Lemma 5

Let Z be a separated manifold of class $C^r$, equipped with a law of left operation $m: G\times Z\rightarrow Z$ of class $C^r$, and $\mu: Z\rightarrow X$ a morphism (of class $C^r$) compatible with the operations of G. Let $z$ be a point of Z, and $x=\mu(z)$. Assume that $\mu$ is étale at $z$, and that the fixer of $z$ in G is equal to the fixer $G_x$ of $x$. Then, there exists an open neighbourhood $\Omega$ of the orbit $Gz$, stable under G, such that $\mu$ induces an isomorphism from $\Omega$ onto an open subset of X.

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).