Editing Differential geometry (section)

===Symplectic geometry===
{{main|Symplectic geometry}}

[[Symplectic geometry]] is the study of [[symplectic manifold]]s. An '''almost symplectic manifold''' is a differentiable manifold equipped with a smoothly varying [[non-degenerate]] [[skew-symmetric matrix|skew-symmetric]] [[bilinear form]] on each tangent space, i.e., a nondegenerate 2-[[Differential form|form]] ''ω'', called the ''symplectic form''. A symplectic manifold is an almost symplectic manifold for which the symplectic form ''ω'' is closed:  {{nowrap|1=d''ω'' = 0}}.

A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a [[symplectomorphism]]. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The [[phase space]] of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of [[Joseph Louis Lagrange]] on [[analytical mechanics]] and later in [[Carl Gustav Jacobi]]'s and [[William Rowan Hamilton]]'s [[Hamiltonian mechanics|formulations of classical mechanics]].

By contrast with Riemannian geometry, where the [[curvature]] provides a local invariant of Riemannian manifolds, [[Darboux's theorem]] states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the [[Poincaré–Birkhoff theorem]], conjectured by [[Henri Poincaré]] and then proved by [[G.D. Birkhoff]] in 1912. It claims that if an area preserving map of an [[annulus (mathematics)|annulus]] twists each boundary component in opposite directions, then the map has at least two fixed points.<ref>The area preserving condition (or the twisting condition) cannot be removed. If one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.</ref>