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.

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