Now that we’ve gotten a little comfortable with the idea of a symplectic vector space, it shouldn’t take a huge leap of the imagination to say there’s a similar notion for (smooth) manifolds, too: they’re called symplectic manifolds (surprise). A (real, smooth) symplectic manifold of dimension consists of a pair
, where
is the manifold, and
is a closed, non-degenerate 2-form on
(i.e., for all
,
is an alternating bilinear form on the tangent space
,
, and
is a volume form on
(equivalently, for all
,
is a non-degenerate bilinear form on
) ). In analogy with the case with symplectic vector spaces,
picks out certain types of submanifolds of
. That is, for a submanifold
, we say
is isotropic (resp., Lagrangian; resp., involutive) if, for all
,
is an isotropic (resp., Lagrangian; resp., involutive) subspace. So, at face value, there doesn’t seem to be much of a change from the ordinary case of symplectic vector spaces, at least regarding these special types of submanifolds. Wrong, I was. I hope to talk about some of these things today.
Example 1. So, for baby’s first example of a symplectic manifold, we look at , equipped with the 2-form
(fyi, we write
to denote the space of (
)
-forms on
) which has constant value
on
, for all
. If you understood proposition 4 from my post about symplectic linear algebra (here: https://brainhelper.wordpress.com/2014/05/04/symplectic-basics/), you shouldn’t have any trouble with this example. The biggest change here is needing to work with differential forms on
, not just a single bilinear form.
Let be a (global) choice of linear coordinates on
, and let
. Then,
for .
Example 2. As far as I’m (currently) concerned, these are the most important examples of symplectic manifolds: given any (real, smooth) manifold of dimension
, the cotangent bundle
has a canonical symplectic structure. This is important, so I’ll go through most of the details on this one.
We’re going to need to conjure up some “canonical” symplectic form . There are essentially two ways to do this: first, a coordinate-free definition of
; after that, we’ll see what
looks like in a system of coordinates to get a better feel for what we’re doing. The gist is that, on any cotangent bundle, we get a really useful 1-form for free, called the canonical 1-form (or tautological 1-form, or Liouville 1-form, or the Poincar\'{e} 1-form…),
.
Let be the projection map,
(i.e.,
is a point, and
is a covector at
). Then, using the pullback
, we define the value of
at
to be
.
This seems stranger/more abstruse than it actually is. From the definition of ,
(this checks out:
is, by definition, a linear map
, and
, so the composition at least makes sense). Finally, we define the “canonical” symplectic form
on
to be
(NB: some authors take
. It doesn’t really matter, since both choices give you a coordinate-independent construction of a symplectic form).
Now, suppose we have a local system of coordinates near
, with associated linear coordinates
. What does
look like in these coordinates? Well, we know
, so we just plug the coordinate expressions in and wind up with:
(since ). Hence, the canonical 1-form looks like
in the local coordinates
; consequently, the symplectic form
has the coordinate expression
near . In particular, since
is exact, it is automatically a closed 2-form, and in these coordinates, it is easy to show that
is non-degenerate, and that (had we started out with using coordinates, the above expression is independent of the coordinates chosen). The rest of the details are yours to check.
What’s the big deal? Why is important/useful? Well, for one, it satisfies the following universal property:
proposition 1: The canonical 1-form is uniquely characterized by the property that, for every 1-form
(i.e.,
is the graph of
), one has
.
proof.
First, we note that , and
is just the expression of in the local coordinates
on
. I’ll leave the proof of uniqueness of
to the reader 🙂
end proof.
Secondly, the canonical 1-form gives a really easy way to produce Lagrangian submanifolds of . Let
be a smooth 1-form on
, and denote by
the graph of
. Then,
proposition 2: is a Lagrangian submanifold of
if and only if
is closed.
proof:
Clearly, is a smooth submanifold of
of dimension
(it’s diffeomorphic to
itself). Then,
,
So vanishes on
if and only if
, i.e., if
is closed.
end proof.
(Aside: for a symplectic vector space (of dimension 2n), a basis
is called a symplectic basis, provided
. In such case, we obtain the relations
,
.
(). Additionally, in a symplectic basis, the Hamiltonian isomorphism
is given by
.
.
for (in the general case,
is defined by the formula
, where
is the canonical pairing, and
). I forgot to talk about this in previous posts, and the Hamiltionian isomorphism has a really neat analogue for symplectic manifolds.
end aside.)
So, I’m giving you fair warning now: for the majority of examples/ situations I’ll talk about, the symplectic manifolds will always be of the form for some smooth manifold
. Okay. I was talking (secretively) about symplectic bases of symplectic vector spaces, and what the Hamiltonian isomorphism looks like when expressed in such a basis. For vector spaces, all we needed was for the (duals of) the basis elements to fit together in a regular way to form the original symplectic form (i.e.,
). But, for symplectic manifolds, the symplectic linear algebra takes place on the tangent space to every point; we don’t have the same freedom to choose “global” symplectic bases. The next best thing would be, of course, if we can at least always locally construct some analogue of symplectic bases, spanned by some frame of vector fields which would be a symplectic basis on each tangent space in a neighborhood of a point. Praise be upon us, for this is in fact the case: Darboux’s theorem guarantees that, for a smooth, 2n-dimensional symplectic manifold
, for all
, there exists an open neighborhood
of
with smooth coordinate chart
(actually, this is a symplectomorphism as well!) such that
for all
(one sometimes calls
a system of Darboux coordinates, instead of/in addition to their symplectic properties). Luckily for us and our cotangent bundles, on
, any local system of cotangent coordinates
on
serves as a system of Darboux coordinates, by our construction of
. The most important thing to take away from Darboux’s theorem, however, is that all symplectic manifolds of a given dimension are (locally) the same! No distinguishing local invariants to be found here.
On , the Hamiltonian isomorphism is the (fiber-wise) linear isomorphism
, and we’ll use it to construct some of the most important objects in symplectic geometry: Hamiltonian vector fields (check out Noether’s theorem, Hamiltonian dynamics, etc.). Precisely, given a smooth (real-valued) function
on an open subset
, we define the Hamiltonian vector field of f, denoted by
, to be the image of the differential
under the Hamiltonian isomorphism
.
If we’ve chosen some local coordinates on
, our above discussion implies the vector field
is given by
Indeed, , so
, and
(for
). Therefore,
from which result is immediate.
I really want to tie all this material together to talk about isotropic,Lagrangian, and involutive submanifolds of , but I’ve already rambled on for quite a while in this post. If you’ve been following along these past few posts, you’ll remember that my original motivation for talking about these symplectic objects in
was for their relationship with the microsupport
of sheaves in
. More precisely, for all
,
is an involutive subset; if
is also
-constructible, then
is a Lagrangian subset of
. A little alarm should be going off in your head right now; there’s no reason at all for
to be a smooth submanifold of
(take, for example,
to be the constant sheaf supported on a singular subset of
), so how do these definitions apply? What details of the definition would we need to modify, and what changes? Let’s find out, next time.
References:
- M. Kashiwara and P. Schapira, Sheaves on Manifolds (Appendix A).
- A. Cannas de Silva, Lectures on Symplectic Geometry (http://www.math.ist.utl.pt/~acannas/Books/symplectic.pdf)
- Alex Zamorzaev, notes from a course on introductory symplectic topology (http://math.stanford.edu/~alzaor/257a.html).