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.

Smooth left actions of Lie groups

Definition

Let $G$ be a Lie group and $M$ a smooth manifold. A smooth left action of $G$ on $M$ is a jointly smooth map

$a : G \times M ⟶ M, (g, x) ⟼ g \cdot x,$

such that, for all $g, h \in G$ and $x \in M$ ,

$e \cdot x = x, (g h) \cdot x = g \cdot (h \cdot x) .$

For each $g$ , the map $a_{g} (x) = g \cdot x$ is then a diffeomorphism with inverse $a_{g^{- 1}}$ . Joint smoothness is part of the definition; separate smoothness of individual orbit maps is not substituted for it.

Depends on

Used by

Equivariant maps and equivariant vector bundles Definition
Free and proper Lie-group actions Definition
Fundamental vector fields for a left action Definition
Homogeneous spaces of Lie groups Definition
Orbits, stabilizers, and orbit maps of smooth actions Definition
The irrational torus flow is free with dense orbits Lemma

Dependency tree · two levels

7 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)