Darboux's theorem

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

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.

Contents 1 Statement and first consequences 2 Comparison with Riemannian geometry 3 See also 4 Notes 5 References 6 External links Statement and first consequences The precise statement is as follows.[3] Suppose that {displaystyle theta } is a differential 1-form on an n dimensional manifold, such that {displaystyle mathrm {d} theta } has constant rank p. If {displaystyle theta wedge left(mathrm {d} theta right)^{p}=0} everywhere, then there is a local system of coordinates {displaystyle x_{1},ldots ,x_{n-p},y_{1},ldots ,y_{p}} in which {displaystyle theta =x_{1},mathrm {d} y_{1}+ldots +x_{p},mathrm {d} y_{p}} .

If, on the other hand, {displaystyle theta wedge left(mathrm {d} theta right)^{p}neq 0} everywhere, then there is a local system of coordinates ' {displaystyle x_{1},ldots ,x_{n-p},y_{1},ldots ,y_{p}} in which {displaystyle theta =x_{1},mathrm {d} y_{1}+ldots +x_{p},mathrm {d} y_{p}+mathrm {d} x_{p+1}} .

Note that if {displaystyle theta wedge left(mathrm {d} theta right)^{p}neq 0} everywhere and {displaystyle n=2p+1} then {displaystyle theta } is a contact form.

In particular, suppose that {displaystyle 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 {displaystyle theta } with {displaystyle mathrm {d} theta =omega } . Moreover, {displaystyle theta } satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart U near p in which {displaystyle theta =x_{1},mathrm {d} y_{1}+ldots +x_{m},mathrm {d} y_{m}} .

Taking an exterior derivative now shows {displaystyle 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 {displaystyle mathbb {R} ^{2m}} with {displaystyle mathbb {C} ^{m}} by letting {displaystyle z_{j}=x_{j}+{textit {i}},y_{j}} . If {displaystyle varphi colon Uto mathbb {C} ^{n}} is a Darboux chart, then {displaystyle omega } is the pullback of the standard symplectic form {displaystyle omega _{0}} on {displaystyle mathbb {C} ^{n}} : {displaystyle 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.

See also Carathéodory–Jacobi–Lie theorem, a generalization of this theorem. Symplectic basis 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 "Proof of Darboux's Theorem". PlanetMath. G. Darboux, "On the Pfaff Problem," transl. by D. H. Delphenich G. Darboux, "On the Pfaff Problem (cont.)," transl. by D. H. Delphenich Categories: Differential systemsSymplectic geometryCoordinate systems in differential geometryTheorems in differential geometryMathematical physics