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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〔Darboux (1882).〕 who established it as the solution of the Pfaff problem.〔Pfaff (1814–1815).〕 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 2''n''-dimensional symplectic manifold can be made to look locally like the linear symplectic space C''n'' 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.〔Sternberg (1964) p. 140–141.〕 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 ''x''1,...,''x''''n''-''p'', ''y''1, ..., ''y''''p'' in which : θ = ''x''1 d''y''1 + ... + ''x''''p'' d''y''''p''. If, on the other hand, : θ ∧ (dθ)p ≠ 0 everywhere, then there is a local system of coordinates ''x''1,...,''x''''n''-''p'', ''y''1, ..., ''y''''p'' in which : θ = ''x''1 d''y''1 + ... + ''x''''p'' d''y''''p'' + d''x''''p''+1. In particular, suppose that ω is a symplectic 2-form on an ''n''=2''m'' 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 = ''x''1 d''y''1 + ... + ''x''''m'' d''y''''m''. Taking an exterior derivative now shows : ω = dθ = d''x''1 ∧ d''y''1 + ... + d''x''''m'' ∧ d''y''''m''. The chart ''U'' is said to be a Darboux chart around ''p''.〔Cf. with McDuff and Salamon (1998) p. 96.〕 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'' → C''n'' is a Darboux chart, then ω is the pullback of the standard symplectic form ω0 on C''n'': : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Darboux's theorem」の詳細全文を読む スポンサード リンク
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