Editing Symplectic geometry (section)

==Comparison with Riemannian geometry==
Symplectic geometry has a number of similarities with and differences from [[Riemannian geometry]], which is the study of [[differentiable manifold]]s equipped with nondegenerate, symmetric 2-tensors (called [[metric tensor]]s). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as [[curvature of Riemannian manifolds|curvature]]. This is a consequence of [[Darboux's theorem]] which states that a neighborhood of any point of a 2''n''-dimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of <math>\mathbb{R}^{2n}</math>. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and [[orientable]]. Additionally, if ''M'' is a closed symplectic manifold, then the 2nd [[de Rham cohomology]] [[group (mathematics)|group]] ''H''<sup>2</sup>(''M'') is nontrivial; this implies, for example, that the only [[n-sphere|''n''-sphere]] that admits a symplectic form is the [[sphere|2-sphere]]. A parallel that one can draw between the two subjects is the analogy between [[geodesics]] in Riemannian geometry and [[pseudoholomorphic curve]]s  in symplectic geometry: Geodesics are curves of shortest length (locally), while pseudoholomorphic curves are surfaces of minimal area. Both concepts play a fundamental role in their respective disciplines.