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,
• 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