TABLE OF CONTENTS PREFACE v NOTATION x v CHAPTER I. INTRODUCTION. THE METHOD OF STATIONARY PHASE 1 APPENDIX I. MORSE'S LEMMA AND SOME GENERALIZATIONS 16 CHAPTER II. DIFFERENTIAL OPERATORS AND ASYMPTOTIC SOLUTIONS 21 § 1. Differential operators 21 §2. Asymptotic sections 27 §3. The Luneburg-Lax-Ludwig technique 30 §4. The methods of characteristics 34 §5. Bicharacteristics 41 §6. The transport equation 50 §7. The Maslov cycle and the Bohr-Sommerfeld quantization conditions... 58 CHAPTER III. GEOMETRICAL OPTICS 71 §1. The laws of refraction and reflection 71 §2. Focusing and magnification 78 §3. Hamilton's method 83 §4. First order optics 89 §5. The Seidel aberrations 95 §6. The asymptotic solution of Maxwell's equations 101 CHAPTER IV. SYMPLECTIC GEOMETRY 109 §1. The Darboux-Weinstein theorem 109 §2. Symplectic vector spaces 115 §3. The cross index and the Maslov class 130 §4. Functorial properties of Lagrangian submanifolds 149 §5. Local parametrizations of Lagrangian submanifolds 153 §6. Periodic Hamiltonian systems 168 §7. Homogeneous symplectic spaces 180 §8. Multisymplectic structures and the calculus of variations 204 CHAPTER V. GEOMETRIC QUANTIZATION 213 §1. Curvature forms and vector bundles 213 §2. The group of automorphisms of an Hermitian line bundle 221 §3. Polarizations 228 §4. Metalinear manifolds and half forms 251 §5. Metaplectic manifolds 261 §6. The pairing of half form sections 273 §7. The metaplectic representation 276 §8. Some examples 290 xvn

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