Comparison With Riemannian Geometry
Unlike Riemannian manifolds, 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 XH, 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 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.
Read more about this topic: Symplectomorphism
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