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.

Free and proper Lie-group actions

Definition

A smooth action of a Lie group $G$ on a manifold $M$ is free if $G_{x} = {e}$ for every $x \in M$ .

It is proper if its action-graph map

$Θ : G \times M ⟶ M \times M, Θ (g, x) = (g \cdot x, x),$

is proper: $Θ^{- 1} (K)$ is compact for every compact $K \subseteq M \times M$ . The reversed coordinate convention $(x, g \cdot x)$ is equivalent by the factor-swap homeomorphism. Freeness and properness are independent conditions; neither is encoded by the other.

Depends on

Used by

A free irrational torus action that is not proper Counterexample
A proper nonfree action is not a principal bundle Counterexample
Integer translations on the line Example
A free action need not have a manifold quotient False statement
Local slice for a free proper action Lemma
Compact Lie-group actions are proper Proposition

Dependency tree · two levels

9 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)