The Quotient Manifold Theorem
This appendix proves Theorem 17.67 of Differentiable Manifolds, Tensors, and Curvature: when a Lie group \(G\) of dimension \(d\) acts smoothly, freely and properly on a manifold \(M_{n}\), the set of orbits carries exactly one smooth structure making the projection Equation (17.156) a submersion, its dimension is \(n-d\), its orbits are embedded copies of \(G\), and a function on it is smooth exactly when its pullback is. It is the second of the two standard ways a manifold is produced — the first, Theorem 17.62, cuts a manifold out with equations; this one makes one by identifying points — and it is the one physics reaches for whenever a symmetry is quotiented away.
The consumer that makes the theorem urgent is Theorem 28.47 of Symplectic Geometry of Phase Space: the Marsden–Weinstein reduced phase space is a level set of a momentum map divided by the isotropy group of that level, and the entire smooth structure of the reduced space comes from the statement proved here. The same theorem is what makes a configuration space of shapes out of a configuration space of frames, and what turns a gauge orbit space into a manifold wherever the action is free.
The proof runs in five movements: the orbit space is Hausdorff and second countable (The orbit space is Hausdorff and second countable); orbits are embedded submanifolds diffeomorphic to \(G\) (Orbits are embedded submanifolds); through every point there is a slice, a transversal meeting nearby orbits at most once (The slice); the slices are the charts (The charts and the smooth structure); and the universal property follows, with it the uniqueness of the structure (The universal property and uniqueness). The heart is the slice, and it is built with the constant rank theorem Theorem 17.63 and the inverse function theorem Corollary A28.7 and nothing else. In particular no Lie algebra, no exponential map and no fundamental vector field is used: only that \(G\) is a manifold on which multiplication and inversion are smooth.
What the derivation uses, and what it must state itself
From Differentiable Manifolds, Tensors, and Curvature: the definition of the action and of freeness and properness (Definition 17.66), smooth maps and their differentials (Definitions 17.49 and 17.52), immersions, submersions and embeddings (Definition 17.53), embedded submanifolds (Definition 17.54), the differentiable structure (Definition 17.47) and the constant rank theorem (Theorem 17.63) together with the observation of Remark 17.65 that it transfers to manifolds by being read in charts. From The Implicit Function Theorem: the inverse function theorem Corollary A28.7. From Topological and Metric Spaces: compactness (Definition 10.9), the continuous image of a compact set (Proposition 10.10) and homeomorphisms (Definition 10.7). From Lie Groups, Lie Algebras, and Fibre Bundles: nothing beyond the definition of a Lie group as a manifold with smooth multiplication and inversion.
Two separation properties are named in the statement of Theorem 17.67 and are defined nowhere in Part II; Topological and Metric Spaces carries neither. They are stated here, and the gap is recorded as a Part II debt in Remark A46.13.
A topological space \(X\) is Hausdorff if any two distinct points have disjoint open neighbourhoods; second countable if its topology has a countable basis, i.e. a countable family of open sets of which every open set is a union; and locally compact if every point has a compact neighbourhood. Rests on Definitions 10.1, 10.2 and 10.9.
Every manifold in the sense of Definition 17.48 is assumed Hausdorff and second countable, as is standard and as the statement of Theorem 17.67 presupposes when it asserts those properties of the quotient. Every manifold is also locally compact and first countable, both because it is locally homeomorphic to \(\R^{n}\): a closed coordinate ball is a compact neighbourhood, and the coordinate balls of rational radius about a point form a countable neighbourhood basis. First countability is what licenses the sequential arguments used below, in which a property is established by testing it on convergent sequences.
Let \(X\) be Hausdorff.
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Every compact \(C\subseteq X\) is closed.
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If \(X\) is in addition locally compact and \(F:Y\longrightarrow X\) is continuous and proper — the preimage of every compact set is compact — then \(F\) carries closed sets to closed sets.
Rests on Definition A46.1, Definition 10.9 and Proposition 10.10.
Derives Lemma A46.2. (1) Let \(q\notin C\). For each \(c\in C\) choose disjoint open \(V_{c}\ni q\) and \(W_{c}\ni c\); the \(W_{c}\) cover \(C\), so finitely many \(W_{c_{1}},\ldots,W_{c_{m}}\) do, and \(V_{c_{1}}\cap\cdots\cap V_{c_{m}}\) is an open neighbourhood of \(q\) disjoint from \(C\). Hence the complement of \(C\) is open.
(2) Let \(A\subseteq Y\) be closed and let \(q\) lie in the closure of \(F(A)\). Choose a compact neighbourhood \(K\) of \(q\). Then \(F^{-1}(K)\) is compact by properness, so \(A\cap F^{-1}(K)\) is compact — a closed subset of a compact set — and \(F\left(A\cap F^{-1}(K)\right)=F(A)\cap K\) is compact by Proposition 10.10, hence closed by (1). Every neighbourhood of \(q\) contained in the interior of \(K\) meets \(F(A)\), and the intersection lies in \(K\); so \(q\) lies in the closure of \(F(A)\cap K\), which is \(F(A)\cap K\) itself. Thus \(q\in F(A)\).
∎Throughout the rest of the section, \(G\) is a Lie group of dimension \(d\), \(M=M_{n}\) a manifold, and the action is smooth, free and proper in the sense of Definition 17.66. We write \(L_{g}:M\to M\) for the diffeomorphism \(P\mapsto g\cdot P\), with inverse \(L_{g^{-1}}\), and
for the map whose properness is the hypothesis. We write \(\theta_{P}:G\to M\), \(\theta_{P}(g)=g\cdot P\), for the orbit map, and \(k=n-d\).
The orbit space is Hausdorff and second countable
For every open \(U\subseteq M\) the set \(\pi(U)\) is open in \(M/G\), and \(\pi\) is continuous and surjective. Rests on Definitions 10.2, 10.6 and 17.66.
Derives Lemma A46.3. Continuity and surjectivity are the definition of the quotient topology. For openness,
a union of open sets, since each \(L_{g}\) is a diffeomorphism. A set whose preimage is open is open in the quotient topology.
∎The orbit space \(M/G\) is Hausdorff and second countable, and every orbit is closed in \(M\). Rests on Lemma A46.2, Lemma A46.3 and Definition 17.66.
Derives Proposition A46.4. The orbit relation is closed. \(M\times M\) is Hausdorff and locally compact, being a manifold, and \(\Theta\) of Equation (A46.1) is continuous and proper by hypothesis. Write
By Lemma A46.2(2) this set is closed — \(G\times M\) being closed in itself. Fixing \(P\) and intersecting with \(M\times\set{P}\) shows each orbit closed in \(M\).
Hausdorff. Let \(\pi(P)\neq\pi(Q)\), so \((Q,P)\notin\mathcal{R}\). Since \(\mathcal{R}\) is closed and the products of open sets form a basis of \(M\times M\), there are open \(U\ni P\) and \(V\ni Q\) with \(\left(V\times U\right)\cap\mathcal{R}=\varnothing\): no point of \(V\) lies on the orbit of a point of \(U\). Then \(\pi(U)\) and \(\pi(V)\) are open by Lemma A46.3, contain \(\pi(P)\) and \(\pi(Q)\), and are disjoint — a common point would be an orbit meeting both \(U\) and \(V\), which is what was excluded.
Second countable. Let \(\set{B_{i}}\) be a countable basis of \(M\). Each \(\pi(B_{i})\) is open. If \(W\subseteq M/G\) is open and \(\pi(P)\in W\), then \(P\in\pi^{-1}(W)\), which is open, so \(P\in B_{i}\subseteq\pi^{-1}(W)\) for some \(i\), whence \(\pi(P)\in\pi(B_{i})\subseteq\pi\left(\pi^{-1}(W)\right)=W\), the last equality by surjectivity of \(\pi\). So \(\set{\pi(B_{i})}\) is a countable basis.
∎Orbits are embedded submanifolds
For every \(P\in M\) the map \(\theta_{P}:G\to M\) has the same rank at every point of \(G\), and that rank is \(d\); that is, \(\theta_{P}\) is an injective immersion. Rests on Theorem 17.63, Definition 17.66 and Definition 17.53.
Derives Lemma A46.5. Constancy. Write \(\ell_{g}:G\to G\) for left translation \(h\mapsto gh\), a diffeomorphism. The associativity of the action gives \(\theta_{P}\left(gh\right)=g\cdot\left(h\cdot P\right)\), that is
Differentiating at the identity \(e\) and using the chain rule for differentials of smooth maps (Definition 17.52), \(\left(\theta_{P}\right)_{*g}\circ\left(\ell_{g}\right)_{*e} =\left(L_{g}\right)_{*P}\circ\left(\theta_{P}\right)_{*e}\). Both \(\left(\ell_{g}\right)_{*e}\) and \(\left(L_{g}\right)_{*P}\) are isomorphisms, so the ranks of \(\left(\theta_{P}\right)_{*g}\) and \(\left(\theta_{P}\right)_{*e}\) agree.
The rank is \(d\). Suppose it were \(\rho<d\). Read in charts, \(\theta_{P}\) is a \(C^{\infty}\) map of an open subset of \(\R^{d}\) into \(\R^{n}\) of constant rank \(\rho\), so by the constant rank theorem (Theorem 17.63, transferred to manifolds as in Remark 17.65) there are coordinates about \(e\) and about \(P\) in which it reads \(\left(u^{1},\ldots,u^{d}\right)\mapsto \left(u^{1},\ldots,u^{\rho},0,\ldots,0\right)\), which is Equation (17.152). Two distinct nearby points differing only in the coordinate \(u^{\rho+1}\) then have the same image, so \(\theta_{P}\) is not injective on any neighbourhood of \(e\) — and \(\theta_{P}\) is injective, because \(g\cdot P=h\cdot P\) gives \(\left(h^{-1}g\right)\cdot P=P\) and hence \(g=h\) by freeness (Definition 17.66). The rank is therefore \(d\), and \(\left(\theta_{P}\right)_{*g}\) is injective for every \(g\).
∎For every \(P\in M\) the orbit map \(\theta_{P}\) is an embedding, so \(G\cdot P\) is an embedded submanifold of \(M\) of dimension \(d\), diffeomorphic to \(G\). Rests on Lemma A46.5, Definition 17.53 and Definition 17.54.
Derives Proposition A46.6. By Lemma A46.5 the map is an injective immersion; by Definition 17.53 what remains is that it be a homeomorphism onto its image with the topology inherited from \(M\). Equivalently, since \(G\) is first countable, that \(g_{j}\cdot P\to g\cdot P\) in \(M\) implies \(g_{j}\to g\) in \(G\).
Let \(g_{j}\cdot P\to Q=g\cdot P\). The set \(C=\set{g_{j}\cdot P\mid j}\cup\set{Q}\) is compact — a convergent sequence with its limit — and so is \(C\times\set{P}\); properness makes \(\Theta^{-1}\left(C\times\set{P}\right)\) compact, and it contains every \(\left(g_{j},P\right)\). Hence the \(g_{j}\) lie in a compact subset \(K\) of \(G\). Since \(G\) is first countable, a compact subset of it is sequentially compact, so every subsequence of \(\left(g_{j}\right)\) has a further subsequence converging in \(K\); and if \(g_{j_{m}}\to g'\), then continuity of the action gives \(g'\cdot P=\lim g_{j_{m}}\cdot P=Q=g\cdot P\), whence \(g'=g\) by freeness. A sequence in a compact set all of whose convergent subsequences have the limit \(g\) converges to \(g\): otherwise some neighbourhood of \(g\) would be left by infinitely many terms, and those terms would have a convergent subsequence whose limit, lying in the complement of that neighbourhood, could not be \(g\). So \(g_{j}\to g\).
The image of an embedding is an embedded submanifold and the map is a diffeomorphism onto it, as recorded after Definition 17.54.
∎The slice
Let \(P\in M\). There are an open neighbourhood \(O\subseteq G\) of \(e\), an embedded \(k\)-dimensional submanifold \(S\subseteq M\) with \(P\in S\) carried by a single chart \(\varsigma:S\to B\subseteq\R^{k}\), and an open neighbourhood \(\Omega=O\cdot S\) of \(P\) in \(M\), such that
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the map \(F:O\times S\to\Omega\), \(F(g,Q)=g\cdot Q\), is a diffeomorphism; and
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\(S\) meets every orbit at most once: if \(Q,Q'\in S\) and \(Q'=g\cdot Q\) for some \(g\in G\), then \(g=e\) and \(Q=Q'\).
Rests on Lemma A46.5, Corollary A28.7 and Definition 17.66.
Derives Proposition A46.7. Choice of a transversal. By Lemma A46.5 the image \(V_{P}=\left(\theta_{P}\right)_{*e}\left(T_{e}G\right)\) is a \(d\)-dimensional subspace of \(T_{P}M\). Choose a chart \((\phi,W)\) about \(P\) with \(\phi(P)=0\); composing \(\phi\) with a linear automorphism of \(\R^{n}\) — itself a change of chart — we may assume \(\phi_{*P}\left(V_{P}\right)=\R^{d}\times\set{0}\). For a small ball \(B\subseteq\R^{k}\) put
an embedded \(k\)-submanifold through \(P\) with \(T_{P}S_{0}\oplus V_{P}=T_{P}M\), since \(\phi_{*P}\) carries the two summands onto the complementary \(\set{0}\times\R^{k}\) and \(\R^{d}\times\set{0}\).
The map \(F\) is a local diffeomorphism. On the \(n\)-manifold \(G\times S_{0}\) define \(F(g,Q)=g\cdot Q\), smooth as a restriction of the action. Its differential at \((e,P)\) acts on \(T_{e}G\oplus T_{P}S_{0}\) by
because \(F(\cdot,P)=\theta_{P}\) and \(F(e,\cdot)\) is the inclusion of \(S_{0}\). By the choice of \(S_{0}\) this is a linear isomorphism onto \(T_{P}M\). Read in charts, \(F\) is a smooth map between open subsets of \(\R^{n}\) with invertible derivative at the point, so the inverse function theorem (Corollary A28.7) makes it a diffeomorphism of a neighbourhood of \((e,P)\) onto an open neighbourhood of \(P\). Shrinking, that neighbourhood contains a product \(O\times S\) with \(O\ni e\) open in \(G\) and \(S\ni P\) open in \(S_{0}\); put \(\Omega=F\left(O\times S\right)=O\cdot S\), open in \(M\). This is (1).
Shrinking to a genuine slice. Suppose (2) failed for every choice of \(S\). Take \(S_{j}=\phi^{-1}\left(\set{0}\times B_{1/j}\right)\), \(j\in\N\), a basis of neighbourhoods of \(P\) in \(S_{0}\). For each \(j\) there would then be \(Q_{j},Q_{j}'\in S_{j}\) and \(g_{j}\neq e\) with \(g_{j}\cdot Q_{j}=Q_{j}'\). Both sequences converge to \(P\), so \(C=\set{Q_{j}}\cup\set{Q_{j}'}\cup\set{P}\) is compact and \(\Theta^{-1}\left(C\times C\right)\) is compact by properness; it contains every \(\left(g_{j},Q_{j}\right)\), so the \(g_{j}\) lie in a compact subset of \(G\) and a subsequence converges, \(g_{j_{m}}\to g\). Continuity of the action gives \(g\cdot P=\lim g_{j_{m}}\cdot Q_{j_{m}}=\lim Q_{j_{m}}'=P\), so \(g=e\) by freeness. Hence \(g_{j_{m}}\in O\) and \(Q_{j_{m}},Q_{j_{m}}'\in S\) for \(m\) large. But then
and the injectivity of \(F\) on \(O\times S\) forces \(g_{j_{m}}=e\), contradicting \(g_{j}\neq e\). So some \(S_{j}\) satisfies (2); replace \(S\) by it and shrink \(O\) and \(\Omega\) accordingly, which preserves (1).
∎The charts and the smooth structure
With \(O,S,\Omega\) as in Proposition A46.7, the set \(\pi(S)=\pi(\Omega)\) is open in \(M/G\) and \(\pi|_{S}:S\to\pi(S)\) is a homeomorphism. Writing \(\varrho:\Omega\to S\) for the second component of \(F^{-1}\), a smooth map, one has \(\varrho=\left(\pi|_{S}\right)^{-1}\circ\pi\) on \(\Omega\). Rests on Proposition A46.7, Lemma A46.3 and Definition 10.7.
Derives Lemma A46.8. Every point of \(\Omega\) is \(g\cdot Q\) with \(Q\in S\), so \(\pi(\Omega)=\pi(S)\), and \(\pi(\Omega)\) is open by Lemma A46.3. The map \(\pi|_{S}\) is continuous, surjective onto \(\pi(S)\) by definition, and injective by Proposition A46.7(2).
For \(R\in\Omega\) write \(R=F(g,Q)=g\cdot Q\) with \((g,Q)\in O\times S\) unique; then \(\varrho(R)=Q\) lies on the orbit of \(R\) and in \(S\), and by (2) it is the only point of \(S\) on that orbit. Hence \(\varrho=\left(\pi|_{S}\right)^{-1}\circ\pi\) on \(\Omega\), and \(\varrho\) is smooth because \(F^{-1}\) is.
It remains to see that \(\left(\pi|_{S}\right)^{-1}\) is continuous. The restriction \(\pi|_{\Omega}:\Omega\to\pi(S)\) is continuous, surjective and open (Lemma A46.3 applied to open subsets of \(\Omega\)), hence a quotient map: a subset of \(\pi(S)\) is open exactly when its preimage is. The map \(\varrho\) is continuous and constant on the fibres of \(\pi|_{\Omega}\), by the previous paragraph; so for \(U\subseteq S\) open the set \(\left(\left(\pi|_{S}\right)^{-1}\right)^{-1}(U)=\pi\left( \varrho^{-1}(U)\right)\) has preimage \(\varrho^{-1}(U)\), which is open, and is therefore open. Thus \(\left(\pi|_{S}\right)^{-1}\) is continuous and \(\pi|_{S}\) is a homeomorphism onto the open set \(\pi(S)\).
∎The maps
one for each slice of Proposition A46.7, form a \(k\)-dimensional atlas on \(M/G\). With the differentiable structure it generates (Definition 17.47), \(M/G\) is a smooth manifold of dimension \(k=n-d\), which is Equation (17.157), and \(\pi\) is a smooth surjective submersion. Rests on Lemma A46.8, Proposition A46.4 and Definition 17.53.
Derives Theorem A46.9. The charts cover and are homeomorphisms. Each \(u\) is a homeomorphism of the open set \(\pi(S)\) onto the open \(B\subseteq\R^{k}\), being the composition of the homeomorphism \(\left(\pi|_{S}\right)^{-1}\) of Lemma A46.8 with the chart \(\varsigma\) of \(S\); and every orbit meets some slice, namely one built at any of its points.
Smooth compatibility. Let \(S\) and \(S'\) be two slices with \(\pi(S)\cap\pi(S')\neq\varnothing\) and let \(\pi(Q_{0})\) lie in the intersection, \(Q_{0}\in S\). The transition map is \(u'\circ u^{-1}=\varsigma'\circ\tau\circ\varsigma^{-1}\), where \(\tau=\left(\pi|_{S'}\right)^{-1}\circ\pi|_{S}\) sends a point of \(S\) to the unique point of \(S'\) on its orbit. Put \(Q_{0}'=\tau(Q_{0})\in S'\) and let \(g_{0}\in G\) with \(Q_{0}'=g_{0}\cdot Q_{0}\). The set \(L_{g_{0}}^{-1}\left(\Omega'\right)\) is an open neighbourhood of \(Q_{0}\) in \(M\), and on it the map \(\varrho'\circ L_{g_{0}}\) is smooth and takes each point to the unique point of \(S'\) on its orbit — which is \(\tau\) where both are defined. Restricted to the embedded submanifold \(S\), a smooth map remains smooth, so \(\tau\) is smooth near \(Q_{0}\) as a map \(S\to S'\), and the transition map is smooth as a map between open subsets of \(\R^{k}\). Exchanging the roles of \(S\) and \(S'\) gives smoothness of the inverse. Together with Proposition A46.4 — Hausdorff and second countable — this makes \(M/G\) a smooth \(k\)-manifold.
\(\pi\) is a submersion. Fix \(P\in M\), take the slice at \(P\) and use \(F^{-1}:\Omega\to O\times S\) to give \(M\) the chart \(\left(\kappa,\varsigma\right)\circ F^{-1}\) near \(P\), where \(\kappa\) is any chart of \(G\) about \(e\). In these coordinates, and with the chart \(u\) downstairs, \(\pi\) reads
because \(\pi\left(F(g,Q)\right)=\pi(Q)\) and \(u\left(\pi(Q)\right)=\varsigma(Q)\). A projection has surjective differential, so \(\pi\) is a submersion at \(P\) (Definition 17.53); and \(P\) was arbitrary.
∎The universal property and uniqueness
Let \(p:M\to X\) be a smooth submersion between manifolds and let \(P\in M\). There are an open \(W\ni p(P)\) in \(X\) and a smooth \(s:W\to M\) with \(p\circ s=\id_{W}\) and \(s\left(p(P)\right)=P\). Rests on Theorem 17.63, Definition 17.53 and Definition 17.49.
Derives Lemma A46.10. A submersion has constant rank \(\dim X\), so Theorem 17.63, read in charts as in Remark 17.65, supplies coordinates \(\left(u^{1},\ldots,u^{\dim M}\right)\) about \(P\) and \(\left(y^{1},\ldots,y^{\dim X}\right)\) about \(p(P)\), both vanishing at the point, in which \(p\) reads \(\left(u^{1},\ldots\right)\mapsto\left(u^{1},\ldots,u^{\dim X}\right)\). In those coordinates set \(s(y)=(y,0)\), which is smooth and satisfies both requirements on the coordinate neighbourhood.
∎Let \(p:M\to X\) be a smooth surjective submersion and \(N\) a manifold. A map \(f:X\to N\) is smooth if and only if \(f\circ p\) is smooth. Rests on Lemma A46.10 and Definition 17.49.
Derives Proposition A46.11. If \(f\) is smooth then so is \(f\circ p\), a composition of smooth maps. Conversely let \(f\circ p\) be smooth and let \(x\in X\); pick \(P\) with \(p(P)=x\), which exists by surjectivity, and a smooth local section \(s\) on \(W\ni x\) from Lemma A46.10. On \(W\), \(f=f\circ p\circ s\) is a composition of smooth maps, hence smooth; and \(f\) is continuous there for the same reason. Smoothness is local, so \(f\) is smooth.
∎There is at most one differentiable structure on the topological space \(M/G\) for which \(\pi\) is a smooth submersion. Rests on Proposition A46.11 and Definition 17.47.
Derives Corollary A46.12. Let \(X\) and \(X'\) denote the same topological space \(M/G\) carrying two such structures. Since \(\pi:M\to X'\) is smooth and \(\pi:M\to X\) is a smooth surjective submersion, Proposition A46.11 applied to \(f=\id\) shows \(\id:X\to X'\) is smooth. Exchanging the two gives \(\id:X'\to X\) smooth. The identity is therefore a diffeomorphism, and two differentiable structures related by the identity diffeomorphism have the same smooth functions and hence the same maximal atlas.
∎Proof of Theorem 17.67. Derives Theorem 17.67. Assembling. The orbit space is Hausdorff and second countable by Proposition A46.4; it carries a smooth structure of dimension \(n-d\) for which Equation (17.156) is a submersion, by Theorem A46.9, and Equation (17.157) is that dimension count; the structure is the only one with that property, by Corollary A46.12; every orbit is an embedded submanifold diffeomorphic to \(G\), by Proposition A46.6; and a map \(f\) on the quotient is smooth exactly when \(f\circ\pi\) is, by Proposition A46.11 applied to the surjective submersion \(\pi\).
∎Reading the result
No theorem from outside this treatise is used, and the two analytic inputs — the inverse function theorem Corollary A28.7 and the constant rank theorem Theorem 17.63 — are proved in The Implicit Function Theorem and in Differentiable Manifolds, Tensors, and Curvature respectively. What is missing from Part II, and is supplied here rather than cited, is point-set vocabulary: Topological and Metric Spaces defines neither the Hausdorff property nor second countability nor local compactness, although Theorem 17.67 asserts the first two of the quotient and Definition 17.48 tacitly assumes all of them of a manifold. They are stated in Definition A46.1, and the two facts about them that the proof consumes — a compact subset of a Hausdorff space is closed, and a proper map into a locally compact Hausdorff space is closed — are proved in Lemma A46.2. That is a Part II debt of about a page, and it is the only one this section incurs.
Two things are deliberately not used, and the omission is what keeps the section self-contained. There is no Lie algebra anywhere: the standard proof shows the orbit map to be an immersion by computing its differential as the fundamental vector field of an element of \(\mathfrak{g}\) and appealing to freeness, which would import the machinery of Lie Groups, Lie Algebras, and Fibre Bundles; instead Lemma A46.5 gets the same conclusion from the constant rank theorem alone, by observing that a constant-rank map of rank less than \(d\) cannot be injective near a point. And there is no partition of unity and no smooth-structure transport theorem: the charts are the slices, and the transitions are computed from the action itself.
Freeness is used twice and properness three times, and it is worth saying where, because Example 17.68 shows what happens when either fails but not which step breaks.
Freeness makes the orbit map injective, which is what forces its rank to be full in Lemma A46.5 — and hence what gives every orbit the same dimension \(d\), without which no dimension count is possible. It is used again in Proposition A46.7 to identify the limiting group element as \(e\). When it fails, orbits through different points have different dimensions, and the quotient is a stratified space: this is the \(\SO(2)\) action of Example 17.68, whose orbit space is a half-line with a bad endpoint.
Properness gives the closedness of the orbit relation (Proposition A46.4), and with it the Hausdorff property of the quotient; it gives the embedding property of the orbit map (Proposition A46.6); and it gives the second shrinking in Proposition A46.7, which is what turns a local transversal into a genuine slice. The irrational flow on the torus in Example 17.68 fails at the first of these, and every later step fails with it: no slice exists, because every orbit returns arbitrarily close to every point. Rests on Proposition A46.7, Proposition A46.4 and Example 17.68.
The Quotient Manifold Theorem discharges the derivation owed at Theorem 17.67 of Differentiable Manifolds, Tensors, and Curvature. The statement is used at once in the same chapter, where it is the second of the two manufacturing theorems beside Theorem 17.62 and Corollary 17.64, and it is what makes Example 17.68 a statement about manifolds rather than about sets. Its principal consumer in the physical parts is Theorem 28.47 of Symplectic Geometry of Phase Space: symplectic reduction divides a level set of a momentum map — a manifold by Theorem 17.62 — by the isotropy group of that level, and every smooth statement about the reduced phase space, from the existence of its symplectic form to the descent of the dynamics, rests on the projection being a submersion with the universal property Proposition A46.11. The one hypothesis to watch there is the same one watched here: the action must be free, and where it is not the reduced space is singular and none of this applies.