Book review

In the last few decades, we assisted at a continous growth of the number and importance of applications of modern differential geometry to theoretical physics. Born in the early 50th, as a geometrical theory of Hamiltonian systems, symplectic geometry proved to be, after only few years, a powerful tool in quantization theory. The theory of geometric quantization was born in the 60th, in the works of Kostant and Souriau. Originated in a graduate course, given by the author at the University of Timisoara, Romania, the book under review intends to present some basic material of the geometry of symplectic and Poisson manifolds, in relation with Hamiltonian mechanics and geometric quantization.

The contents of the book are as follows. The first chapter is dealing with the foundations of symplectic algebra and geometry. Chapter 2 is an introduction to finite and infinite dimensional Hamiltonian mechanics. The chapter 3, one of the key chapters, is concerned with the fundamental notion of momentum mapping and the celebrated Marsden-Weinstein reduction theorem. The fourth chapter generalizes Hamiltonian mechanics, constructing the so-called Hamilton-Poisson mechanics, while the fifth one is devoted to some results obtained by Holm, Marsden, Ratiu and Weinstein in the theory of stability of Hamiltonian mechanical systems. The chapters 6 and 7 are concerned with geometric prequantization and quantization in the sense of Kostant and Souriau. The chapter 8 presents some facts about foliated cohomology, and its relation with geometric quantization, while the Chapter 9 discusses the relation between Marsden-Weinstein reduction and quantization and some applications to constrained mechanical systems. The last chapter attempts to generalize the geometric quantization theory to Poisson manifolds, by means of the theory of symplectic groupoids.

Every chapter is completed with a number of exercises and problems (all of them accompanied by their solutions). Some of problems contain important theorem that can be obtained applying the results from the text.

Especially the seccud half of the book is largely based on the author's works and his research interest (the list of references includes more than thirty papers of the author and his collaborators).

The book is well written, In a fairly pedagogical style and I would like to recommend it to anyone who wish to learn geometric quantization and, more generally, symplectic geometry. There are, of course, many other books on these topics (e.g. Woodhouse - Geometric Quantization, Oxford University Press, 1980), but this one has some incontestable advantages. First of all, it is quite comprehensible and the range of subjects included is much broader than in the case of the Woodhouse's book. On the other hand, the book includes many results obtained only in the last few years and presents, in some details, the new directions of research in this dynamical field (e.g. symplectic groupoids, Poisson manifolds, quantization of constrained systems which are at the very frontier of research). The book can be useful to graduate students in physics and mathematics, but, as I said before, it also contains a lot of advanced topics, of interest for experts, too. It is very suitable for a course in symplectic geometry and geometric quantization.

The book includes an extended list of references and a detailed index.