A symplectic manifold is
a smooth manifold $X$ of even dimension $dim X = 2 n$;
equipped with a symplectic form:
a closed smooth 2-form $\omega \in \Omega^2_{cl}(X)$;
such that $\omega$ is non-degenerate, which means equivalently that
$\omega^{\wedge n}=\omega\wedge\omega\wedge\cdots\wedge\omega$ has the maximal rank at every point $p\in X$;
$(\wedge^2 T^*_p X,\omega_p)$ is a symplectic vector space for every point $p\in X$.
A $2n$-dimensional topological manifold $X$ is
a real symplectic manifold
equipped with a symplectic atlas:
an atlas consisting of smooth charts $\phi_i:U_i\to X$ as usual,
such that the transition functions $\phi_j^{-1}\circ\phi_i:\phi_i^{-1}(\phi_i(U_i)\cap\phi_j(U_j))\to \phi_j^{-1}(\phi_i(U_i)\cap\phi_j(U_j))$ preserve the standard symplectic form $\omega_0=\sum_{i=1}^n dx_i\wedge dp_i$ on $\mathbb{R}^{2n}$ with the basis $(x_1,\ldots,x_n,p_1,\ldots,p_n)$.
The non-degenracy of the symplectic form implies that it defines an isomorphism
between sections of the tangent bundle – vector fields – and sections of the cotangent bundle – differential 1-forms – on $X$ by the map
The vector fields in the image of the exact 1-forms under the isomorphism, remark 1, are called Hamiltonian vector fields.
This means that for $H \in C^\infty(X)$ a smooth function and $d H$ its differential 1-form, the corresponding Hamiltonian vector field $v_H \in \Gamma(T X)$ is the unique vector field such that
Equivalently, for $\phi \mathbb{R}^{2n} \to X$ a coordinate chart of $X$ and $\phi^*\omega = \omega_{i j} d x^i \wedge d x^j$ the symplectic form on this patch, the Hamiltonian vector field $v_H$ is
By Darboux's theorem every symplectic manifold has an atlas by coordinate charts $\mathbb{R}^{2n} \simeq U \hookrightarrow X$ on which the symplectic form takes the canonical form $\omega|_U = \sum_{k = 1}^n d x^{2k} \wedge d x^{2 k+1}$.
The existence of a 2-form $\omega \in \Omega^2(X)$ which is non-degenerate (but not necessarily closed) is equivalent to the existence of a Sp-structure on $X$, 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 $X$. Adding the extra condition that $d \omega = 0$ – the condition for integrability of G-structures – makes it a genuine symplectic structure. See at integrability of G-structures – Examples – Symplectic structure.
A metaplectic structure on a symplectic or almost symplectic manifold is in turn lift of the structure group to the metaplectic group.
By the above, a symplectic manifold structure is an integrable $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$-structure. Further reduction of the structure group along the maximal compact subgroup inclusion of the unitary group $U(n) \hookrightarow Sp(2n,\mathbb{R})$ 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.
For $(X, \omega)$ a symplectic manifold, the vector fields $v \in \Gamma(T X)$ that generate diffeomorphisms that preserve the symplectic structure are precisely the locally Hamiltonian vector fields.
The condition in question is that the Lie derivative
vanishes. By Cartan's magic formula and using that $d \omega = 0$ this is equivalently
By the Poincare lemma it follows that there is locally a function $H$ with $d H = \iota_v \omega$.
For $(X,\omega)$ a symplectic manifold, define a bilinear skew-symmetric map
by
In a coordinate chart this says that
The bracket $\{-,-\}$ makes $C^\infty(X)$ a Poisson algebra.
every Kähler manifold is canonically also a symplectic manifold;
the critical locus over any local action functional becomes a symplectic manifold after dividing out symmetries: the reduced phase space.
The notion of symplectic manifold is equivalent to that of symplectic Lie n-algebroid for $n = 0$. (See there.)
type of subspace $W$ of inner product space | condition on orthogonal space $W^\perp$ | |
---|---|---|
isotropic subspace | $W \subset W^\perp$ | |
coisotropic subspace | $W^\perp \subset W$ | |
Lagrangian subspace | $W = W^\perp$ | (for symplectic form) |
symplectic space | $W \cap W^\perp = \{0\}$ | (for symplectic form) |
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
See the references at symplectic geometry.
Discussion of the torsion-invariants of almost symplectic structures includes
The generalization of the notion of symplectic manifolds to dg-manifolds is sometimes known as PQ-supermanifolds , due to