Theorem 6

Let G be a Lie group germ, X a manifold of class $C^r$ ($r \geq 2$) and $x_0$ a point of X. Let $a \mapsto D_a$ be a law of left infinitesimal operation of class $C^{r-1}$ of $L(G)$ on X.

(i) There exists an open neighbourhood $X'$ of $x_0$ in X and a law chunk of left operation of class $C^{r-1}$ of G on $X'$ such that the associated law of infinitesimal operation is $a \mapsto D_a|X'$.

(ii) Let there be two law chunks of left operation of class $C^{r-1}$ of G on an open neighbourhood $X''$ of $x_0$; if they admit $a \mapsto D_a|X''$ as associated law of infinitesimal operation, they coincide on a neighbourhood of $(e, x_0)$.

Assertion (ii) follows from the Corollary to Proposition 11. We prove (i). For all $(g, x) \in G \times X$ and all $a \in L(G)$, we write

Let $S_{(g, x)}$ be the set of $Q_a(g, x)$ for $a \in L(G)$. By Lemma 5 of no. 6, the $S_{(g, x)}$ are the fibres of a vector subbundle S of $T(G) \times T(X)$. Let $a, b$ be in $L(G)$; then and hence S is integrable (Differentiable and Analytic Manifolds, R, 9.3.3, (iv)).

By Differentiable and Analytic Manifolds, R, 9.3.7, there exist an open neighbourhood $G_1$ of e in G, an open neighbourhood $X_1$ of $x_0$ in X and a mapping $(g, x) \mapsto gx$ of class $C^{r-1}$ of $G_1 \times X_1$ into X such that $ex = x$ for all $x \in X_1$ and

In particular Let $G_2$ be an open neighbourhood of e in $G_1$ and $X_2$ an open neighbourhood of $x_0$ in $X_1$ such that $gg'$ is defined and belongs to $G_1$ for $g, g'$ in $G_2$ and $gx$ is defined and belongs to $X_1$ for $(g, x) \in G_2 \times X_2$. Consider the mappings $\alpha_1, \alpha_2$ of $G_2 \times (G_2 \times X_2)$ into $X$ defined by They are of class $C^{r-1}$. Then On the other hand By Differentiable and Analytic Manifolds, R, 9.3.7, $\alpha_1$ and $\alpha_2$ coincide on a neighbourhood of $(e, (e, x_0))$. Then (i) follows from (7) and Proposition 23 of § 1, no. 11.