Let $x\in A$; there exist open subsets $U_x$ of T and $U'_x$ of $T'$ such that $\{x\} \times A'\subset U_x\times U'_x\subset W:$ indeed, the compact subset $\{x\} \times A'$ of $T\times T'$ can be covered by a finite number of open sets contained in W, of the form $U_i\times U'_i$, with $x\in U_i$; it suffices to take $U_x=\bigcap_iU_i$ and $U'_x=\bigcup_iU'_i$.

Since A is compact, there exist points $x_1, . . . , x_m$ of A such that $A\subset \bigcup_iU_{x_i}$; put $U =\bigcup_iU_{x_i}$ and $U'=\bigcap_iU'_{x_i}$. Then $A\times A'\subset U\times U'\subset W$, hence the lemma.

In the remainder of this number, Y denotes a separated manifold of class $C^r$.

(i) The restriction of $\varphi$ to A is injective, and $\varphi$ is an immersion at every point of A;

(ii) there exists an open neighbourhood U of A such that $\varphi$ induces an embedding of U into Y.

When these conditions are satisfied, $\varphi$ is said to be an embedding in the neighbourhood of A.

We prove that (i) implies (ii), the other implication being clear. Assuming (i), there exists an open neighbourhood V of X containing A such that the restriction of $\varphi$ to V is an immersion (Differentiable and Analytic Manifolds, Results, 5.7.1). Denote by $\Gamma$ the set of points $(x, y)$ in $V\times V$ such that $\varphi (x) =\varphi (y)$, and by $\Delta$ the diagonal in $V\times V$. Then $\Delta$ is an open subset of $\Gamma :$ indeed, for all $x\in V$, there exists an open neighbourhood $U_x$ of $x$ such that the restriction of $\varphi$ to $U_x$ is injective, that is, such that $\Gamma \cap (U_x\times U_x) =$ $\Delta \cap (U_x\times U_x)$.

Since Y is separated, $\Gamma$ is closed in $V\times V$; consequently the complement W of $\Gamma$ -- $\Delta$ in $V\times V$ is open. By assumption, W contains $A\times A$; Lemma 1 implies that there exists an open subset $U'$ of V containing A such that $U'\times U'\subset W$, that is, such that the restriction of $\varphi$ to $U'$ is injective. Moreover, there exists an open neighbourhood U of A whose closure is compact and contained in $U'($General Topology, Chap. I, §9, no. 7, Prop. 10). Then $\varphi$ induces a homeomorphism from U to $\varphi$(U), and consequently from U to $\varphi (U)$; it follows that the restriction of $\varphi$ to U is an embedding (Differentiable and Analytic Manifolds, Results, 5.8.3).

Restricting X and Y if necessary, we can assume that $\varphi$ is étale and surjective. Denote by $\sigma :\varphi (A)\rightarrow A$ the inverse homeomorphism of $\varphi |A$. Since Y is metrizable (Differentiable and Analytic Manifolds, Results, 5.1.6), $\varphi (A)$ admits a fundamental system of paracompact neighbourhoods; hence, by General Topology, Chap. XI, there exist an open neighbourhood V of $\varphi (A)$ in Y and a continuous map $s: V\rightarrow X$, that coincides with $\sigma$ on $\varphi (A)$ and is such that $\varphi (s(y)) =y$ for all $y\in V$. Moreover, $s$ is topologically étale, so $s(V)$ is an open set U containing A. Then $\varphi$ induces a homeomorphism $\varphi '$ from U to V; by Differentiable and Analytic Manifolds, Results$, 5.7.8,\varphi '$ is an isomorphism.

In the remainder of this number, we assume that $r\not=\omega$.

Clearly, it suffices to prove the proposition for $r= 1$.

a) We show first that the subset J of $\mathscr{C}^1(X;Y)$ consisting of the morphisms that are immersions at every point of A is open. Consider the map $j_A:\mathscr{C}^1(X;Y)\times A\rightarrow J^1(X;Y)$ such that $j_A(\varphi , x) =j_x^1(\varphi ) ($Differentiable and Analytic Manifolds, Results, 12.1).

By definition of the topology on $\mathscr{C}^1$(X;Y), the map $\widetilde{j}_A:\varphi \rightarrow j_A(\varphi , .)$ from $\mathscr{C}^1(X;Y)$ to $\mathscr{C}(A;J^1(X;Y))$ is continuous; it now follows from General Topology, Chap. X, §3, no. 4, Th. 3, that $j_A$ is continuous.

On the other hand, let M be the set of jets $j$ in $J^1(X;Y)$ whose tangent map $T(j) : T_{\mathbf{s}(j)}(X)\rightarrow T_{\mathbf{b}(j)}(Y) ($Differentiable and Analytic Manifolds, Results, 12.3.4) is injective. The set M is open in $J^1(X;Y)$; indeed, it suffices to verify this assertion when X is an open subset of a finite dimensional vector space E, and Y is an open subset of a Banach space F; we are then reduced (Differentiable and Analytic Manifolds, Results, 12.3.1) to proving that the set of injective continuous linear maps is open in $\mathscr{L}$(E;F), which follows from Spectral Theory, Chap. III, §2, no. 7, Prop. 16.

We conclude from the preceding that the set $j_A^{-1}(M)$ is open in $\mathscr{C}^1(X;Y)\times A$, hence that its complement $\mathscr{F}$ is closed. Since A is compact, the projection pr$_1:\mathscr{C}^1(X;Y)\times A\rightarrow \mathscr{C}^1(X;Y)$ is a proper morphism, hence closed; consequently, the set J, which is equal to $\mathscr{C}^1(X;Y)$ -- pr$_1(\mathscr{F})$, is open in $\mathscr{C}^1(X;Y)$.

b) Let H be the subset of $J\times A\times A$ consisting of the elements $(f, x, y)$ such that $f(x) =f(y)$. It is clear that H contains $J\times \Delta$, where $\Delta$ denotes the diagonal in the product $A\times A$; we show that $H'= H$ -- $(J\times \Delta )$ is closed in $J\times A\times A$. Since $\mathscr{P}$ is the complement in J of the image of $H'$ under the proper projection pr$_1: J\times A\times A\rightarrow J$, this will imply the proposition.

The topology of $\mathscr{C}^1(X;Y)$ being finer than the topology of compact convergence, the map $(\varphi , x) \rightarrow \varphi (x)$ from $\mathscr{C}^1(X;Y)\times A$ to Y is continuous (General Topology, Chap. X, §3, no. 4, Cor. 1 of Th. 3); it follows that H is closed in $J\times A\times A$. Hence, it suffices to show that $J\times \Delta$ is open in H, in other words, that for all $\varphi \in J$ and all $x\in A$ there exists a neighbourhood $\Omega$ of $\varphi$ in J and a neighbourhood B of $x$ in X such that, for any morphism $\psi$ in $\Omega$, the restriction of $\psi$ to $A\cap B$ is injective.

Thus, the proposition follows from the following lemma:

Let U be a relatively compact open neighbourhood of $x$ isomorphic to a finite dimensional vector space, and such that $\varphi (U)$ is contained in the domain V of a chart. The set $\Omega_0$ of $\psi \in \mathscr{C}^1(X;Y)$ such that $\psi (U)\subset V$ is open in $\mathscr{C}^1$(X;Y), and the restriction map $\Omega_0\rightarrow \mathscr{C}^1(U;V)$ is continuous; we are thus reduced to proving the lemma when X = U and Y = V, in other words, we can assume that X is a finite dimensional vector space and Y is a Banach space. Choose norms on X and Y.

The linear map $D\varphi (x) : X\rightarrow$ Y is injective; denote its conorm by $q($Spectral Theory, Chap. III, §2, no. 6), so that, by definition, we have $\|D\varphi (x).t\| \geq q\|t\|$ for all $t\in X$. Let $\varepsilon \in \mathbf{R}$ be such that $0< \varepsilon < q/2$, and let B be a closed ball with centre $x$ such that $\|D\varphi (u)-D\varphi (x)\| \leq \varepsilon$ for all $u\in B$. Denote by $\Omega$ the subset of $\mathscr{C}^1(X;Y)$ consisting of the morphisms $\psi$ such that $\|D\psi (u)-D\varphi (u)\| \leq \varepsilon$ for all $u\in B$; it is open by definition of the topology of $\mathscr{C}^1(X;Y)$. For $\psi \in \Omega$, put $\psi_0=\psi -D\varphi (x)$. We have $\|D\psi_0(u)\| \leq 2\varepsilon$ for all $u\in B$, and consequently $\|\psi_0(u)-\psi_0(v)\| \leq 2\varepsilon \|u-v\|$ for all $u$ and $v$ in B (Differentiable and Analytic Manifolds, Results, 2.2.3). It follows that

Consequently, the restriction of $\psi$ to B is injective, hence the lemma.

Let $(U_i, \varphi_i,E_i)_{i\in I}$ be a finite family of charts of X whose domains cover A. We extend $\varphi_i$ to a map from X to $E_i$ (also denoted by $\varphi_i$) by putting $\varphi_i(x) = 0$ for $x \notin U_i$. Let $(V_i)_{i\in I}$ be a covering of A by open subsets of X such that $\overline{V}_i\subset U_i$ for all $i\in I$ (the existence of such a covering follows from General Topology, Chap. IX, §4, no. 3, Cor. 1 of Th. 3, applied to the compact space $X'$ obtained by adjoining to X a point at infinity and the covering of $X'$ consisting of the open sets $U_i(i\in I)$ and $X'$ -- A). For all $i\in I$, let $\alpha_i$ be a numerical function of class $C^r$ on X, equal to 1 at every point of $V_i$, and with support contained in $U_i($Differentiable and Analytic Manifolds, Results, 5.3.6).

Consider the map $\varphi : X\rightarrow \bigoplus_{i\in I}(E_i\oplus \mathbf{R})$ defined by

For all $i\in I$, the map $\alpha_i\varphi_i$ is of class $C^r$ (since its restrictions to $U_i$ and to the complement of the support of $\alpha_i$ are), and its restriction to $V_i$ is an embedding; it follows that $\varphi$ is a morphism of class $C^r$ and is an immersion at every point of A. We show that the restriction of $\varphi$ to A is injective. Let $x, y$ be two points of A such that $\varphi (x) =\varphi (y)$, and let $i\in I$ be such that $x\in V_i$. Then $\alpha_i(x) = 1$, so $\alpha_i(y) = 1$, which implies that $y\in U_i$; but we also have $\varphi_i(x) =\varphi_i(y)$, so $x=y$ since $\varphi_i$ induces an embedding of $U_i$ into $E_i$.

It can be shown$^9$ that every separated manifold, countable at infinity and of

pure dimension $n$, embeds in $\mathbf{R}^{2n}$; for a weaker result, cf. Exercise 2.

$^9$ See H. WHITNEY, The self-intersections of a smooth $n$-manifold in $2n$-space,

Ann. of Math., 45 (1944), pp. 220-246.

Put F = X -- W and V = X-- GF. Then V is open (General Topology, Chap. III, §4, no. 1, Cor. 1 of Prop. 1), stable under G, and $A\subset V\subset W$.

Replacing A by the compact set GA, we are reduced to the case in which A is stable under G.

Let $E_0$ be a finite dimensional vector space such that there exists an element of $\mathscr{C}^r(X;E_0)$ that is an embedding in the neighbourhood of A (no. 1, Prop. 4); the set $\mathscr{P}$ of morphisms having this property is a non-empty open subset of $\mathscr{C}^r(X;E_0)$ (no. 1, Prop. 3). Consider the continuous linear representation of the compact group G on the space $\mathscr{C}^r(X;E_0) ($§6, no. 4, Lemma 4). By the Peter-Weyl theorem (Spectral Theory, in preparation), the union of the finite dimensional subspaces stable under G is dense in $\mathscr{C}^r(X;E_0)$; hence, there exists an element $\varphi_0$ of $\mathscr{P}$ such that the maps $x \rightarrow \varphi_0(gx)$, for $g\in G$, generate a finite dimensional vector subspace $E_1$ of $\mathscr{C}^r(X;E_0)$, which is clearly stable under the operation of G.

Take E to be the space Hom$_{\mathbf{R}}(E_1,E_0),\rho$ to be the representation of G on E induced by the action on $E_1$, and $\varphi : X\rightarrow E$ to be the map that associates to $x\in X$ the linear map $\psi \rightarrow \psi (x)$ from $E_1$ to $E_0$. This is a morphism of class $C^r$; for $x\in X,g\in G,\psi \in E_1$, we have (denoting by $\tau (g)$ the automorphism $x \rightarrow gx$ of X):

Let $\alpha :$ Hom$_{\mathbf{R}}(E_1,E_0)\rightarrow E_0$ be the linear map $u \rightarrow u(\varphi_0)$; we have $\alpha \circ \varphi =\varphi_0$, so $\varphi$ is an embedding in the neighbourhood of A because $\varphi_0$ is one. Hence, there exists an open neighbourhood U of A such that the restriction of $\varphi$ to U is an embedding; we can choose U stable under G by Lemma 3, hence the theorem.

Apply Cor. 1 to the canonical operation of G on the compact manifold $G/H$. This gives an analytic linear representation $\rho : G\rightarrow \mathbf{G}\mathbf{L}(E)$ and an embedding $\varphi : G/H\rightarrow E$ such that $\varphi (gx) =\rho (g)\varphi (x),g\in G, x\in G/H$. Let $\overline{e}\in G/H$ be the class of $e\in G$, and $v=\varphi (\overline{e})$ its image. For all $g\in G$, we have $\rho (g)v=v\Leftarrow \Rightarrow \varphi (g\overline{e}) =\varphi (\overline{e})\Leftarrow \Rightarrow g\overline{e}= \overline{e}\Leftarrow \Rightarrow g\in H$.

The space $X/G$ is locally compact (General Topology, Chap. III, §4, no. 5, Prop. 11). Its connected components are the images of the connected components of X, which are countable at infinity (General Topology, Chap. I, §9, no. 10, Th. 5); thus, they are themselves countable at infinity, which implies that $X/G$ is paracompact (loc. cit.). Hence, there exists a locally finite covering $(U'_{\alpha})_{\alpha\in I}$ of $X/G$ by relatively compact open sets, and a covering $(V'_{\alpha})_{\alpha\in I}$ such that $\overline{V}'_{\alpha}\subset U'_{\alpha}$ for all $\alpha \in I ($General Topology, Chap. IX, §4, no. 3, Cor. 1 of Th. 3); taking the inverse image, we obtain two locally finite coverings $(U_{\alpha})_{\alpha\in I}$ and $(V_{\alpha})_{\alpha\in I}$ of X by relatively compact open sets stable under G, such that $\overline{V}_{\alpha}\subset U_{\alpha}$ for all $\alpha \in I$.

For all $\alpha \in I$, there exists a representation $\rho_{\alpha}$ of G on a finite dimensional real vector space $E_{\alpha}$ and a morphism $\varphi_{\alpha}\in \mathscr{C}^r(X;E_{\alpha})$, compatible with the operations of G, whose restriction to $U_{\alpha}$ is an embedding (Th. 1). For $\alpha \in I$, let $a_{\alpha}$ be a numerical function of class $C^r$ on X, equal to 1 on $V_{\alpha}$ and to 0 outside $U_{\alpha}($Differentiable and Analytic Manifolds, Results, 5.3.6). Put $b_{\alpha}(x) =\int_Ga_{\alpha}(gx)dg$ for $x\in X$. The function $b_{\alpha}$ is of class $C^r$, invariant under G (§6, no. 4, Cor. 2), equal to 1 on $V_{\alpha}$ and to 0 outside $U_{\alpha}$. Give each $E_{\alpha}$ a Hilbert scalar product invariant under G (§1, no. 1), and $\mathbf{R}$ its canonical Hilbert structure; let E be the Hilbert sum of the family $(E_{\alpha}\oplus \mathbf{R})_{\alpha\in I}$, and let $\rho$ be the representation of G on E induced by the $\rho_{\alpha}$ and the trivial action of G on $\mathbf{R}$. For $x\in X$, put $\varphi (x) = (b_{\alpha}(x)\varphi_{\alpha}(x), b_{\alpha}(x))_{\alpha\in I}$. Then $\varphi$ is a morphism of class $C^r$ from X to E, compatible with the operations of G; we verify as in the proof of Prop. 4 (no. 1) that $\varphi$ is an embedding, which implies the corollary.

$^{10}$That is (Spectral Theory, in preparation) a continuous linear representation (Integration, Chap. VIII, §2, no. 1) such that the operator $\rho (g)$ is unitary for all

Choose a scalar product on V invariant under H (§1, no. 1). There exists a real number $r >0$ such that the open ball B of radius $r$ is contained in W; it is clearly stable under H. Put $u(v) =r(r^2+\|v\|^2)^{-1/2}v$ for all $v\in V$; then $u$ is a bijective analytic map from V to B, compatible with the operations of H, and its inverse map $w \rightarrow r(r^2- \|w\|^2)^{-1/2}w$ is analytic. Moreover, $u(0) = 0$ and $Du(0) =$ Id$_V$.

Let $(U, \psi ,E)$ be a chart of X at $x$, such that U is stable under H (no. 2, Lemma 3) and such that $\psi (x) = 0$. Identify E with T by means of $T_x(\psi )$, and put

$\psi^{\sharp}(y) =\int_Hh.\psi (h^{-1}y)dh$ for $y\in U$,

where $dh$ is the Haar measure on H of total mass 1.

Then (§6, no. 4, Cor. $1$)$\psi^{\sharp}$ is a morphism of class $C^r$ from U to T, compatible with the operations of H, such that $\psi^{\sharp}(x) = 0$ and $d_x\psi^{\sharp}=$ Id$_T$. Hence, there exists an open set $U'\subset U$ containing $x$, and an open neighbourhood V of 0 in T, such that $\psi^{\sharp}$ induces an isomorphism $\theta : U'\rightarrow V$. Restricting $U'$ and V if necessary, we can assume that they are stable under H and that there exists an isomorphism $u: T\rightarrow V$ compatible with the operations of H (Lemma 4). It now suffices to take $\varphi =\theta^{-1}\circ u$.

Recall (Differentiable and Analytic Manifolds, Results, 6.5.1) that if G is a Lie group, H a Lie subgroup of G and Y a manifold on which H operates on the left, we denote by $G\times^HY$ the quotient of the product manifold $G\times Y$ by the right operation $((g, y), h) \rightarrow (gh, h^{-1}y)$ of H; this is a manifold on which the Lie group G operates naturally on the left; the projection $G\times^HY\rightarrow G/H$ is a bundle with fibre Y. Further, if Y is a finite dimensional vector space on which H operates linearly, $G\times^HY$ has a natural structure of vector G-bundle with base $G/H ($Differentiable and Analytic Manifolds, Results, 7.10.2).

Let G be a Lie group operating properly on the manifold X (General Topology, Chap. III, §4, no. 1, Def. 1) such that the law of operation $(g, x) \rightarrow gx$ is of class $C^r$. Then, for every point $x$ of X, the orbit $Gx$ of $x$ is a closed submanifold of X, isomorphic to the Lie homogeneous space $G/G_x$, where $G_x$ is the fixer of $x$ in G (cf. Chap. III, §1, no. 7, Prop. 14 (ii), and General Topology, Chap. III, §4, no. 2, Prop. 4); this is a compact Lie group (loc. cit.).

Put $T = T_x(X)$. Let W be a complementary subspace of $T_x(Gx)$ in T, stable under $G_x$ (for example, the orthogonal complement of $T_x(Gx)$ with respect to a scalar product on T invariant under $G_x$). On the other hand, let $\varphi : T\rightarrow X$ be a morphism with the properties stated in Prop. 5 (relative to $H = G_x$). Consider the morphism $\lambda : G\times W\rightarrow X$ defined by $\lambda (g, w) =g\varphi (w)$. It induces by passage to the quotient a morphism $\mu: G\times^{G_x}W\rightarrow X$ of class $C^r$, compatible with the operations of G, that maps the class $z$ of $(e,0)$ to $x$.

We show that $\mu$ is étale at the point $z$. We have

dim(G $\times^{G_x}W$) $=$ dim(G) + dim(W) $-$ dim(G$_x$)

= dim(G$x$) $+$ dim(W) = dim(T),

so it suffices to show that $\mu$ is submersive at $z$, or equivalently that $\lambda$ is submersive at $(e,0)$. But, the tangent map $T_{(e,0)}(\lambda ) : T_e(G)\oplus W\rightarrow T$ is equal to $T_e(\rho (x)) +i$, where $\rho (x)$ is the orbital map $g \rightarrow gx$ and $i$ the canonical injection from W to T; since Im $T_e(\rho (x)) = T_x(Gx)$, the map $T_{(e,0)}(\lambda )$ is surjective, and $\mu$ is étale at $z$.

We are going to show that there exists an open neighbourhood $\Omega$ of $Gz$ in $G\times^{G_x}W$, stable under G, such that $\mu$ induces an isomorphism from $\Omega$ onto an open subset of X. This will imply the proposition: indeed, the inverse image of $\Omega$ in $G\times W$ is stable under G, and hence is of the form $G\times B$, where B is an open subset of W containing the origin and stable under $G_x$; restricting $\Omega$ if necessary, we can assume that there exists an isomorphism $u: W\rightarrow B$, compatible with the operations of $G_x$ (Lemma 4). It is clear that the composite morphism $\alpha : G\times^{G_x}W\overset{(Id,u)}{\longrightarrow}G\times^{G_x}B\longrightarrow^\mu$ X satisfies the conditions in the statement of the proposition.

Thus, the proposition is a consequence of the following lemma:

Since $\mu$ is compatible with the operations of G, it is étale at every point of $Gz$; since the canonical map $G/G_x\rightarrow Gx$ is a homeomorphism, so is the map from $Gz$ to $Gx$ induced by $\mu$. Hence, it follows from Prop. 2 of no. 1 that there exists an open neighbourhood U of $Gz$ in Z such that $\mu$ induces an open embedding of U into X.

Since G operates properly on X, there exists an open neighbourhood V of $x$ and a compact subset K of G such that $gV\cap V =\emptyset$ for $g \notin K ($General Topology, Chap. III, §4, no. 4, Prop. 7); in particular, $e\in K$. The set $W_1$ of points $y\in Z$ such that $Ky\subset U$ is open in Z: indeed, Z-- $W_1$ is the image of the closed set $(K\times Z)$ -- $m^{-1}(U)$ under the proper projection pr$_2: K\times Z\rightarrow Z$. Put $W = W_1\cap \mu^{-1}(V)$; this is an open subset of Z, containing $z$, and satisfying the following conditions:

(i) KW $\subset U$, and in particular $W\subset U$;

(ii) $\mu(W)\subset V$.

Put $\Omega =$ GW and consider the restriction of $\mu$ to $\Omega$. This is an étale morphism, since every point of $\Omega$ is conjugate under G to a point of U. We show that it is injective: let $g, h$ in G and $u, v$ in W be such that $\mu(gu) =$ $\mu(hv)$. Put $k=g^{-1}h$; then $\mu(u) =k\mu(v)$, so $k\in K$ by (ii). But $kv$ and $u$ belong to U by (i); thus, $u=kv$ because the restriction of $\mu$ to V is injective, so $gu=hv$. Hence, the restriction of $\mu$ to $\Omega$ is injective, and consequently (Differentiable and Analytic Manifolds, Results, 5.7.8) is an isomorphism onto an open submanifold of X, which completes the proof.

Under the conditions of Prop. 6, the image of $\alpha$ is an open neighbourhood T of the orbit A of $x$, equipped with the structure of vector bundle with base A, for which the zero section is the orbit A itself. Such a neighbourhood is called a linear tube (around the orbit in question). For each point $a\in A$, the fibre $Y_a$ of this vector bundle is a submanifold of X, stable under the fixer $G_a$ of $a$, and such that the morphism from $G\times^{G_a}Y_a$ to X that maps the class of $(g, y)\in G\times Y_a$ to $gy\in X$ induces a morphism of class $C^r$ from $G\times^{G_a}Y_a$ to T. Then $Y_a$ is said to be the transversal at $a$ of the tube T. We remark that the tangent space at $a$ of $Y_a$ is canonically isomorphic to $Y_a$ and that it is a complement of $T_a(A)$ in $T_a(X)$; thus, the vector bundle T with base A is canonically isomorphic to the normal bundle of A in X (Differentiable and Analytic Manifolds, Results, 8.1.3).

a) Every decreasing sequence of compact subgroups of G is stationary.

b) Let H and $H'$ be two compact subgroups of G such that $H\subset H'$ and such that there exists an isomorphism (of topological groups) from $H'$ to H. Then $H = H'$.

c) With the relation $t\leq t'$, the set $\mathscr{T}$ is a noetherian ordered set (Theory of Sets, Chap. III, §6, no. 5, text preceding Prop. 7).

a) Let $(H_i)_{i\geq 1}$ be a decreasing sequence of compact subgroups of G; these are Lie subgroups of G (Chap. III, §8, no. 2, Th. 2). The sequence of integers (dim $H_i$)$_{i\geq 1}$ is decreasing, hence stationary, so there exists an integer N such that the subgroups $H_i$ have the same identity component for $i\geq N$. Then the decreasing sequence of positive integers $(H_i: (H_i)_0)_{i\geq N}$ is stationary, so $H_i= H_{i+1}$ for $i$ sufficiently large.

b) Let $f$ be an isomorphism from $H'$ to H. The sequence $(f^n(H))_{n\geq 0}$ is a decreasing sequence of compact subgroups of G, so $f^n(H) =f^{n+1}(H)$ for $n$ sufficiently large, by a). Since $f$ is an isomorphism, this implies that $f(H) = H =f(H')$, so $H = H'$.

c) Let $t, t'\in \mathscr{T}$ be such that $t\leq t'$ and $t'\leq t$. Then, there exist $H,H_1\in t$ and $H',H'_1\in t'$ such that $H\supset H'$ and $H_1\subset H'_1$. Let $g$ and $g'$ be two elements of G such that $H_1=gHg^{-1}$ and $H'_1=g'H'g^{'-1}$; put $u=g^{'-1}g$. Then

by b), this implies that $uHu^{-1}= H$, so $H'= H$ and $t'=t$. Thus, the set $\mathscr{T}$ is ordered, and it is noetherian by a).

a) The map which associates to any point of X its orbit type has the following semi-continuity property: let $x\in X$ and let $t\in \mathscr{T}$ be its orbit type; there exists a stable open neighbourhood U of $x$ such that, for any $u\in U$, the type of $u$ is $\geq t$.

b) For all $t\in \mathscr{T},X_{(t)}$ is a submanifold of X, the equivalence relation on $X_{(t)}$ induced by the operation of G is regular (Differentiable and Analytic Manifolds, Results, 5.9.5), and the morphism $X_{(t)}\rightarrow X_{(t)}/G$ is a bundle.

c) Assume that $X/G$ is connected. Then the set of orbit types of elements of X has a largest element $\tau$; moreover, $X_{(\tau)}$ is a dense open subset of X and $X_{(\tau)}/G$ is connected.

Let $x$ be a point of X and $t\in \mathscr{T}$ its type. To prove a) and b), we can replace X by a stable open set containing $x$, and hence (Prop. 6) can assume that X is of the form $G\times^HW$, where W is the space of a finite dimensional analytic linear representation of a compact subgroup H of G, the point $x$ being the image $p(e,0)$ of $(e,0)\in G\times W$ under the canonical projection $p: G\times W\rightarrow G\times^HW$. If $u=p(g, y)\in G\times^HW$ and $a\in G$, then $au=u$ if and only if there exists $h\in H$ with $(ag, y) = (gh^{-1}, hy)$, that is, if $a\in gH_yg^{-1}$. Thus, $G_u=gH_yg^{-1}$; in particular, $G_x= H$, so $G_u$ is conjugate to a subgroup of $G_x$, which proves that the type of $u$ is $\geq t$, hence a).

Moreover, $u$ is of type $t$ if and only if $G_u$ is conjugate to H in G, or equivalently that $H_y$ is conjugate to H in G; by Lemma $6b)$, this means that $H_y= H$, and hence that $y$ is fixed by H. If $W'$ is the vector subspace of W consisting of the elements fixed by H, it follows that $X_{(t)}$ can be identified with $G\times^HW'$, and hence also with $G/H\times W'$, hence b).

To prove c), observe that the assumption that $X/G$ is connected implies that X is pure of finite dimension: indeed, for all $k\geq 0$, denote by $X_k$ the set of points $x\in X$ such that dim$_xX =k$; then $X_k$ is open and closed in X, and stable under G, so X is equal to one of the $X_k$.

We now prove c) by induction on the dimension of X, the assertion being clear for dim X = 0. Let $\tau$ be a maximal element among the orbit types of the points of X (such an element exists by Lemma $6c$)$)$. We shall prove the following:

$c')$ For every subset A of $X_{(t)}$, open and closed in $X_{(\tau)}$ and stable under G, the closure $\overline{A}$ of A in X is open.

This assertion implies c). Indeed, note first that $X_{(\tau)}$ is open in X, by a); assertion $c')$ implies that $\overline{X}_{(\tau)}$ is open and closed in X, hence equal to X since it is stable under G and $X/G$ is connected. Let A be a non-empty open and closed subset of $X_{(\tau)}$ stable under G; by $c'),\overline{A}$ is open and closed in X and stable under G, hence equal to X; this implies that A is dense in $X_{(\tau)}$, hence equal to $X_{(\tau)}$. Consequently, every non-empty open and closed subset of $X_{(\tau)}/G$ is equal to $X_{(\tau)}/G$, which proves that $X_{(\tau)}/G$ is connected. Finally, since $X_{(\tau)}$ is dense in X, it follows from a) that every point of X is of type $\leq \tau$; in other words, $\tau$ is the largest element among the orbit types of the points of X.

We now prove $c')$. We can assume that A is non-empty; let $x\in A$. It suffices to prove that A is a neighbourhood of $x$. For this we can, as above, assume that $X = G\times^HW$ with H compact, $x$ being the canonical image of $(e,0)$. Assume first that H operates trivially on W: then X can be identified with $(G/H)\times W$, and $X_{(\tau)}/G = X/G$ is homeomorphic to W, hence connected; thus, $A/G = X/G$, so A = X. Assume from now on that H does not operate trivially on W. Choose a scalar product on W invariant under the compact group H; let S be the unit sphere in W (the set of vectors of norm 1). Note that $S/H$ is connected: indeed, if dim(W) $\geq 2, S$ is connected, and if dim(W) = 1, S is a space of two points on which H operates non-trivially. Put $Y = G\times^HS$; this is a closed submanifold of X, stable under G, of codimension 1, and $Y/G$, which is homeomorphic to $S/H$, is connected. Thus, by the induction hypothesis, there exists a maximal orbit type $\theta$ for Y, the set $Y_{(\theta)}$ is open and dense in Y, and $Y_{(\theta)}/G$ is connected.

Consider the operation of $\mathbf{R}^*_+$ on X induced by passage to the quotient by the law of operation $(\lambda ,(g, w)) \rightarrow (g, \lambda w)$ of $\mathbf{R}^*_+$ on $G\times W$. Two points of X conjugate under this operation are of the same orbit type; consequently, $X_{(\theta)}$ contains $\mathbf{R}^*_+Y_{(\theta)}$, which is a dense open subset of X. But $X_{(\tau)}$ is open by a), and hence meets $X_{(\theta)}$, so $\theta =\tau$.

On the other hand, the homeomorphism $(\lambda , w) \rightarrow \lambda w$ from $\mathbf{R}^*_+\times S$ to W -- $\{0\}($General Topology, Chap. VI, §2, no. 3, Prop. 3) induces a homeomorphism from $\mathbf{R}^*_+\times (S/H)$ to $(\mathbf{R}^*_+S)/H$, hence also from $\mathbf{R}^*_+\times (Y/G)$ to $(\mathbf{R}^*_+Y)/G$, and from $\mathbf{R}^*_+\times (Y_{(\theta)}/G)$ to $(\mathbf{R}^*_+Y_{(\theta)})/G$. Thus, $(\mathbf{R}^*_+Y_{(\theta)})/G$ is connected, and $X_{(\tau)}/G$, which contains a connected dense subset, is itself connected (General Topology, Chap. I, §11, no. 1, Prop. 1). Consequently, A is equal to $X_{(\tau)}$, hence is dense in X, and A is a neighbourhood of $x$. This completes the proof of the theorem.

With the notations in Th. $2c)$, the points of $X_{(\tau)}$ are said to be principal and their orbits are called principal orbits. If $x$ is a point of X, and if $G\times^{G_x}W$ is a linear tube in X around the orbit of $x$, the point $x$ is principal if and only if $G_x$ operates trivially on W, that is, if the tube is of the form $(G/G_x)\times W$.