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 Encyclopedia of Keywords > Topology > Topological Spaces > Manifolds > Symplectic Manifold > Symplectic Form Michael Charnine

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### DefinitionsUpDw('Definitions','-Abz-');

1. If the symplectic form is exact on (e.g.
2. Because a symplectic form is nondegenerate, so is the associated bilinear form.

### Cubic FormUpDw('CUBIC_FORM','-Abz-');

1. The symplectic form identifies the normal to with the dual, so by contracting indices, one obtains a cubic form on.

### Associated FormUpDw('ASSOCIATED_FORM','-Abz-');

1. 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.

### Symmetric FormUpDw('SYMMETRIC_FORM','-Abz-');

1. In this case every symplectic form is a symmetric form, but not vice-versa.

### Matrices PreservingUpDw('MATRICES_PRESERVING','-Abz-');

1. The symplectic group Sp 2 n(R) consists of all 2 n × 2 n matrices preserving a nondegenerate skew-symmetric bilinear form on R 2 n (the symplectic form).
2. The group Sp 2 n(R) of all matrices preserving a symplectic form is a Lie group called the symplectic group.

### Symplectic Form ΩUpDw('SYMPLECTIC_FORM_ω','-Abz-');

1. 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.
2. 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.

### ExistenceUpDw('EXISTENCE','-Abz-');

1. Obstructions to the existence of a symplectic form.

### Linear TransformationUpDw('LINEAR_TRANSFORMATION','-Abz-');

1. That is, it is a linear transformation which preserves the symplectic form.

### ParticularUpDw('PARTICULAR','-Abz-');

1. In particular, it is important in the study of “regular systems” (a notion linked to the non-degeneracy of the symplectic form).

### TorusUpDw('TORUS','-Abz-');

1. Two cases must be considered, depending on the cohomology class of the symplectic form on the torus.

### Volume FormUpDw('VOLUME_FORM','-Abz-');

1. More generally, the n th exterior power of the symplectic form on a symplectic manifold is a volume form.

### Curvature FormUpDw('CURVATURE_FORM','-Abz-');

1. In particular, the symplectic form should be the curvature form of a connection of a Hermitian line bundle, which is called a prequantization.

### ManifoldUpDw('MANIFOLD','-Abz-');

1. That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form.
2. A pair, where is a manifold and is a symplectic form on.

### MathematicsUpDw('MATHEMATICS','-Abz-');

1. In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form.

### DifferentialUpDw('DIFFERENTIAL','-Abz-');

1. 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.

### InvariantsUpDw('INVARIANTS','-Abz-');

1. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form.

### Hamiltonian Vector FieldUpDw('HAMILTONIAN_VECTOR_FIELD','-Abz-');

1. Considering with the symplectic structure given by the standard symplectic form (see [a1]), is the Hamiltonian vector field generated by.

### Kähler ManifoldsUpDw('KÄHLER_MANIFOLDS','-Abz-');

1. A Kähler form is a symplectic form, and so Kähler manifolds are naturally symplectic manifolds.

### SubmanifoldUpDw('SUBMANIFOLD','-Abz-');

1. There are symplectic submanifolds (potentially of any even dimension), where the symplectic form is required to induce a symplectic form on the submanifold.

### ClosedUpDw('CLOSED','-Abz-');

1. 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.
2. 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.

### Lie DerivativeUpDw('LIE_DERIVATIVE','-Abz-');

1. This follows from the closedness of the symplectic form and Cartan's formula for the Lie derivative in terms of the exterior derivative.

### SymplecticUpDw('SYMPLECTIC','-Abz-');

1. Alternatively, a vector field is symplectic if its interior product with the symplectic form is closed.

### Cotangent BundleUpDw('COTANGENT_BUNDLE','-Abz-');

1. If, the cotangent bundle of a configuration space, with local coordinates, then the symplectic form is called canonical.
2. The claim now is that is a symplectic form on the cotangent bundle.

### OmegaUpDw('OMEGA','-Abz-');

1. A symplectic space is a 2 n -dimensional manifold with a nondegenerate 2-form, ω, called the symplectic form.
2. 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.
3. Then, its symplectic space is the cotangent bundle T*S with the canonical symplectic form ω.

### Differential TopologyUpDw('DIFFERENTIAL_TOPOLOGY','-Abz-');

1. Differential topology is the study of (global) geometric invariants without a metric or symplectic form.

### Differentiable ManifoldUpDw('DIFFERENTIABLE_MANIFOLD','-Abz-');

1. A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2-form).
2. A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2- form).

### Symplectic ManifoldUpDw('SYMPLECTIC_MANIFOLD','-Abz-');

1. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: d ω = 0.
2. A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a non-degenerate, bilinear, skew-symmetric and closed 2- form).
3. A symplectic manifold endowed with a metric that is compatible with the symplectic form is a Kähler manifold.

### Symplectic ManifoldsUpDw('SYMPLECTIC_MANIFOLDS','-Abz-');

1. Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable complex structure compatible with the symplectic form.

### Symplectic FormUpDw('SYMPLECTIC_FORM','-Abz-');

1. In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form.
2. In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form.
3. A symplectic manifold is a pair (M,ω) where M is a smooth manifold and ω is a closed, nondegenerate, 2-form on M called the symplectic form.

### CategoriesUpDw('Categories','-Abz-');

1. Topology > Topological Spaces > Manifolds > Symplectic Manifold
2. Nondegenerate
3. Smooth Manifold
4. Information > Science > Mathematics > Volume Form
5. Physics > Theoretical Physics > Differential Geometry > Differentiable Manifold
6. Books about "Symplectic Form" in Amazon.com  Short phrases about "Symplectic Form"   Originally created: April 04, 2011.   Please send us comments and questions by this Online Form   Please click on to move good phrases up.
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