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

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Every orbit is an injectively immersed homogeneous space

Statement

Assume ${A C}_{ω}$ . For a smooth action of $G$ on $M$ and $x \in M$ , the map

${\overline{Φ}}_{x} : G / G_{x} ⟶ M, g G_{x} ⟼ g \cdot x,$

is a $G$ -equivariant injective immersion with image $G \cdot x$ . Transporting the quotient structure through this map gives the orbit its canonical immersed homogeneous-space structure. The immersion need not be an embedding.

Facts & Assumptions

Given: A smooth left action of $G$ on $M$ and a point $x \in M$ .

[F1]

The stabilizer is a closed embedded Lie subgroup. Stabilizers are closed embedded Lie subgroups.

[F2]

For a closed subgroup, $G / G_{x}$ is a smooth homogeneous manifold and the quotient map is a submersion. Quotient manifold by a closed Lie subgroup.

[F3]

The kernel of the orbit-map differential at the identity is $T_{e} G_{x}$ , and its tangent image is the infinitesimal orbit. Kernel of the infinitesimal orbit map.

[F4]

Constant-rank normal forms describe immersed images locally. The constant-rank theorem for manifolds.

[F5]

The preceding closed-subgroup and quotient results carry countable choice. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Proof

technique · factor the orbit map through its stabilizer cosets

1.1

If $g_{1} G_{x} = g_{2} G_{x}$ , then $g_{2}^{- 1} g_{1} \in G_{x}$ and $g_{1} \cdot x = g_{2} \cdot x$ , so the formula is well defined. Conversely, equality of the two orbit points puts $g_{2}^{- 1} g_{1}$ in $G_{x}$ , proving injectivity. Every orbit point is $g \cdot x$ , so the image is exactly $G \cdot x$ .

F1algebra

2.1

For $a, g \in G$ , ${\overline{Φ}}_{x} (a \cdot g G_{x}) = a g \cdot x = a \cdot {\overline{Φ}}_{x} (g G_{x})$ , so the map is $G$ -equivariant.

step 1.1algebra

3.1

By [F2], the quotient map $q : G \to G / G_{x}$ is a submersion and has smooth local sections. Since $Φ_{x} = {\overline{Φ}}_{x} \circ q$ , on the domain of such a section $s$ one has ${\overline{Φ}}_{x} = Φ_{x} \circ s$ , proving smoothness. If $v \in T_{e G_{x}} (G / G_{x})$ and $d {\overline{Φ}}_{x} (v) = 0$ , choose $X \in g$ with $d q_{e} X = v$ . Then $d Φ_{x} (X) = 0$ , so [F3] gives $X \in T_{e} G_{x} = \ker d q_{e}$ and hence $v = 0$ . Equivariance from step 2.1 transports this injectivity to every coset. Thus the factor is an injective immersion.

F2F3step 1.1step 2.1

4.1

The local form [F4] now makes the image locally an immersed coordinate plane, and transport along the bijection in step 1.1 gives the canonical intrinsic orbit structure. The induced $G$ -action is smooth and transitive by step 2.1. If $G_{x} = G$ , the orbit is a zero-dimensional point; if $G_{x} = {e}$ , its dimension is $\dim G$ . Self-accumulating immersed orbits are allowed, which is why embeddedness is not asserted. ${A C}_{ω}$ is used through [F1]--[F3], and no further choice is made.

F1F2F3F4F5step 1.1step 2.1step 3.1∎

Depends on

Used by

Transitive smooth actions identify M with G/H Corollary

Dependency tree · two levels

28 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)