examples of symplectic manifolds

Examples of symplectic manifolds:The most basic example of a symplectic manifold is ${ℝ}^{2n}$. If we choose coordinate functions$x_1,\mathrm{\dots },x_n,y_1,\mathrm{\dots }{y}_n$, then

$$\omega =\sum _{m=1}^{n}d{x}_m\wedge d{y}_m$$

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

Any orientable $2$-manifold is symplectic. Any volume form is a symplectic form.

If $M$ is any manifold, then the cotangent bundle $T^{*}M$ is symplectic.If $x_1,\mathrm{\dots },x_n$ are coordinates on a coordinate patch $U$ on $M$, and ${\xi }_1,\mathrm{\dots },{\xi }_n$ are the functions $T^{*}(U)\to ℝ$

$${\xi }_i(m,\eta )=\eta (\frac{\partial }{\partial x_i})(m)$$

at any point $(m,\eta )\in T^{*}(M)$, then

$$\omega =\sum _{i=1}^{n}d{x}_i\wedge d{\xi }_i.$$

(Equivalently, using the notation $\alpha$ from the entry Poincare 1-form, we can define $\omega =-d\alpha$.)

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.

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 compact, $2n$ dimensional and $M$ is a closed 2-form, consider the form ${\omega }^n$. If this form is exact, then ${\omega }^n$ must be 0 somewhere, and so $\omega$ is somewhere degenerate. Since the wedge of a closed and an exact form is exact, no power ${\omega }^m$ of $\omega$ can be exact. In particular, $H^{2m}(M)\ne 0$ for all $0\le m\ne n$, for any compact symplectic manifold.

Thus, for example, $S^n$ 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 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.