Darboux's Theorem - Statement and First Consequences

Statement and First Consequences

The precise statement is as follows. Suppose that θ is a differential 1-form on an n dimensional manifold, such that dθ has constant rank p. If

θ ∧ (dθ)p = 0 everywhere,

then there is a local system of coordinates x₁,...,x_n-p, y₁, ..., y_p in which

θ = x₁ dy₁ + ... + x_p dy_p.

If, on the other hand,

θ ∧ (dθ)p ≠ 0 everywhere,

then there is a local system of coordinates x₁,...,x_n-p, y₁, ..., y_p in which

θ = x₁ dy₁ + ... + x_p dy_p + dx_p+1.

In particular, suppose that ω is a symplectic 2-form on an n=2m dimensional manifold M. In a neighborhood of each point p of M, by the Poincaré lemma, there is a 1-form θ with dθ=ω. Moreover, θ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart U near p in which

θ = x₁ dy₁ + ... + x_m dy_m.

Taking an exterior derivative now shows

ω = dθ = dx₁ ∧ dy₁ + ... + dx_m ∧ dy_m.

The chart U is said to be a Darboux chart around p. The manifold M can be covered by such charts.

To state this differently, identify R2m with Cm by letting z_j = x_j + i y_j. If φ : U → Cn is a Darboux chart, then ω is the pullback of the standard symplectic form ω₀ on Cn: