Symplectic manifold - Knowlpedia

A symplectic manifold is a pair $(M,\omega)$ where $M$ is a smooth manifold and $\omega\in\Omega^2(M)$ is a differential $2$ -form satisfying:

Nondegeneracy: for every $p\in M$ , the bilinear form $\omega_p:T_pM\times T_pM\to\mathbb R$ is nondegenerate. Equivalently, $\iota_v\omega_p=0\quad\Longrightarrow\quad v=0,$ so $v\mapsto\iota_v\omega_p$ identifies $T_pM$ with $T_p^*M$ .

Closedness: $d\omega=0$ .

Nondegeneracy is pointwise linear algebra, while closedness is a differential condition. A nondegenerate two-form that is not closed defines only an almost-symplectic manifold.

Closedness also gives the basic integral consequence

\int_{\partial C}\omega=\int_C d\omega=0

for every smooth singular $3$ -chain $C$ , by Stokes' theorem.

Dimension, orientation, and local form

The dimension of $M$ is even, say $2n$ , and

\frac{\omega^n}{n!}

is a nowhere-vanishing top-degree form. It supplies a canonical orientation and volume form. The symplectic Darboux theorem says that every point has local coordinates $(q_1,\ldots,q_n,p_1,\ldots,p_n)$ in which

\omega=\sum_{i=1}^n dq_i\wedge dp_i.

Thus symplectic forms of a fixed dimension have no local invariants analogous to Riemannian curvature; their distinguishing phenomena are global.

Maps and dynamics

A symplectic map $f:(M,\omega_M)\to(N,\omega_N)$ satisfies $f^*\omega_N=\omega_M$ . Such maps form the category of symplectic manifolds, and its isomorphisms are symplectomorphisms.

Given $H\in C^\infty(M)$ , nondegeneracy determines the Hamiltonian vector field $X_H$ from the house convention

\iota_{X_H}\omega=dH.

Its flow preserves $\omega$ . More generally, symplectic isotopies are generated by time-dependent vector fields whose contractions with $\omega$ are closed; Hamiltonian isotopies require those one-forms to be exact.

Examples

The standard form on $\mathbb R^{2n}$ is $\sum_i dq_i\wedge dp_i$ . Every cotangent bundle carries its canonical symplectic form. A Kähler manifold is symplectic after retaining its fundamental form and forgetting the metric and complex structure. A closed two-form may fail to be symplectic if it is degenerate; the zero form is the simplest example.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: Lectures 1–2, symplectic manifolds and Darboux coordinates.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapters 1 and 3.