DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Isotropic, coisotropic, symplectic, and Lagrangian submanifolds

Definition

Let $(M, ω)$ be symplectic and let $i : S ↪ M$ be a smooth embedded submanifold of constant dimension. It is isotropic, coisotropic, symplectic, or Lagrangian when $T_{p} S$ has the corresponding property from Isotropic, coisotropic, symplectic, and Lagrangian subspaces inside the symplectic vector space $(T_{p} M, ω_{p})$ for every $p \in S$ . Equivalently, the symplectic case says $i^{*} ω$ is nondegenerate, while the Lagrangian case says $T_{p} S = (T_{p} S)^{ω}$ at every point.

Depends on

Used by

Symplectic normal bundle of a symplectic submanifold Definition
Product and opposite symplectic manifolds Example
The zero section and cotangent fibres as Lagrangians Example
Every half-dimensional submanifold is Lagrangian False statement
A graph of a one-form is Lagrangian exactly when the form is closed Proposition
Characteristic distribution of a coisotropic submanifold is involutive Proposition
Lagrangian submanifolds have half dimension Proposition
Regular common level sets are Lagrangian submanifolds Proposition

Dependency tree · two levels

4 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)