习题 8, § 9

令X为类$C^r(1\leq r\leq \infty )$的仿紧流形，并且G李群在X上正确运行，使得运算$(g, x) \rightarrow gx$的律为类$C^r$。

a) 证明在希尔伯特空间 V 上存在 G 的酉表示 $\rho$ 以及 V 中X 的嵌入 $\varphi$（类 $C^r$），使得 $\varphi (gx) =\sigma (g)\varphi (x)$ 为$g\in G, x\in X$。 （使用命题 6、练习 7，并按照命题 4 的证明进行论证。）

b) 添加以下假设：

(i) G 承认有限维线性表示；

(ii) X 在无穷大处可数，且维数有界；

(iii) 对于 G 的运算，X 仅具有轨道型的有限数。

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).