LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-14

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The tautological one-form is intrinsic and smooth

Statement

Assume ${A C}_{ω}$ . The tautological formula is coordinate independent and defines a smooth one-form. In cotangent coordinates $(q^{1}, \dots, q^{n}, p_{1}, \dots, p_{n})$ ,

$λ = \sum_{i = 1}^{n} p_{i} d q^{i} .$

Facts & Assumptions

Given: ${A C}_{ω}$ , a smooth $n$ -manifold $Q$ , and its canonical smooth cotangent bundle.

[A1]

${A C}_{ω}$ is countable choice. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

The tautological formula uses only the bundle projection and the natural covector--vector evaluation. Tautological one-form on a cotangent bundle.

Proof

technique · direct

1.1

The expression $p (d π ξ)$ in [F1] involves intrinsic maps and their natural pairing, so changing coordinates cannot change its value. It is linear in $ξ$ , hence defines a covector at every $(q, p)$ .

2.1

Write $p = \sum_{i} p_{i} d q^{i} ∣_{q}$ and $ξ = \sum_{i} a^{i} \partial_{q^{i}} + \sum_{i} b_{i} \partial_{p_{i}}$ . Since $d π (ξ) = \sum_{i} a^{i} \partial_{q^{i}}$ , [F1] gives $λ (ξ) = \sum_{i} p_{i} a^{i} = (\sum_{i} p_{i} d q^{i}) (ξ)$ . The displayed coefficients are smooth, so $λ$ is smooth.

A1F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

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Sources

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