Editing Symplectic manifold (section)

== Definition ==
A '''symplectic form''' on a smooth [[manifold]] <math> M </math> is a closed non-degenerate differential [[2-form]] <math> \omega </math>.<ref name="Gosson">{{cite book |first=Maurice |last=de Gosson |title=Symplectic Geometry and Quantum Mechanics |year=2006 |publisher=Birkhäuser Verlag |location=Basel |isbn=3-7643-7574-4 |page=10 }}
</ref><ref name="Arnold">{{Cite book|first1=V. I.|last1=Arnold|first2=A. N.|last2=Varchenko|first3=S. M.|last3=Gusein-Zade|author-link1=Vladimir Arnold|author-link3=Sabir Gusein-Zade|author-link2=Alexander Varchenko|title=The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1|publisher=Birkhäuser|year=1985|isbn=0-8176-3187-9}}</ref> Here, non-degenerate means that for every point <math> p \in M </math>, the skew-symmetric pairing on the [[tangent space]] <math> T_p M </math> defined by <math> \omega </math> is non-degenerate. That is to say, if there exists an <math> X \in T_p M </math> such that <math> \omega( X, Y ) = 0 </math> for all <math> Y \in T_p M </math>, then <math> X = 0 </math>. Since in odd dimensions, [[skew-symmetric matrices]] are always singular, the requirement that <math> \omega </math> be nondegenerate implies that <math> M </math> has an even dimension.<ref name="Gosson"/><ref name="Arnold"/> The closed condition means that the [[exterior derivative]] of <math> \omega </math> vanishes. A '''symplectic manifold''' is a pair <math> (M, \omega) </math> where <math> M </math> is a smooth manifold and <math> \omega </math> is a symplectic form. Assigning a symplectic form to <math> M </math> is referred to as giving <math> M </math> a '''symplectic structure'''.