Symplectic manifolds
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1 Introduction
A symplectic manifold is a smooth manifold together with a differential two-form
that is nondegenerate and closed. The form
is called a symplectic form. The nondegeneracy means that the highest nonzero power of
is a volume form on
It follows that a symplectic manifold is even dimensional.
Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system.
2 Examples
The most basic example of a symplectic manifold is
equipped with the form
A theorem of Darboux [McDuff-Salamon] states that locally every
symplectic manifold if of this form. More precisely, if is a symplectic
-manifold
then for every point
there exists an open neighbourhood
of
and a
diffeomorphism
such that the restriction of
to
is equal to the pull-back
This implies that symplectic manifolds have
no local invariants.
An area form on an oriented surface is symplectic.
Let
be a smooth manifold and let
be a one-form on the cotangent bundle
defined as follows.
If
is a vector tangent to
at a point
then
where
is the projection. In local coordinates the form
can be expressed as
The differential
is a symplectic form on the cotangent bundle
If
is a closed, i.e. compact and without boundary, symplectic
-manifold then the cohomology classes
are non-zero for
This follows from the fact that the cohomology class of the volume
form
is nonzero on a closed manifold. This necessary condition implies that spheres of dimension greater than
two are not symplectic. More generally, no closed manifold of the form
is symplectic for
The complex projective space
is symplectic with respect to its K\"ahler form.
Its pull back to a complex projective smooth manifold
is also symplectic.
More generally, every K\"ahler manifold is symplectic.
3 Symmetries
A diffeomorphism of a symplectic manifold
is called symplectic if it preserves
the symplectic form,
Sometimes such a diffeomorphism is called a symplectiomorphism.
The group of all symplectic diffeomorphisms of
is denoted by
It follows from the nondegeneracy of the symplectic form the map
defines an isomorphism
between the vector fields and the one-forms on a symplectic manifold
If the flow of a vector field
preserves the symplectic form we have that
Then the closedness
of the symplectic form implies that the one-form
is closed. It follows that the Lie algebra of
the group of symplectic diffeomorhism consists of the vector fields
for which the one-form
is closed. Hence it can be identified with the space of closed one-forms.
If the one-form is exact, i.e.
for some function
then the vector field
is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form
a group
called the group of Hamiltonian diffeomorphism. Its Lie algebra can be
identified with the quotient of the space of smooth functions on
by the constants.
4 Constructions
1 Products
The product of symplectic manifolds and
is a symplectic manifold with
respect to the form
for nonzero real numbers
Here
is the projection.
2 Bundles
A locally trivial bundle is called symplectic (resp. Hamiltonian) if its structure
group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms.
Example. The product of the Hopf bundle with the circle is a symplecti bundle
Indeed, the structure group is a group of rotations of the torus and hence it preserves the area.
As we have seen above the product
does not admit a symplectic form. This example
shows that, in general, the total space of a symplectic bundle is not symplectic.
Let is a compact symplectic bundle over a symplectic base.
According to a theorem of Thurston, if there exists a cohomology class
such that
its pull back to every fibre is equal to the class of the symplectic form of the fibre
then there exists a representative
of the class
such that
is a symplectic form on
for every big enough
3 Symplectic reduction
4 Surgery
5 Invariants
...
6 Classification/Characterization
...
7 Further discussion
...