Proposition 27

Let G be a Lie group and X a manifold of class $C^r$. Suppose that a law of left (resp. right) operation of class $C^r$ of G on X is given. For all $a \in L(G)$, let $D_a$ be the field of point distributions defined by a on X.

(i) The mapping $(a, x) \mapsto D_a(x)$ is a morphism of class $C^{r-1}$ of the trivial vector bundle $L(G) \times X$ into the vector bundle $T(X)$.

(ii) Let I be an open subset of K containing 0 and $\gamma : I \to G$ a mapping of class $C^r$ such that $\gamma(0) = e$. Let $a = T_0(\gamma)1 \in L(G)$. If f is a function of class $C^r$ on an open subset of X, then

(iii) If $r \geq 2$, the mapping $a \mapsto D_a$ is a law of infinitesimal left (resp. right) operation of class $C^{r-1}$ of $L(G)$ on X.

Suppose that G operates on X on the left. Let $\phi : G \times X \to X$ be the law of operation. Then $T(\phi)$ is a $\phi$-morphism of class $C^{r-1}$ of the vector bundle $T(G) \times T(X)$ into the vector bundle $T(X)$ (Differentiable and Analytic

Manifolds, R, 8.1.2). The induced vector bundle $(\mathrm{T}(G) \times \mathrm{T}(X))|_{\{\epsilon\} \times X}$ is identified with $E = L(G) \times T(X)$. Hence $T(\phi)|E$ is a vector bundle morphism of class $C^{r-1}$. For $(a, x) \in L(G) \times X, T(\phi)(a, x) = D_a(x)$, whence (i).

The formula giving $(D_a f)(x)$ follows from § 2, the end of no. 2, and Differentiable and Analytic Manifolds, R, 8.4.5.

Suppose that $r \geqslant 2$. Let $a, b$ be in $L(G)$ and $f$ be a function of class $C^r$ on an open subset of $X$. Then

Let $x \in X$. By taking $f$ to be a chart on an open neighbourhood of $x$, it follows that $D_{[a, b]}(x) = [D_b, D_a](x)$, whence (iii). The argument is similar if $G$ operates on $X$ on the right.

When $r \geqslant 2$, the mapping $a \mapsto D_a$ is called the law of infinitesimal operation associated with the given law of operation.