Editing Symplectic manifold (section)

== Examples ==
=== Symplectic vector spaces ===
{{main|Symplectic vector space}}

Let <math>\{v_1, \ldots, v_{2n}\}</math> be a basis for <math>\R^{2n}.</math> We define our symplectic form <math>\omega</math> on this basis as follows:

:<math>\omega(v_i, v_j) = \begin{cases} 1 & j-i =n \text{ with } 1 \leqslant i \leqslant n \\ -1 & i-j =n \text{ with } 1 \leqslant j \leqslant n \\ 0 & \text{otherwise} \end{cases}</math>

In this case the symplectic form reduces to a simple [[quadratic form]]. If <math>I_n</math> denotes the <math>n\times n</math> [[identity matrix]] then the matrix, <math>\Omega</math>, of this quadratic form is given by the <math>2n\times 2n</math> [[block matrix]]:

:<math>\Omega = \begin{pmatrix} 0 & I_n  \\ -I_n & 0 \end{pmatrix}. </math>

=== Cotangent bundles ===
Let <math>Q</math> be a smooth manifold of dimension <math>n</math>. Then the total space of the [[cotangent bundle]] <math>T^* Q</math> has a natural symplectic form, called the Poincaré two-form or the [[canonical symplectic form]]

:<math>\omega = \sum_{i=1}^n dp_i \wedge dq^i </math>

Here <math>(q^1, \ldots, q^n)</math> are any local coordinates on <math>Q</math> and <math>(p_1, \ldots, p_n)</math> are fibrewise coordinates with respect to the cotangent vectors <math>dq^1, \ldots, dq^n</math>. Cotangent bundles are the natural [[phase space]]s of classical mechanics.  The point of distinguishing upper and lower indexes is driven by the case of the manifold having a [[metric tensor]], as is the case for [[Riemannian manifold]]s. Upper and lower indexes transform contra and covariantly under a change of coordinate frames.  The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momenta <math>p_i</math> are "[[solder form|soldered]]" to the velocities <math>dq^i</math>. The soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.

=== Kähler manifolds ===
A [[Kähler manifold]] is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of [[complex manifold]]s. A large class of examples come from complex [[algebraic geometry]]. Any smooth complex [[projective variety]] <math>V \subset \mathbb{CP}^n</math> has a symplectic form which is the restriction of the [[Fubini-Study metric|Fubini—Study form]] on the [[projective space]] <math>\mathbb{CP}^n</math>.

=== Almost-complex manifolds ===
[[Riemannian manifolds]] with an <math>\omega</math>-compatible [[almost complex structure]] are termed [[almost-complex manifold]]s. They generalize Kähler manifolds, in that they need not be [[integrable]].  That is, they do not necessarily arise from a complex structure on the manifold.