CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

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Transitive smooth actions identify M with G/H

Statement

Assume ${A C}_{ω}$ . If a Lie group $G$ acts smoothly and transitively on a smooth manifold $M$ , then for every $x \in M$ the map

$G / G_{x} ⟶ M, g G_{x} ⟼ g \cdot x$

is a $G$ -equivariant diffeomorphism.

Facts & Assumptions

Given: ${A C}_{ω}$ , a transitive smooth left action of $G$ on $M$ , and $x \in M$ .

[A1]

The induced map $f : G / G_{x} \to M$ is a smooth equivariant injective immersion; transitivity makes it bijective. The Axiom of Countable Choice ( ${A C}_{ω}$ ), Every orbit is an injectively immersed homogeneous space, Quotient manifold by a closed Lie subgroup.

[F1]

Critical values of a smooth map are null, and a null set cannot be all of a positive-dimensional manifold. Morse-Sard for smooth manifolds, A null set has dense complement in a positive-dimensional manifold.

[F2]

An invertible differential gives a local diffeomorphism. The smooth inverse function theorem on manifolds.

Proof

technique · use Sard to prove equality of dimensions, then apply the inverse function theorem

1.1

Put $m = \dim (G / G_{x})$ and $n = \dim M$ . Since $f$ is an immersion, its differential is injective everywhere, so $m \leq n$ . If $n = 0$ , this forces $m = 0$ . Assume $n > 0$ .

A1algebra

2.1

If $m < n$ , no differential of $f$ is surjective, so every point in its image is a critical value. But $f$ is surjective by transitivity, so all of $M$ would be a null set by Morse–Sard [F1]. The dense-complement result [F1] would then say that the empty complement of $M$ is dense, impossible because $M$ contains $x$ . Therefore $m = n$ .

A1F1step 1.1contradiction

3.1

The injective differential of $f$ is now an isomorphism everywhere. By [F2], $f$ is a local diffeomorphism. A bijective local diffeomorphism has a smooth inverse, since its local inverses agree with its unique set-theoretic inverse. Thus $f$ is a diffeomorphism; its equivariance was already proved in [A1]. Zero-dimensional and disconnected cases are included. Countable choice is inherited through [A1].

A1F2step 1.1step 2.1∎

Depends on

Used by

Grassmannians and flag manifolds as homogeneous spaces Example
Real and complex projective spaces as homogeneous spaces Example
Spheres as SO(n+1)/SO(n) Example

Dependency tree · two levels

31 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)