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

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G to G/H is a smooth principal H-bundle

Statement

Assume ${A C}_{ω}$ . For every closed subgroup $H \leq G$ , the canonical map $q : G \to G / H$ is a smooth principal $H$ -bundle: it is locally $H$ -equivariantly diffeomorphic to $U \times H$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , a finite-dimensional real Lie group $G$ , and a closed subgroup $H$ .

[A1]

The principal-bundle candidate, including the right-action convention, is fixed. The Axiom of Countable Choice ( ${A C}_{ω}$ ), The canonical principal-bundle candidate G to G/H.

[F1]

The quotient map is a surjective submersion, and the closed subgroup $H$ has its embedded Lie-subgroup structure. Quotient manifold by a closed Lie subgroup. Cartan closed subgroup theorem.

[F2]

A submersion admits a smooth local section near each point in its image. The constant-rank theorem for manifolds.

Proof

technique · trivialize using a local quotient section

1.1

By [F1], $q$ is a surjective submersion. For any $x_{0} \in G / H$ , [F2] therefore supplies an open neighborhood $U$ of $x_{0}$ and a smooth section $s : U \to G$ with $q \circ s = {id}_{U}$ .

A1F1F2

2.1

Define $Φ : U \times H ⟶ q^{- 1} (U), Φ (x, h) = s (x) h .$ It is smooth. It is bijective: every $g$ over $x$ has $s (x)^{- 1} g \in H$ , and that element is unique. Its inverse is $g ⟼ (q (g), s (q (g))^{- 1} g),$ which is smooth because the second component is a smooth $G$ -valued expression whose values lie in the embedded subgroup $H$ and, in local product coordinates, is exactly the smooth $H$ -coordinate. Thus $Φ$ is a diffeomorphism over $U$ .

A1step 1.1algebra

3.1

For $k \in H$ , $Φ (x, h k) = s (x) h k = Φ (x, h) k$ , so $Φ$ is equivariant for the required right action. Translating the identity chart by each $g \in G$ supplies such a chart around every coset $g H$ .

A1step 2.1algebra

4.1

These charts prove the principal-bundle assertion. If $H = G$ , this is the principal $G$ -bundle $G \to {*}$ ; if $H = {e}$ it is the identity bundle. No connectedness, normality, or effectiveness condition is needed. Countable choice is inherited through [A1] and [F1].

A1F1F2step 3.1∎

Depends on

Used by

Associated bundles Definition
The tangent bundle of G/H as an associated bundle Example

Dependency tree · two levels

34 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)