PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

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Products and opposites of symplectic manifolds

Statement

If $(M, ω)$ and $(N, η)$ are symplectic, then $M^{-} = (M, - ω)$ is symplectic and

$Ω = {pr}_{M}^{*} ω + {pr}_{N}^{*} η$

is symplectic on $M \times N$ .

Facts & Assumptions

Given: Symplectic manifolds $(M, ω)$ and $(N, η)$ .

[F1]

A symplectic form is closed and pointwise nondegenerate. Symplectic form and symplectic manifold.

[F2]

Exterior differentiation commutes with pullback. The exterior derivative commutes with pullback.

Proof

technique · direct

1.1

The form $- ω$ is closed and has the same radical as $ω$ , so it is symplectic. By [F2], $d Ω = {pr}_{M}^{*} d ω + {pr}_{N}^{*} d η = 0$ .

F1F2algebra

2.1

Under $T_{(p, q)} (M \times N) = T_{p} M \oplus T_{q} N$ , if $Ω ((u, v), (u^{'}, v^{'})) = 0$ for all $(u^{'}, v^{'})$ , taking $v^{'} = 0$ and then $u^{'} = 0$ gives $u = 0$ and $v = 0$ by [F1]. Thus $Ω$ is nondegenerate, including when either factor has dimension zero.

F1step 1.1∎

Depends on

Used by

Product and opposite symplectic manifolds Example

Dependency tree · two levels

6 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)