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
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