Almost symplectic manifold

In differential geometry, an almost symplectic structure on a differentiable manifold $M$ is a two-form $\omega$ on $M$ that is everywhere non-singular.^[1] If in addition $\omega$ is closed then it is a symplectic form.

An almost symplectic manifold is an Sp-structure; requiring $\omega$ to be closed is an integrability condition.

References

↑ Ramanan, S. (2005), Global calculus, Graduate Studies in Mathematics, vol. 65, Providence, RI: American Mathematical Society, p. 189, ISBN 0-8218-3702-8, MR 2104612.

Further reading

Alekseevskii, D.V. (2001) [1994], "Almost-symplectic structure", Encyclopedia of Mathematics, EMS Press

Manifolds (Glossary)

Basic concepts

Main results (list)

Types of
manifolds

Vectors	Distribution Lie bracket Pushforward Tangent space bundle Torsion Vector field Vector flow
Covectors	Closed/Exact Covariant derivative Cotangent space bundle De Rham cohomology Differential form Vector-valued Exterior derivative Interior product Pullback Ricci curvature flow Riemann curvature tensor Tensor field density Volume form Wedge product
Bundles	Adjoint Affine Associated Cotangent Dual Fiber (Co) Fibration Jet Lie algebra (Stable) Normal Principal Spinor Subbundle Tangent Tensor Vector
Connections	Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Parallel transport

Related

Generalizations

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