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

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Nondegeneracy is equivalent to a nonvanishing top wedge

Statement

Let $M$ have dimension $2 n$ and let $ω$ be a smooth two-form. Then $ω$ is pointwise nondegenerate if and only if the top-degree form $ω^{n}$ is nowhere zero.

Facts & Assumptions

Given: A smooth $2 n$ -manifold $M$ and $ω \in Ω^{2} (M)$ .

[F1]

Wedge products of differential forms are defined pointwise. The wedge product of differential forms.

[F2]

Every alternating form has the symplectic-pair/radical normal form. Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even.

[F3]

Pointwise nondegeneracy is the linear clause in the definition of a symplectic form. Symplectic form and symplectic manifold.

Proof

technique · direct

1.1

Fix $p \in M$ . If $ω_{p}$ is nondegenerate, [F2] supplies a basis $e_{1}, \dots, e_{n}, f_{1}, \dots, f_{n}$ with $ω_{p} = \sum_{i} e^{i} \land f^{i}$ . Hence $ω_{p}^{n} = n! e^{1} \land f^{1} \land \dots \land e^{n} \land f^{n} \neq 0$ .

F1F2

1.2

Conversely, if $0 \neq v$ lies in the radical of $ω_{p}$ , then the graded contraction rule gives $ι_{v} (ω_{p}^{n}) = n (ι_{v} ω_{p}) \land ω_{p}^{n - 1} = 0$ . A nonzero top covector has nonzero contraction by every nonzero vector: extend $v$ to a basis and evaluate on the remaining basis vectors. Therefore $ω_{p}^{n} = 0$ .

F1algebra

2.1

Steps 1.1--1.2 prove the equivalence at every $p$ , which is exactly [F3]. For $n = 0$ , $ω^{0} = 1$ and the zero tangent space is nondegenerate, so the same conclusion holds. Closedness is irrelevant to this pointwise equivalence.

F3step 1.1step 1.2∎

Depends on

Used by

Symplectic manifolds have a canonical orientation and volume form Corollary

Dependency tree · two levels

11 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)