somewhat parallel to that of the symplectic manifolds. The chapter will
then take a glance at new research by considering the assignment of invari-
ants to symplectic manifolds; in particular, the symplectic capacities and
the pseudoholomorphic curves will be given.
In the course of the second chapter, we will present several examples of
symplectic manifolds:
First, the example which forms the origin of the theory and remains
the primary application to physics is the cotangent bundle T*Q of
a given manifold Q.
Second, the general Kahler manifold.
Third, the coadjoint orbits. This description of symplectic mani-
folds with the operation of a Lie group G can be taken as the sec-
ond major result of this chapter. We describe a theorem of Kostant
and Souriau that says that for a given Lie group G with Lie alge-
bra g satisfying the condition that the first two cohomology groups
vanish, that is
= 0, there is, up to covering, a
one-to-one correspondence between the symplectic manifolds with
transitive G-action and the G-orbits in the dual space g* of g.
Here we will need several facts from the theory of Lie algebras and
systems of differential equations, and we will at least cover some of
the rudiments we require. This will then offer yet another means
for introducing one of the central concepts of the field, namely the
moment map. This will, however, be somewhat postponed so that
In the fourth and last example, complex projective space can be
presented as a symplectic manifold; this will be seen as a specific
example of the third example, as well as the second; that is, as a
coadjoint orbit as well as as a Kahler manifold.
As preparation for the higher level construction of symplectic manifolds,
Chapter 3 will introduce the standard concepts of a Hamiltonian vector
field and a Poisson bracket. With the aid of these ideas, we can give the
Hamiltonian formulation of classical mechanics and establish the following
fundamental short exact sequence:
0 IR - T(M) -+ Ham M -• 0,
where T{M) is the space of smooth functions / defined on the symplectic
manifold and given the structure of Lie algebra via the Poisson bracket, and
Ham M is the Lie algebra of Hamiltonian vector fields on the manifold.
The third chapter continues with a brief introduction to contact man-
ifolds. A theory for these manifolds in odd dimension can be developed
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