# examples of symplectic manifolds

Examples of symplectic manifolds:
The most basic example of a symplectic manifold^{} is ${\mathbb{R}}^{2n}$. 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 ${T}^{*}(U)\to \mathbb{R}$

$${\xi}_{i}(m,\eta )=\eta (\frac{\partial}{\partial {x}_{i}})(m)$$ |

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 |