DefinitionDefinition: Literature-sourcedProof: 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.

Orbits, stabilizers, and orbit maps of smooth actions

Definition

For a smooth left action of $G$ on $M$ and $x \in M$ , the stabilizer or isotropy subgroup, the orbit, and the orbit map are

$G_{x} = {g \in G : g \cdot x = x}, G \cdot x = {g \cdot x : g \in G},$

$Φ_{x} : G ⟶ M, Φ_{x} (g) = g \cdot x .$

The action laws make $G_{x}$ a subgroup, and joint smoothness makes $Φ_{x}$ smooth. No embeddedness, closedness, or manifold structure on the orbit is included in this definition.

Depends on

Smooth left actions of Lie groups

Used by

Kernel of the infinitesimal orbit map Proposition
Stabilizers are closed embedded Lie subgroups Theorem

Dependency tree · two levels

3 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

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)