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.

Equivariant maps and equivariant vector bundles

Definition

If $X$ and $Y$ are smooth left $G$ -manifolds, a smooth map $f : X \to Y$ is $G$ -equivariant if

$f (g \cdot x) = g \cdot f (x)$

for all $g \in G$ and $x \in X$ .

A smooth vector bundle $π : E \to M$ is a $G$ -equivariant vector bundle if $G$ acts smoothly on $E$ and $M$ , the projection is equivariant, and each map $E_{x} \to E_{g \cdot x}$ , $v \mapsto g \cdot v$ , is linear. Thus the total-space action is jointly smooth and consists of fibrewise-linear bundle maps covering the base action.

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed.