Abstract
In this paper we will give a proof of the classical Moser's lemma. Using it, we give a proof of the main Darboux theorem, which states that every point in a symplectic manifold has a neighborhood with Darboux coordinates.
Highlights
We will study in this note a theorem which plays a central role in symplectic geometry namely the Darboux theorem : the symplectic manifolds (M, ω) of dimension 2m are locally isomorphic to (R2m, ω)
By the Recovery Theorem 1, it follows that there exist local coordinates x1, xm+1, z1, Z2, ..., z2m−2 on a neighborhood U1 ⊂ U of x such that :
Remark 4 Suppose that the variety M is compact and connected. we show if ωt = ω0, M
Summary
We will study in this note a theorem which plays a central role in symplectic geometry namely the Darboux theorem : the symplectic manifolds (M, ω) of dimension 2m are locally isomorphic to (R2m, ω). If (M, ω) is a symplectic manifold of dimension 2m, in the neighborhood of each point of M , there exist local coordinates (x1, ..., x2m) such that : Lemma 1 Let {ωt}, 0 ≤ t ≤ 1, be a family of symplectic forms, differentiable in t.
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