Editing Symplectomorphism (section)

==Comparison with Riemannian geometry==
Unlike [[Riemannian manifold]]s, symplectic manifolds are not very rigid: [[Darboux's theorem]] shows that all symplectic manifolds of the same dimension are locally isomorphic. In contrast, isometries in Riemannian geometry must preserve the [[Riemann curvature tensor]], which is thus a local invariant of the Riemannian manifold. 
Moreover, every function ''H'' on a symplectic manifold defines a [[Hamiltonian vector field]] ''X''<sub>''H''</sub>, which exponentiates to a [[one-parameter group]] of Hamiltonian diffeomorphisms. It follows that the group of symplectomorphisms is always very large, and in particular, infinite-dimensional. On the other hand, the group of [[isometry|isometries]] of a Riemannian manifold is always a (finite-dimensional) [[Lie group]]. Moreover, Riemannian manifolds with large symmetry groups are very special, and a generic Riemannian manifold has no nontrivial symmetries.