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.

Smooth left actions of Lie groups

Definition

Depends on

Dependency tree · 2 levels

Sources