DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Symplectic form and symplectic manifold

Definition

A symplectic form on a smooth manifold $M$ is a smooth two-form $ω$ such that

$d ω = 0$ , and
$(T_{p} M, ω_{p})$ is a symplectic vector space for every $p \in M$ .

The pair $(M, ω)$ is a symplectic manifold. Thus both closedness and pointwise nondegeneracy are required. A zero-dimensional manifold with its zero two-form satisfies the definition.

Depends on

Symplectic vector space

Used by

A nondegenerate nonclosed two-form in dimension at least four Counterexample
Action and angle coordinates Definition
Hamiltonian vector field and Hamiltonian function Definition
Isotropic, coisotropic, symplectic, and Lagrangian submanifolds Definition
Liouville vector field on an exact symplectic manifold Definition
Symplectic vector field Definition
Symplectomorphisms, local symplectomorphisms, and symplectic embeddings Definition
Every nondegenerate two-form is symplectic False statement
Products and opposites of symplectic manifolds Proposition
Nondegeneracy is equivalent to a nonvanishing top wedge Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)