examples of symplectic manifolds
is a symplectic form, and one can easily check that it is closed.
If is any manifold, then the cotangent bundle is symplectic. If are coordinates on a coordinate patch on , and are the functions
at any point , then
(Equivalently, using the notation from the entry Poincare 1-form, we can define .)
One can check that this behaves well under coordinate transformations, and thus defines a form on the whole manifold. One can easily check that this is closed and non-degenerate.
Examples of non-symplectic manifolds: Obviously, all odd-dimensional manifolds are non-symplectic.
More subtlely, if is compact, dimensional and is a closed 2-form, consider the form . If this form is exact, then must be 0 somewhere, and so is somewhere degenerate. Since the wedge of a closed and an exact form is exact, no power of can be exact. In particular, for all , for any compact symplectic manifold.
Thus, for example, for is not symplectic. Also, this means that any symplectic manifold must be orientable.
Finally, it is not generally the case that connected sums of compact symplectic manifolds are again symplectic: Every symplectic manifold admits an almost complex structure (a symplectic form and a Riemannian metric 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|
|Date of creation||2013-03-22 13:12:31|
|Last modified on||2013-03-22 13:12:31|
|Last modified by||mathcam (2727)|