Editing Symplectic manifold (section)

== Lagrangian fibration ==
A '''Lagrangian fibration''' of a symplectic manifold ''M'' is a [[fibration]] where all of the [[Fiber bundle#Formal definition|fibres]] are Lagrangian submanifolds. Since ''M'' is even-dimensional we can take local coordinates {{nowrap|1=(''p''<sub>1</sub>,...,''p''<sub>''n''</sub>, ''q''<sup>1</sup>,...,''q''<sup>''n''</sup>),}} and by [[Darboux's theorem]] the symplectic form ''ω'' can be, at least locally, written as {{nowrap|1=''ω'' = &sum; d''p''<sub>''k''</sub> &and; d''q''<sup>''k''</sup>}}, where d denotes the [[exterior derivative]] and ∧ denotes the [[exterior product]]. This form is called the [[Poincaré two-form]] or the canonical two-form. Using this set-up we can locally think of ''M'' as being the [[cotangent bundle]] <math>T^*\R^n,</math> and the Lagrangian fibration as the trivial fibration <math>\pi: T^*\R^n \to \R^n.</math> This is the canonical picture.