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