دانلود کتاب Symplectic Geometry and Secondary Characteristic Classes

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This book unites two important mathematical subjects: symplectic geometry and the theory of secondary characteristic classes, two subjects which are also of independent and much larger interest, and which, until now, were not treated together in the same work. This is a good framework for a study of the Maslov class, a fundamental invariant in the asymptotic analysis of partial differential equations in quantum physics. The Maslov class is the first of a sequence of secondary characteristic classes. The characteristic classes which generalize the Maslov class are discussed and their properties established. They are then computed in the most interesting cases, namely for Lagrangian and Legendrian submanifolds of cotangent bundles. This exposition, incorporating results of the author's original research, will be of interest to researchers and graduate students in differential geometry, differential topology, mathematical physics, and quantum mechanics. TABLE OF CONTENTS CHAPTER 1. INTRODUCTION AND MOTIVATION. . . . . Equations of the Hamilton-Jacobi type Some more symplectic geometry . Some more mathematical physics CHAPTER 2. SYMPLECTIC VECTOR SPACES Symplectic vector spaces and their automorphisms Subspaces of symplectic vector spaces Complex structures in real symplectic spaces CHAPTER 3. SYMPLECTIC GEOMETRY ON MANIFOLDS Symplectic vector bundles Symplectic vector bundles of geometric structures Lagrangian subbundles . . . . . . . . . . . . Local equivalence theorems in symplectic geometry CHAPTER 4. TRANSVERSALITY OBSTRUCTIONS OF LAGRANGIAN SUB BUNDLES (MASLOV CLASSES) Connections on principal bundles Secondary characteristic classes The Maslov class and index Maslov secondary characteristic classes Computations in cotangent bundles . . . REFERENCES INDEX