A symplectic manifold is
A -dimensional topological manifold is
The non-degenracy of the symplectic form implies that it defines an isomorphism
The existence of a 2-form which is non-degenerate (but not necessarily closed) is equivalent to the existence of a Sp-structure on , a reduction of the structure group of the tangent bundle along the inclusion of the symplectic group into the general linear group
Such an Sp(2n)-structure is also called an almost symplectic structure on . Adding the extra condition that – the condition for integrability of G-structures – makes it a genuine symplectic structure. See at integrability of G-structures – Examples – Symplectic structure.
By the above, a symplectic manifold structure is an integrable -structure. Further reduction of the structure group along the maximal compact subgroup inclusion of the unitary group yields is an almost Hermitian structure. If that is again first order integrable then it is Kähler structure.
Such a refinement from symplectic to Kähler structure is also called a choice of Kähler polarization.
The condition in question is that the Lie derivative
vanishes. By Cartan's magic formula and using that this is equivalently
By the Poincare lemma it follows that there is locally a function with .
For a symplectic manifold, define a bilinear skew-symmetric map
In a coordinate chart this says that
The bracket makes a Poisson algebra.
every Kähler manifold is canonically also a symplectic manifold;
The notion of symplectic manifold is equivalent to that of symplectic Lie n-algebroid for . (See there.)
|type of subspace of inner product space||condition on orthogonal space|
|Lagrangian subspace||(for symplectic form)|
|symplectic space||(for symplectic form)|
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||discussed in:|
|0||symplectic manifold||symplectic manifold||symplectic geometry||Lagrangian submanifold||–||ordinary space of states (in geometric quantization)||geometric quantization|
|1||Poisson Lie algebroid||symplectic groupoid||2-plectic geometry||Poisson sigma-model||coisotropic submanifold (of underlying Poisson manifold)||brane of Poisson sigma-model||2-module = category of modules over strict deformation quantiized algebra of observables||extended geometric quantization of 2d Chern-Simons theory|
|2||Courant Lie 2-algebroid||symplectic 2-groupoid||3-plectic geometry||Courant sigma-model||Dirac structure||D-brane in type II geometry|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
See the references at symplectic geometry.
Discussion of the torsion-invariants of almost symplectic structures includes