Editing Symplectic manifold (section)

== Special cases and generalizations ==
* A symplectic manifold <math>(M, \omega)</math> is '''exact''' if the symplectic form <math>\omega</math> is [[closed and exact differential forms|exact]]. For example, the cotangent bundle of a smooth manifold is an exact symplectic manifold if we use the [[canonical symplectic form]].  The area 2-form on the 2-sphere is a symplectic form that is not exact.
* A symplectic manifold endowed with a [[metric tensor|metric]] that is [[Almost complex manifold#Compatible triples|compatible]] with the symplectic form is an [[almost Kähler manifold]] in the sense that the tangent bundle has an [[almost complex structure]], but this need not be [[integrability condition|integrable]].
* Symplectic manifolds are special cases of a [[Poisson manifold]].
* A '''multisymplectic manifold''' of degree ''k'' is a manifold equipped with a closed nondegenerate ''k''-form.<ref>{{cite journal |first1=F. |last1=Cantrijn |first2=L. A. |last2=Ibort |first3=M. |last3=de León |title=On the Geometry of Multisymplectic Manifolds |journal=J. Austral. Math. Soc. |series=Ser. A |volume=66 |year=1999 |issue=3 |pages=303–330 |doi=10.1017/S1446788700036636 |doi-access=free }}</ref>
* A '''polysymplectic manifold''' is a [[Legendre bundle]] provided with a polysymplectic tangent-valued <math>(n+2)</math>-form; it is utilized in [[Hamiltonian field theory]].<ref>{{cite journal |first1=G. |last1=Giachetta |first2=L. |last2=Mangiarotti |first3=G. |last3=Sardanashvily |author-link3=Gennadi Sardanashvily |title=Covariant Hamiltonian equations for field theory |journal=Journal of Physics |volume=A32 |year=1999 |issue=38 |pages=6629–6642 |doi=10.1088/0305-4470/32/38/302 |arxiv=hep-th/9904062 |bibcode=1999JPhA...32.6629G |s2cid=204899025 }}</ref>