examples of symplectic manifolds

Examples of symplectic manifolds: The most basic example of a symplectic manifoldMathworldPlanetmath is 2n. If we choose coordinate functions x1,,xn,y1,yn, then


is a symplectic form, and one can easily check that it is closed.

Any orientable 2-manifoldMathworldPlanetmath is symplectic. Any volume formMathworldPlanetmath is a symplectic form.

If M is any manifold, then the cotangent bundleMathworldPlanetmath T*M is symplectic. If x1,,xn are coordinates on a coordinate patch U on M, and ξ1,,ξn are the functions T*(U)


at any point (m,η)T*(M), then


(Equivalently, using the notation α from the entry Poincare 1-form, we can define ω=-dα.)

One can check that this behaves well under coordinate transformationsMathworldPlanetmath, and thus defines a form on the whole manifold. One can easily check that this is closed and non-degenerate.

All orbits in the coadjoint action of a Lie group on the dual of it Lie algebra are symplectic. In particular, this includes complex Grassmannians and complex projective spaces.

Examples of non-symplectic manifolds: Obviously, all odd-dimensional manifolds are non-symplectic.

More subtlely, if M is compactPlanetmathPlanetmath, 2n dimensional and M is a closed 2-form, consider the form ωn. If this form is exact, then ωn must be 0 somewhere, and so ω is somewhere degenerate. Since the wedge of a closed and an exact formMathworldPlanetmath is exact, no power ωm of ω can be exact. In particular, H2m(M)0 for all 0mn, for any compact symplectic manifold.

Thus, for example, Sn for n>2 is not symplectic. Also, this means that any symplectic manifold must be orientable.

Finally, it is not generally the case that connected sumsMathworldPlanetmathPlanetmath of compact symplectic manifolds are again symplectic: Every symplectic manifold admits an almost complex structureMathworldPlanetmath (a symplectic form and a Riemannian metricMathworldPlanetmath on a manifold are sufficient to define an almost complex structure which is compatible with the symplectic form in a nice way). In the case of a connected sum of two symplectic manifolds, there does not necessarily exist such an almost complex structure, and hence connected sums cannot be (generically) symplectic.

Title examples of symplectic manifolds
Canonical name ExamplesOfSymplecticManifolds
Date of creation 2013-03-22 13:12:31
Last modified on 2013-03-22 13:12:31
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Example
Classification msc 53D05