nondegenerate implies that has an even dimension. The closed condition means that the exterior derivative of vanishes. A symplectic manifold is a pair where is a smooth manifold and is a symplectic form. Assigning a symplectic form to is referred to as giving a symplectic structure. Examples Symplectic vector spaces Let be a basis for We define our symplectic form ω on this basis as follows: In this case the symplectic form reduces to a simple quadratic form. If In denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the block matrix: Cotangent bundles Let be a smooth manifold of dimension . Then the total space of the cotangent bundle has a natural symplectic form, called the Poincaré two-form or the canonical symplectic form Here are any local coordinates on and are fibrewise coordinates with respect to the cotangent vectors . Cotangent bundles are the natural phase spaces 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 manifolds. 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 are "soldered" to the velocities . 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 manifolds. A large class of examples come from complex algebraic geometry. Any smooth complex projective variety has a symplectic form which is the restriction of the Fubini—Study form on the projective space . Almost-complex manifolds Riemannian manifolds with an -compatible almost complex structure are termed almost-complex manifolds. 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. Lagrangian and other submanifolds There are several natural geometric notions of submanifold of a symplectic manifold : Symplectic submanifolds of (potentially of any even dimension) are submanifolds such that is a symplectic form on . Isotropic submanifolds are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an isotropic