The purpose of this post is to discuss a few basic facts about differentiable manifolds and state the Darboux theorem, which I will prove next time. (People who are looking for a more ambitious leap into symplectic geometry might want to try lewallen’s two posts over at Concrete Nonsense.)
A symplectic manifold is a smooth manifold equipped with a closed symplectic 2-form . In other words, is alternating and nondegenerate on each tangent space .
The basic example of a symplectic form is
on with coordinates .
This can also be written in a more invariant form, which will also give an invariant manner of making the cotangent bundle of any manifold into a symplectic manifold. First, we define a 1-form on . Let be the projection downwards. Given lying above , define
To make this clearer, here is an interpretation in local coordinates. Let be local coordinates for . Then coordinates for . Then
as is easily checked by working through the definitions. So we can define a canonical 2-form as ; this makes into a symplectic manifold.
The Darboux theorem
Unlike the case for Riemannian manifolds, symplectic manifolds are always locally isomorphic—or more precisely, symplectomorphic, i.e. diffeomorphic in a manner preserving the symplectic forms. There is thus no analog of the Riemann curvature tensor for symplectic manifolds.
The proof will rely on the Moser trick, which constructs the diffeomorphism using an isotopy induced by a certain time-dependent vector field.
Theorem 1 Let be a symplectic manifold. If , then there is a neighborhood containing and a diffeomorphism for such that , where is the canonical 2-form.
Lie derivatives and Cartan’s magic formula
Recall the operation of Lie derivative. Given a vector field on a manifold generating a local flow and a tensor , define
This is easily seen to be a derivation operator, i.e.
and one that commutes with contractions. It also commutes with symmetrization and alternation, so preserves forms.
Proposition 2 On vector fields , the Lie derivative is just the Lie bracket: .
This is proved by some fairly involved means in Spivak or Kobayashi-Nomizu. I don’t know why they don’t use the following argument, which I learned from Volume one of Partial Diffential Equations by Michael Taylor.
The idea is that since this is an invariant assertion, we can choose the local coordinates wisely. For instance, let’s suppose that we want to establish this at , and . Then (using the flow —this is a well-known lemma), we can choose local coordinates to straighten out , i.e. such that . Then if , it is clear from the definitions that
which is easily checked to coincide with the Lie bracket.
When , then there are two cases. If is a limit point of nonzero points of , then what we’ve already proved implies the claim by continuity. If not, then vanishes in a neighborhood of , and the flows are locally the identity, so this result becomes immediate.
We now recall one more operation, on -forms. If is a -form and a vector field, define the interior product
Proposition 3 (Cartan’s magic formula)We haveon -forms.
Kobayashi-Nomizu prove this using arguments about derivations and skew-derivations, but again it is not necessary. We repeat the same argument, and note that it suffices to treat the case . So pick local coordinates such that . Write
Adding these gives the result, since
Time-dependent vector fields
A time-dependent vector field is a map from an open subset of containing into such that . This can be regarded as a generalization of a time-independent vector field. These too generate differential equations: we can talk about an integral curve of a time-dependent vector field as one that satisfies
This is an ordinary differential equation. In particular, we can locally draw integral curves through a given point, and generate an analog of a “flow” in tthe time-independent case—the difference is that there is no semigroup property .
Proposition 4 Let be a manifold, an open subset, be an isotopy—that is is a diffeomorphism of into . Then corresponds in this manner to a time-dependent vector-field
Indeed, take .