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

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Regular common level sets are Lagrangian submanifolds

Statement

For a completely integrable system $F = (F_{1}, \dots, F_{n})$ on a $2 n$ -manifold, every nonempty regular common level $N = F^{- 1} (c)$ is an $n$ -dimensional Lagrangian submanifold. At each point,

$T_{p} N = span {X_{F_{1}} (p), \dots, X_{F_{n}} (p)} .$

Facts & Assumptions

Given: A completely integrable system and a nonempty regular fibre.

[F1]

At a regular value, $N$ is an embedded codimension- $n$ submanifold and $T_{p} N = \ker d F_{p}$ . A regular level set is an embedded submanifold, The tangent space of a regular level set is the kernel.

[F2]

The functions pairwise Poisson commute, and the $d F_{i}$ are independent on the dense open regular locus. Completely integrable Hamiltonian system.

[F3]

In a $2 n$ -dimensional symplectic vector space an isotropic $n$ -plane is Lagrangian, and a submanifold is Lagrangian exactly when its tangent spaces are Lagrangian subspaces. Equivalent characterizations of Lagrangian subspaces, Isotropic, coisotropic, symplectic, and Lagrangian submanifolds.

Proof

technique · direct

1.1

By [F1], $N$ has dimension $2 n - n = n$ and $T_{p} N = ⋂_{j} \ker d F_{j}$ . For every $i, j$ , $d F_{j} (X_{F_{i}}) = ω (X_{F_{j}}, X_{F_{i}}) = - {F_{j}, F_{i}} = 0$ , so each $X_{F_{i}} (p)$ is tangent.

F1F2given

2.1

The bundle isomorphism $ω^{♭}$ sends $X_{F_{i}}$ to $d F_{i}$ . Since $N$ is a regular fibre, [F1] says $d F_{p}$ has rank $n$ , so these $n$ vectors are independent. By dimension they span $T_{p} N$ . Their mutual symplectic pairings are the zero brackets ${F_{i}, F_{j}}$ from [F2], so $T_{p} N$ is isotropic.

F1F2step 1.1

3.1

Apply [F3] at every point: the $n$ -dimensional isotropic tangent spaces are Lagrangian. Thus $N$ is a Lagrangian submanifold and the displayed spanning formula holds.

F3step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

15 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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)