Symplectic geometry and secondary characteristic classes / Izu Vaisman

The aim of this book is to provide a self-consistent treatment of the Maslov class, which is a fundamental invariant in the asymptotic analysis of partial differential equations of quantum physics. The characteristic classes which generalize the Maslov class are discussed and their properties established. These classes are computed in some important cases, namely for Lagrange and Legendre submanifolds of cotangent bundles. The general method of computation used by the author shows that the Maslov classes depend on a generalized second fundamental form, and on the curvature of the Lagrange (Legendre) submanifold. Because the framework for the secondary characteristic classes is the symplectic geometry, some notions as general and natural symplectic vector bundles that appear on symplectic manifolds and their submanifolds, or on contact manifolds, are presented in detail in the first part of the book. The book is of interest to researchers and graduate students in differential geometry, differential topology, mathematical physics and quantum physics. (Zentralblatt)