
Review of Short Phrases and Links 
This Review contains major "Symplectic Form" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 If the symplectic form is exact on (e.g.
 Because a symplectic form is nondegenerate, so is the associated bilinear form.
 The symplectic form identifies the normal to with the dual, so by contracting indices, one obtains a cubic form on.
 Moreover, the associated form is preserved by J if and only if the symplectic form and if ω is tamed by J then the associated form is positive definite.
 In this case every symplectic form is a symmetric form, but not viceversa.
 The symplectic group Sp 2 n(R) consists of all 2 n × 2 n matrices preserving a nondegenerate skewsymmetric bilinear form on R 2 n (the symplectic form).
 The group Sp 2 n(R) of all matrices preserving a symplectic form is a Lie group called the symplectic group.
 If M is a 2 n dimensional manifold with symplectic form ω, then ω n is nowhere zero as a consequence of the nondegeneracy of the symplectic form.
 With a Euclidean inner product g, we have g(v, v) 0 for all nonzero vectors v, whereas a symplectic form ω satisfies ω(v, v) = 0.
 Obstructions to the existence of a symplectic form.
 That is, it is a linear transformation which preserves the symplectic form.
 In particular, it is important in the study of “regular systems” (a notion linked to the nondegeneracy of the symplectic form).
 Two cases must be considered, depending on the cohomology class of the symplectic form on the torus.
 More generally, the n th exterior power of the symplectic form on a symplectic manifold is a volume form.
 In particular, the symplectic form should be the curvature form of a connection of a Hermitian line bundle, which is called a prequantization.
 That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form.
 A pair, where is a manifold and is a symplectic form on.
 In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form.
 But there the one form defined is the sum of y i d x i, and the differential is the canonical symplectic form, the sum of.
 The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form.
 Considering with the symplectic structure given by the standard symplectic form (see [a1]), is the Hamiltonian vector field generated by.
 A Kähler form is a symplectic form, and so Kähler manifolds are naturally symplectic manifolds.
 There are symplectic submanifolds (potentially of any even dimension), where the symplectic form is required to induce a symplectic form on the submanifold.
 If the interior product of a vector field with the symplectic form is exact (and in particular, closed), it is called a Hamiltonian vector field.
 The meaning of the symplectic form being closed is not visible on the level of tangent spaces and is not relevant for the physics at hand.
 This follows from the closedness of the symplectic form and Cartan's formula for the Lie derivative in terms of the exterior derivative.
 Alternatively, a vector field is symplectic if its interior product with the symplectic form is closed.
 If, the cotangent bundle of a configuration space, with local coordinates, then the symplectic form is called canonical.
 The claim now is that is a symplectic form on the cotangent bundle.
 A symplectic space is a 2 n dimensional manifold with a nondegenerate 2form, ω, called the symplectic form.
 With a euclidean form g, we have g(v, v) 0 for all nonzero vectors v, whereas a symplectic form ω satisfies ω(v, v) = 0.
 Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form ω.
 Differential topology is the study of (global) geometric invariants without a metric or symplectic form.
 A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed nondegenerate 2form).
 A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed nondegenerate 2 form).
 A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: d ω = 0.
 A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a nondegenerate, bilinear, skewsymmetric and closed 2 form).
 A symplectic manifold endowed with a metric that is compatible with the symplectic form is a Kähler manifold.
 Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable complex structure compatible with the symplectic form.
Symplectic Form
 In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2form ω called the symplectic form.
 In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skewsymmetric, bilinear form ω called the symplectic form.
 A symplectic manifold is a pair (M,ω) where M is a smooth manifold and ω is a closed, nondegenerate, 2form on M called the symplectic form.
Categories
 Topology > Topological Spaces > Manifolds > Symplectic Manifold
 Nondegenerate
 Smooth Manifold
 Information > Science > Mathematics > Volume Form
 Physics > Theoretical Physics > Differential Geometry > Differentiable Manifold

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