Symplectic manifold

Definition
A symplectic manifold is a smooth even‑dimensional manifold $M$ equipped with a closed, non‑degenerate differential 2‑form $\omega$, called a symplectic form. Formally, $(M,\omega)$ satisfies

1. $d\omega = 0$ (closedness), and
2. For every point $p\in M$, the bilinear map $\omega_p : T_pM \times T_pM \to \mathbb{R}$ is non‑degenerate (i.e., $\omega_p(v,\cdot)=0$ implies $v=0$).

Overview
Symplectic manifolds provide the natural geometric framework for Hamiltonian mechanics, where $\omega$ encodes the canonical pairing between positions and momenta. The non‑degeneracy of $\omega$ forces the dimension of $M$ to be even, commonly expressed as $\dim M = 2n$. In local coordinates $(q^1,\dots,q^n,p_1,\dots,p_n)$, Darboux’s theorem guarantees the existence of a chart in which $\omega$ takes the standard form $\omega = \sum_{i=1}^n dq^i \wedge dp_i$. This local normal form underpins many structural results in symplectic geometry and topology, including the study of symplectomorphisms (diffeomorphisms preserving $\omega$), Lagrangian submanifolds, and Hamiltonian vector fields.

Etymology/Origin
The adjective “symplectic” derives from the Greek word symplekton (συμπλεκτόν), meaning “woven together” or “intertwined”. The term was introduced in the mathematical literature by Hermann Weyl in 1939 to describe structures where a bilinear form intertwines coordinates, building on earlier work by Élie Cartan on exterior differential systems and by André Weil on classical mechanics. The formal definition of a symplectic manifold emerged in the mid‑20th century through contributions of mathematicians such as André Weil, Vladimir Arnold, and Michael Atiyah.

Characteristics

• Even dimension: The non‑degeneracy condition forces $\dim M = 2n$.
• Closed form: The symplectic form satisfies $d\omega = 0$, making $(M,\omega)$ a special case of a closed differential form on a manifold.
• Darboux’s theorem: Locally, any symplectic manifold is symplectomorphic to $(\mathbb{R}^{2n},\sum dq^i\wedge dp_i)$.
• Symplectomorphisms: Diffeomorphisms $\phi : M \to M$ with $\phi^*\omega = \omega$ preserve the symplectic structure; they form an infinite‑dimensional group denoted $\mathrm{Symp}(M,\omega)$.
• Hamiltonian vector fields: For a smooth function $H : M \to \mathbb{R}$, there exists a unique vector field $X_H$ satisfying $\iota_{X_H}\omega = dH$; the flow of $X_H$ consists of symplectomorphisms.
• Compatibility with almost complex structures: A symplectic manifold often admits compatible almost complex structures $J$ such that $\omega(\cdot,J\cdot)$ defines a Riemannian metric, leading to the notion of a Kähler manifold when $J$ is integrable.

Related Topics

• Symplectic form – the closed, non‑degenerate 2‑form defining the structure.
• Hamiltonian mechanics – physical theory formulated on symplectic manifolds.
• Poisson manifold – a generalization where the bivector field may be degenerate; every symplectic manifold is a Poisson manifold with an invertible Poisson tensor.
• Lagrangian submanifold – a submanifold $L\subset M$ of half the dimension on which $\omega$ restricts to zero.
• Symplectomorphism group – the group of diffeomorphisms preserving $\omega$.
• Darboux theorem – local classification result for symplectic forms.
• Symplectic topology – the study of global properties of symplectic manifolds, including Gromov’s pseudoholomorphic curves and Floer homology.
• Kähler manifold – a complex manifold with a compatible symplectic (Kähler) form.

These elements collectively define the modern theory of symplectic manifolds and their applications across mathematics and theoretical physics.