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

1.1

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].

A1F1

2.1

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].

A2step 1.1∎

Depends on

Used by

Kernel of the infinitesimal orbit map Proposition
Every orbit is an injectively immersed homogeneous space Theorem

Dependency tree · two levels

24 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)