Exercise 6, § 1

Let G be a real Lie group operating properly on a finite-dimensional Hausdorff real analytic manifold V. Let $p \in V$, let H be the stabilizer of $p$ and let $Gp$ be the orbit of $p$. The group H is compact and the manifold $Gp$ is identified with the Lie homogeneous space $G/H$.

(a) Show that there exists a submanifold S of V, passing through $p$, stable under H and such that $T_p(V)$ is the direct sum of $T_p(Gp)$ and $T_p(S)$. (Choose a supplement of $T_p(Gp)$ in $T_p(V)$ which is stable under H and apply Exercise 4.)

(b) Suppose that S is chosen as above. The group H operates properly and freely on $G \times S$ by $h . (g, s) = (gh, h^{-1}s)$; let E = $(G \times S)/H$ be the corresponding quotient manifold (cf. Differentiable and Analytic Manifolds, R, 6.5.1). The mapping $(g, s) \mapsto gs$ defines when passing to the quotient a morphism $\varphi : E \to V$. Show that $\varphi$ commutes with the action of G on E and V and that there exists an open submanifold S' of S containing $p$, stable under H and such that $\varphi$ induces a manifold isomorphism of $E' = (G \times S')/H$ onto a saturated neighbourhood of the orbit $Gp$.

(c) Let $N_p = T_p(V)/T_p(Gp)$ be the transversal space at $p$ to $Gp$ (Differentiable and Analytic Manifolds, R, 5.8.8); the group H operates on $N_p$. Deduce from (b) that knowing H and the linear representation of H on $N_p$ determines the G-space V in a neighbourhood of the orbit of $p$.

(d) If $x \in N_p$, let $H_x$ be the stabilizer of $x$ in H. Show that there exists a saturated neighbourhood $V'$ of $Gp$ with the following property: For all $p' \in V'$, there exists $x \in N_p$ such that the stabilizer of $p'$ in G is conjugate (in G) to $H_x$.

(e) Show that the $H_x$ are finite in number, up to conjugation in H. (Argue by induction on $\dim N_p = n$, using the action of H on a sphere of dimension $n - 1$ which is stable under H.)†