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

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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

1.1F1F2algebra

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

2.1F1step 1.1∎

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.

Depends on

Used by

Product and opposite symplectic manifolds Example
Products and opposites of symplectic moment maps Proposition
Reduction commutes with products Proposition

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)