User:ReiVaX/GDT

Generalized Darboux's theorem is a theorem in symplectic topology which generalizes the Darboux's theorem.

The statement is as follows. Let M be a 2n-dimensional symplectic manifold with symplectic form &omega;. Let $$f_1, f_2, \ldots, f_r (r \leq n)$$ functions linearly invariant $$(df_1(p) \wedge \ldots \wedge df_r(p) \neq 0)$$ at each point such that {fi, fj} = 0 (they are within involution, {-,-} is the Poisson bracket). Then there are functions $$f_{i+1}, \ldots, f_n, g_1, \ldots, g_n$$ such that (fi, gi) is a symplectic chart of M, i.e.


 * $$\omega = \sum_{i=1}^{n} df_i(p) \wedge dg_i(p)$$.

Category:Symplectic topology Category:Mathematical theorems