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# 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 $x_{1},\ldots,x_{{n}},y_{1},\ldots y_{n}$, then

$\omega=\sum_{{m=1}}^{n}dx_{m}\wedge dy_{m}$ |

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 $x_{1},\ldots,x_{n}$ are coordinates on a coordinate patch $U$ on $M$, and $\xi_{1},\ldots,\xi_{n}$ 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

$\omega=\sum_{{i=1}}^{n}dx_{i}\wedge d\xi_{i}.$ |

(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, $H^{{2m}}(M)\neq 0$ for all $0\leq m\neq n$, 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.

## Mathematics Subject Classification

53D05*no label found*

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