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

Orbits, stabilizers, and orbit maps of smooth actions

Definition

Depends on

Dependency tree · 2 levels

Sources