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

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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.

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Compact connected regular fibres are tori

Statement

This item assumes ${A C}_{ω}$ , namely countable choice. In the propagated dependency chain, that assumption is required through Stabilizer of the $R^{n}$ -action on a compact connected regular fibre is a full lattice; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

Every compact connected regular fibre of a completely integrable system on a $2 n$ -dimensional symplectic manifold is diffeomorphic to the torus $T^{n} = R^{n} / Z^{n}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ and such a compact connected regular fibre $N$ .

[A1]

${A C}_{ω}$ is countable choice and is required here through Stabilizer of the $R^{n}$ -action on a compact connected regular fibre is a full lattice; after those supplied interfaces are fixed, the remaining local or finite calculation makes no additional countable-family choice.

[F1]

$N ≅ R^{n} / Γ$ for a full lattice $Γ$ . Stabilizer of the $R^{n}$ -action on a compact connected regular fibre is a full lattice.

Proof

technique · direct

1.1

Choose a lattice basis $γ_{1}, \dots, γ_{n}$ of $Γ$ . The linear isomorphism $A : R^{n} \to R^{n}$ sending the standard basis to this basis carries $Z^{n}$ onto $Γ$ .

A1F1

2.1

Therefore $A$ descends to a diffeomorphism $R^{n} / Z^{n} \to R^{n} / Γ$ , which composed with [F1] identifies $T^{n}$ with $N$ . For $n = 0$ , both are a point.

F1step 1.1∎

Depends on

Used by

Liouville–Arnold action–angle theorem Theorem

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)