Xll

Preface

needed in symplectic geometry. Furthermore, more advanced topics will

continue to rely heavily on other disciplines, in particular on results from

the study of differential equations.

Specifically, the text tries to reach the following two goals:

• To present the idea of the formalism of symplectic forms, to in-

troduce the symplectic group, and especially to describe the sym-

plectic manifolds. This will be accompanied by the presentation of

many examples of how they come to arise; in particular the quotient

manifolds of group actions will be described,

and

• To demonstrate the connections and interworking between math-

ematical objects and the formalism of theoretical mechanics; in

particular, the Hamiltonian formalism, and that of the quantum

formalism, namely the process of quantization.

The pursuit of these goals proceeds according to the following plan. We

begin in Chapter 0 with a brief introduction of a few topics from theoretical

mechanics needed later in the text. The material of this chapter will already

be familiar to physics students; however, for the majority of mathematics

students, who have not learned the connections of their subject to physics,

this material will perhaps be new.

We are constrained, in the first chapter, to consider symplectic (and a

little later Kdhler) vector spaces. This is followed by the introduction of the

associated notion of a symplectic group Sp(V) along with its generation. We

continue with the introduction of several specific and theoretically important

subspaces, the isotropic, coisotropic and Lagrangian subspaces, as well as

the hyperbolic planes and spaces and the radical of a symplectic space.

Our first result will be to show that the symplectic subspaces of a given

dimension and rank are fixed up to symplectic isomorphism. A consequence

is then that the Lagrangian subspaces form a homogeneous space C(V) for

the action of the group Sp(V). The greatest effort will be devoted to the

description of the spaces of positive complex structures compatible with the

given symplectic structure. The second major result will be that this space

is a homogeneous space, and is, for dim V = 2n, isomorphic to the Siegel

half space fin = Spn(M)/U(n).

The second chapter is dedicated to the central object of the book, namely

symplectic manifolds. Here the consideration of differential forms is unavoid-

able. In Appendix A their calculus will be given. The first result of this

chapter is then the derivation of a theorem by Darboux that says that the

symplectic manifolds are all locally equivalent. This is in sharp contrast

to the situation with Riemannian manifolds, whose definition is otherwise