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.

Homogeneous spaces of Lie groups

Definition

A homogeneous $G$ -space is a smooth manifold $M$ with a smooth transitive left action of a Lie group $G$ : for every $x, y \in M$ there is $g \in G$ with $g \cdot x = y$ .

If $H \leq G$ is a subgroup, $G / H$ denotes the set of left cosets $g H$ with the quotient topology induced by $q (g) = g H$ . It carries the set-theoretic left action $a \cdot (g H) = (a g) H$ . The notation alone does not assert that $G / H$ is a Hausdorff smooth manifold; that conclusion will require $H$ to be closed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)
Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)