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.

Symplectomorphisms, local symplectomorphisms, and symplectic embeddings

Definition

Let $(M, ω_{M})$ and $(N, ω_{N})$ be symplectic manifolds. A smooth map $F : M \to N$ satisfying $F^{*} ω_{N} = ω_{M}$ , with pullback as in The pullback of a differential form, is

a symplectomorphism if $F$ is a diffeomorphism;
a local symplectomorphism if $F$ is a local diffeomorphism; and
a symplectic embedding if $F$ is a smooth embedding.

The pullback equality is literal; none of the three terms means merely volume preservation.

Depends on

Used by

A cohomology class obstructs a global symplectomorphism Counterexample
Canonical transformation Definition

Dependency tree · two levels

5 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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)