Almost symplectic manifold
Encyclopedia
In differential geometry, an almost symplectic structure on a differentiable manifold
M is a two-form
ω on M which is everywhere non-singular. If, in addition, ω is closed
, then it is a symplectic form
.
An almost symplectic manifold is an Sp-structure
; requiring ω to be closed is an integrability condition.
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
M is a two-form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
ω on M which is everywhere non-singular. If, in addition, ω is closed
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
, then it is a symplectic form
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
.
An almost symplectic manifold is an Sp-structure
G-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
; requiring ω to be closed is an integrability condition.