberg*

Symplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years—the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence.

Chapters included in this book are: Chapter I, Introduction. The method of stationary phase; Appendix I, Morse's lemma and some generalizations; Chapter II, Differential operators and asymptotic solutions; Chapter III, Geometrical optics; Chapter IV, Symplectic geometry; Chapter V, Geometric quantization; Chapter VI, Geometric aspects of distribution; Appendix to Chapter VI, The Plancherel formula for the complex semisimple Lie groups; Chapter VII, Compound Asymptotics; Appendix II, Various functorial constructions; Index.

#### Table of Contents

# Table of Contents

## Geometric Asymptotics

- TABLE OF CONTENTS xvii18
- PREFACE v6 free
- NOTATION xv16 free
- CHAPTER I. INTRODUCTION. THE METHOD OF STATIONARY PHASE 120 free
- CHAPTER II. DIFFERENTIAL OPERATORS AND ASYMPTOTIC SOLUTIONS 2140
- CHAPTER III. GEOMETRICAL OPTICS 7190
- CHAPTER IV. SYMPLECTIC GEOMETRY 109128
- §1. The Darboux-Weinstein theorem 109128
- §2. Symplectic vector spaces 115134
- §3. The cross index and the Maslov class 130149
- §4. Functorial properties of Lagrangian submanifolds 149168
- §5. Local parametrizations of Lagrangian submanifolds 153172
- §6. Periodic Hamiltonian systems 168187
- §7. Homogeneous symplectic spaces 180199
- §8. Multisymplectic structures and the calculus of variations 204223

- CHAPTER V. GEOMETRIC QUANTIZATION 213232
- §1. Curvature forms and vector bundles 213232
- §2. The group of automorphisms of an Hermitian line bundle 221240
- §3. Polarizations 228247
- §4. Metalinear manifolds and half forms 251270
- §5. Metaplectic manifolds 261280
- §6. The pairing of half form sections 273292
- §7. The metaplectic representation 276295
- §8. Some examples 290309

- CHAPTER VI. GEOMETRIC ASPECTS OF DISTRIBUTIONS 305324
- §1. Elementary functorial properties of distributions 305324
- §2. Traces and characters 316335
- §3. The wave front set 324343
- §4. Lagrangian distributions 342361
- §5. The symbol calculus 354373
- Appendix to Section 5 363382
- §6. Fourier intergral operators 364383
- §7. The transport equation 373392
- §8. Some applications to spectral theory 379398
- APPENDIX TO CHAPTER VI. THE PLANCHEREL FORMULA FOR THE COMPLEX SEMI-SIMPLE LIE GROUPS 388407

- CHAPTER VII. COMPOUND ASYMPTOTICS 399418
- §0. Introduction 399418
- §1. The asymptotic Fourier transform 400419
- §2. The frequency set 404423
- §3. Functorial properties of compound asymptotics 409428
- §4. The symbol calculus 414433
- §5. Pointwise behavior of compound asymptotics and Bernstein's theorem 425444
- Appendix to Section 5 of Chapter VII 429448
- §6. Behavior near caustics 434453
- §7. Iterated S[sub(1)] and S[sub(2,0)] singularities, computations 447466
- §8. Proofs of the normal forms 456475
- §9. Behavior near caustics (continued) 462481

- APPENDIX II. VARIOUS FUNCTORIAL CONSTRUCTIONS 469488
- INDEX 477496

#### Reviews

The topic of this nice book can be defined as a geometric approach to the investigation of some analytic problems, especially to the study of Fourier integral operators. These operators are now widely used for the analysis of singularities of solutions of linear partial differential equations and for the study of the spectra of the corresponding operators. In general the book is very interesting and useful for specialists both in analysis and in differential geometry.

-- Mathematical Reviews