Lecture I. Symplectic Manifolds and Lagrangi•n Submanifolds, Examples

There are three levels on which one may define the notion of a symplectic structure:

(A) Algebra. A symplectic structure on a finite-dimensional real vector space Vis an

antisymmetric bilinear form

n

on

v

such that the associated map

n:

v-+ v•

defined by

Il(u) (w)

=

U(u, w) is an isomorphism.

(A/G) Algebra/Geometry. A symplectic structure on a [smooth] vector bundle E over

a space [manifold] Xis a continuous [smooth] family

n

=

{Ux} of symplectic structures

on the fibres of E. The associated object

fi =

{flx} is then a bundle isomorphism from E

to E*.

(G) Geometry. A symplectic structure on a smooth manifold

Pis

a symplectic struc-

ture

n

in sense (A/G) on the tangent bundle

TP

which, considered as an exterior differen-

tial 2-form on

P,

is closed, i.e.

dU

=

0.

REMARK.

There is a common generalization of (A) and (A/G): one may consider

symplectic structures on modules over arbitrary commutative rings (see [N] ; except for the

ring of complex numbers, we will not use this generalization, nor will we consider symplectic

structures on Banach spaces (see (C], (MD], [SW]).

An isomorphism between symplectic objects in any of the three categories is called a

symplectomorphism, or canonical transformation.

Historical note. The word symplectic was invented by Hermann Wey! [WE} , who sub-

stituted Greek for Latin roots in the word complex to obtain a term which would describe a

group related to line complexes but which would not be confused with complex numbers.

(I owe this reference to Souriau, who is himself the inventor of the term symplectomorphism.)

Here are some examples of symplectic objects.

In (A), let Ube any real vector space. Then U EB

U*

has a "canonical" symplectic

structure

nu

defined by

Uu(ul $ uj, U2

e

ui)

=

u~(u1)

- uj(u2·

Any symplectic vector space is symplectomorphic to one of these. (See Lecture 2.)

Analogously, in (A/G), the Whitney sum E EB E*, where E is any vector bundle over X,

is a symplectic vector bundle. Not every symplectic vector bundle is isomorphic to one of

these.

In (G), the simplest example of a symplectic manifold is R2

"

with the symplectic

structure

n

=

l:~=l

dx; I\

d~i•

where (x

1

, ... ,

xn,

~ 1 ,

... ,

~n)

are the coordinates. Note

3