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

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Stabilizers are closed embedded Lie subgroups

Statement

Assume ${A C}_{ω}$ . For a smooth action of a Lie group $G$ on a Hausdorff smooth manifold $M$ , every stabilizer $G_{x}$ is a closed embedded Lie subgroup of $G$ .

Facts & Assumptions

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

[A1]

The stabilizer is a subgroup and the orbit map $Φ_{x} (g) = g \cdot x$ is smooth. Orbits, stabilizers, and orbit maps of smooth actions.

[A2]

A closed subgroup has its unique embedded Lie-subgroup structure under countable choice. The Axiom of Countable Choice ( ${A C}_{ω}$ ), Cartan closed subgroup theorem.

[F1]

The manifold convention is Hausdorff, so singletons are closed. Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces.

Proof

technique · realize the stabilizer as a closed fibre

1.1A1F1

By [F1], ${x}$ is closed in $M$ . Since $Φ_{x}$ is continuous by [A1], $G_{x} = Φ_{x}^{- 1} ({x})$ is closed in $G$ . It is a subgroup by the action laws in [A1].

2.1A2step 1.1∎

Apply [A2] to the closed subgroup from step 1.1. It receives its unique embedded Lie-subgroup structure. If the action is trivial then $G_{x} = G$ ; if it is free then $G_{x} = {e}$ . No properness, transitivity, connectedness, or effectiveness is used. Countable choice is used only through [A2].

Stabilizers are closed embedded Lie subgroups

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources