Darboux's theorem

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This article is about Darboux's theorem in symplectic topology. For Darboux's theorem related to the intermediate value theorem, see Darboux's theorem (analysis).

Darboux's theorem is a theorem in symplectic topology which states that every symplectic manifold (of fixed dimension) is locally symplectomorphic. That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cⁿ with its canonical symplectic form. Darboux also proved a contact geometry analogue.

The precise statement is as follows. Let M be a 2n-dimensional symplectic manifold with symplectic form ω. Then around every point p in M there exists a coordinate chart U containing p with coordinates $(x_1, y_1, x_2, y_2, \ldots, x_n, y_n)$ such that on U, ω is of the form

$\omega = \sum_{i=1}^{n} dx^i \wedge dy^i$ .

Stated differently, if φ : U → Cⁿ is the coordinate chart then ω is the pullback of the standard form ω₀ on Cⁿ:

$\omega = \phi^{*}\omega_0\,$ .

The chart U is said to be a Darboux chart around p. The manifold M can be covered by such charts. The transition functions in such an atlas will be given by symplectic matrices.

[edit] Comparison with Riemannian geometry

This result implies that there are no local invariants in symplectic geometry: a Darboux basis can always be taken, valid near any given point. This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares.

It should be emphasized that the difference is that Darboux's theorem states that ω can be made to take the standard form in an entire neighborhood around p. In Riemannian geometry, the metric can always be made to take the standard form at any given point, but not always in a neighborhood around that point.