Exercise 8, § 9

Let X be a paracompact manifold of class $C^r(1\leq r\leq \infty )$, and G a Lie group operating properly on X such that the law of operation $(g, x) \rightarrow gx$ is of class $C^r$.

a) Show that there exists a unitary representation $\rho$ of G on a Hilbert space V and an embedding $\varphi$ (of class $C^r$) of X in V, such that $\varphi (gx) =\sigma (g)\varphi (x)$ for $g\in G, x\in X$. (Use Prop. 6, Exerc. 7, and argue as in the proof of Prop. 4.)

b) Make the following additional assumptions:

(i) G admits a finite dimensional linear representation;

(ii) X is countable at infinity, and of bounded dimension;

(iii) X has only a finite number of orbit types for the operation of G.

Prove that there exists a finite covering of X by open sets $(U_i)_{i\in I}$ that are stable under G, compact subgroups $(H_i)_{i\in I}$ of G, and for all $i\in I$ a closed submanifold $S_i$ of $U_i$, stable under $H_i$, such that the map $(g, s) \rightarrow gs$ induces by passage to the quotient an isomorphism from $G\times^{H_i}S_i$ to $U_i$ (show that the open subsets of $X/G$ whose inverse image in X admits such a covering form a quasi-full subset of the set of open subsets of $X/G$, cf. General Topology, Chap. IX, §4, Exerc. 27). c) With the assumptions in b), prove that there exists a representation of G on a finite dimensional real vector space V and an embedding of X in V of class $C^r$ compatible with the operations of G (argue by induction on the set of subgroups of G, using $b$), Exerc. 2 and Exerc. 7).