December 26, 2009

Proof of the Darboux theorem

Posted by Akhil Mathew under differential geometry | Tags: , , , |
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Now I’ll actually give the proof of the Darboux theorem that a symplectic manifold is locally symplectomorphic to \mathbb{R}^{2n} with the usual form.

Proof of the Darboux theorem

We will prove the equivalent:

Theorem 1 Let {M} be a manifold with closed symplectic forms {\omega_0, \omega_1}, and {p \in M} with {\omega_0(p) = \omega_1(p)}. Then there are neighborhoods {U,V} of {p} and a diffeomorphism {f: U \rightarrow V} with {f^*\omega_1 = \omega}.

 

The idea is to consider the continuously varying family of 2-forms

\displaystyle \omega_t = (1-t) \omega_0 + t \omega_1 = \omega_0 + t \alpha

where

\displaystyle \alpha = \omega_1 - \omega_0 .

We will consider a small neighborhood {U} of {p} and a smooth map { G: U \times [0,1] \rightarrow M } such that {G_t:=G(\cdot, t)} is a diffeomorphism, {G_0 = id}, and

\displaystyle G_t^*(\omega_t) = \omega_0 \ (*). (more…)

 

December 26, 2009

Symplectic manifolds, time-dependent vector fields, and Cartan’s magic formula

Posted by Akhil Mathew under differential geometry | Tags: , , , , , , |
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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 {M} equipped with a closed symplectic 2-form {\omega}. In other words, {\omega} is alternating and nondegenerate on each tangent space {T_p(M)}. 

The basic example of a symplectic form is 

\displaystyle \sum_i dx^i \wedge d\xi^i  

on {\mathbb{R}^{2n}} with coordinates {x^i, \xi^i, 1 \leq i \leq n}. 

This can also be written in a more invariant form, which will also give an invariant manner of making the cotangent bundle {T^*M} of any manifold {M} into a symplectic manifold. First, we define a 1-form {\alpha} on {T^*M}. Let {p: T^*M \rightarrow M} be the projection downwards. Given {v \in T^*M} lying above {\xi \in T^*M}, define 

\displaystyle \alpha(v) = \xi( p_*(v)). 

To make this clearer, here is an interpretation in local coordinates. Let {x^1, \dots, x^n} be local coordinates for {M}. Then {x^1, \dots, x^n, \xi^1, \dots, \xi^n} coordinates for {T^*M}. Then 

\displaystyle \alpha = \sum \xi^i dx^i  

as is easily checked by working through the definitions. So we can define a canonical 2-form {\omega} as {\omega = - d \alpha}; this makes {T^*M} into a symplectic manifold. (more…)