Exercise 14, § 9

Assume that the manifold X is paracompact and locally finite dimensional. Let G be a Lie group operating properly on X, H a compact subgroup of G, and $t$ the conjugacy class of H.

a) Show that the set $X_H$ of points of X whose fixer is equal to H is a locally closed submanifold of X.

b) Show that the map $(g, x) \rightarrow gx$ from $G\times X_H$ to X induces an isomorphism (of class $C^r$) from $G\times^{N(H)}X_H$ to $X_{(t)}$.