Darboux's theorem

Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry. The theorem is named after Jean Gaston Darboux^[1] who established it as the solution of the Pfaff problem.^[2]

One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cⁿ with its canonical symplectic form. There is also an analogous consequence of the theorem as applied to contact geometry.

Statement and first consequences

The precise statement is as follows.^[3] Suppose that $\theta$ is a differential 1-form on an n dimensional manifold, such that $\mathrm {d} \theta$ has constant rank p. If

\theta \wedge \left(\mathrm {d} \theta \right)^{p}=0

everywhere,

then there is a local system of coordinates $x_{1},\ldots ,x_{n-p},y_{1},\ldots ,y_{p}$ in which

\theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{p}\,\mathrm {d} y_{p}

If, on the other hand,

\theta \wedge \left(\mathrm {d} \theta \right)^{p}\neq 0

everywhere,

then there is a local system of coordinates ' $x_{1},\ldots ,x_{n-p},y_{1},\ldots ,y_{p}$ in which

\theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{p}\,\mathrm {d} y_{p}+\mathrm {d} x_{p+1}

In particular, suppose that $\omega$ 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 $\theta$ with $\mathrm {d} \theta =\omega$ . Moreover, $\theta$ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart U near p in which

\theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{m}\,\mathrm {d} y_{m}

Taking an exterior derivative now shows

\omega =\mathrm {d} \theta =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\ldots +\mathrm {d} x_{m}\wedge \mathrm {d} y_{m}

The chart U is said to be a Darboux chart around p.^[4] The manifold M can be covered by such charts.

To state this differently, identify $\mathbb {R} ^{2m}$ with $\mathbb {C} ^{m}$ by letting $z_{j}=x_{j}+{\textit {i}}\,y_{j}$ . If $\varphi \colon U\to \mathbb {C} ^{n}$ is a Darboux chart, then $\omega$ is the pullback of the standard symplectic form $\omega _{0}$ on ${\mathbb {C}}^{{n}}$ :

\omega =\phi ^{{*}}\omega _{0}.\,

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 of coordinate differentials.

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.

Notes

↑ Darboux (1882).
↑ Pfaff (1814–1815).
↑ Sternberg (1964) p. 140–141.
↑ Cf. with McDuff and Salamon (1998) p. 96.

References

Darboux, Gaston (1882). "Sur le problème de Pfaff". Bull. Sci. Math. 6: 14–36, 49–68.
Pfaff, Johann Friedrich (1814–1815). "Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi". Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin: 76–136.
Sternberg, Shlomo (1964). Lectures on Differential Geometry. Prentice Hall.
McDuff, D.; Salamon, D. (1998). Introduction to Symplectic Topology. Oxford University Press. ISBN 0-19-850451-9.

External links

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