Proposition 5

Let H be a compact Lie group, $(h, x) \rightarrow hx$ a law of left operation of class $C^r$ of H on X, and $x$ a point of X fixed under the operation of H. Then the group H operates linearly on the vector space $T = T_x(X)$; there exists an open embedding $\varphi : T\rightarrow X$ of class $C^r$, compatible with the operations of H, such that $\varphi (0) =x$ and $T_0(\varphi )$ is the identity map of T.

Let $(U, \psi ,E)$ be a chart of X at $x$, such that U is stable under H (no. 2, Lemma 3) and such that $\psi (x) = 0$. Identify E with T by means of $T_x(\psi )$, and put

$\psi^{\sharp}(y) =\int_Hh.\psi (h^{-1}y)dh$ for $y\in U$,

where $dh$ is the Haar measure on H of total mass 1.

Then (§6, no. 4, Cor. $1$)$\psi^{\sharp}$ is a morphism of class $C^r$ from U to T, compatible with the operations of H, such that $\psi^{\sharp}(x) = 0$ and $d_x\psi^{\sharp}=$ Id$_T$. Hence, there exists an open set $U'\subset U$ containing $x$, and an open neighbourhood V of 0 in T, such that $\psi^{\sharp}$ induces an isomorphism $\theta : U'\rightarrow V$. Restricting $U'$ and V if necessary, we can assume that they are stable under H and that there exists an isomorphism $u: T\rightarrow V$ compatible with the operations of H (Lemma 4). It now suffices to take $\varphi =\theta^{-1}\circ u$.

Recall (Differentiable and Analytic Manifolds, Results, 6.5.1) that if G is a Lie group, H a Lie subgroup of G and Y a manifold on which H operates on the left, we denote by $G\times^HY$ the quotient of the product manifold $G\times Y$ by the right operation $((g, y), h) \rightarrow (gh, h^{-1}y)$ of H; this is a manifold on which the Lie group G operates naturally on the left; the projection $G\times^HY\rightarrow G/H$ is a bundle with fibre Y. Further, if Y is a finite dimensional vector space on which H operates linearly, $G\times^HY$ has a natural structure of vector G-bundle with base $G/H ($Differentiable and Analytic Manifolds, Results, 7.10.2).

Let G be a Lie group operating properly on the manifold X (General Topology, Chap. III, §4, no. 1, Def. 1) such that the law of operation $(g, x) \rightarrow gx$ is of class $C^r$. Then, for every point $x$ of X, the orbit $Gx$ of $x$ is a closed submanifold of X, isomorphic to the Lie homogeneous space $G/G_x$, where $G_x$ is the fixer of $x$ in G (cf. Chap. III, §1, no. 7, Prop. 14 (ii), and General Topology, Chap. III, §4, no. 2, Prop. 4); this is a compact Lie group (loc. cit.).