Proposition 6

Assume that the manifold X is paracompact; let $x$ be a point of X, $G_x$ its fixer. There exists a finite dimensional analytic linear representation $\tau : G_x\rightarrow \mathbf{G}\mathbf{L}(W)$, and an open embedding $\alpha : G\times^{G_x}W\rightarrow X$ of class $C^r$, compatible with the operations of G, that maps the class of $(e,0)\in G\times W$ to $x$.

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: