Knowlpedia

Definition

Exact symplectic manifold

A symplectic manifold whose symplectic form is the exterior derivative of a global one-form.

Definition

An exact symplectic manifold is a (M,ω)(M,\omega) for which there exists a global one-form λ∈Ω1(M)\lambda\in\Omega^1(M) satisfying

ω=dλ.\omega=d\lambda.

Equivalently, the symplectic form is an . The one-form λ\lambda is called a primitive or Liouville form for ω\omega, but it is additional, noncanonical data: the assertion that (M,ω)(M,\omega) is exact requires only that at least one such primitive exist.

Choice of primitive

If λ\lambda and λ′\lambda' are primitives of the same symplectic form, then λ′−λ\lambda'-\lambda is closed. They differ by an exact one-form only when their de Rham cohomology classes agree. Many constructions in Liouville and Weinstein geometry therefore concern a specified pair (ω,λ)(\omega,\lambda), not just the property that ω\omega is exact.

Canonical example

On a T∗QT^*Q, the tautological one-form θ\theta gives a canonical exact symplectic form, written dθd\theta or −dθ-d\theta according to convention. Cotangent bundles model the local behavior of many exact symplectic and contact-geometric constructions.

Global restriction

A positive-dimensional compact symplectic manifold without boundary cannot be exact. Indeed, if ω=dλ\omega=d\lambda on a 2n2n-manifold with n≥1n\geq1, then

ωn=d(λ∧ωn−1),\omega^n=d(\lambda\wedge\omega^{n-1}),

so would force ∫Mωn=0\int_M\omega^n=0, contradicting that ωn\omega^n is a volume form. Compact exact symplectic manifolds are consequently studied with boundary and additional behavior of the primitive near that boundary.

References
  1. K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back, American Mathematical Society, 2012. AMS DOI record. Relevant: §2 and the conventions for Liouville forms.
  2. D. McDuff and D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Oxford DOI record. Relevant: exact symplectic forms and cotangent bundles.