nLab
symplectic manifold

Definition

A $2n$-dimensional topological manifold $M$ is a real symplectic manifold if it has a symplectic atlas; a symplectic atlas consists of smooth charts ${\varphi }_i:U_i\to M$ as usual, such that the transition functions ${\varphi }_j^{-1}\circ {\varphi }_i:{\varphi }_i^{-1}({\varphi }_i(U_i)\cap {\varphi }_j(U_j))\to {\varphi }_j^{-1}({\varphi }_i(U_i)\cap {\varphi }_j(U_j))$ are preserve the standard symplectic form ${\omega }_0={\sum }_{i=1}^n{\mathrm{dx}}_i\wedge {\mathrm{dp}}_i$ on ${ℝ}^{2n}$ with the basis $(x_1,\dots ,x_n,p_1,\dots ,p_n)$.

Equivalently, a symplectic manifold is a differentiable manifold equipped with a smooth 2-form $\omega \in \Gamma ({\Lambda }^2{T}^{*}M)$ which is nondegenerate in the sense that ${\omega }^{\wedge n}=\omega \wedge \omega \wedge \cdots \wedge \omega$ has the maximal rank at every point $p\in M$, or equivalently such that $({\Lambda }^2{T}_p^{*}M,{\omega }_p)$ is a symplectic vector space for every point $p\in M$. We call such $\omega$ a symplectic form.

Higher versions

The categorification of the notion of symplectic manifold is n-symplectic manifold.

Shouldn’t there be an explanation how a symplectic manifold is a 0-symplectic manifold?

Urs Schreiber: such an explanation is currently at symplectic ∞-Lie algebroid. Should eventually be merged here into the $n$Lab proper.