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 1Let 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 2On 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

Then

Also,

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 .

Conversely:

Proposition 4Let be a manifold, an open subset, be anisotopy—that is is a diffeomorphism of into . Then corresponds in this manner to a time-dependent vector-field

Indeed, take .

December 26, 2009 at 12:55 pm

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