Theorem 5

Let G be a Lie group germ, M a manifold of class $C^r$ ($r \geq 2$) and $\alpha$ a differential form of class $C^{r-1}$ and degree 1 on M with values in $L(G)$, such that $d\alpha + [\alpha]^2 = 0$. For all $x \in M$ and all $g \in G$, there exists a mapping $f$, defined and of class $C^{r-1}$ on an open neighbourhood of $x$, with values in $G$, such that $f(x) = g$ and $f^{-1}.df = \alpha$. Two mappings which satisfy these conditions coincide on a neighbourhood of $x$.

Let $x \in M$ and $g \in G$. By Lemma 6 (whose notation we adopt) and Differentiable and Analytic Manifolds, R, 9.3.7, there exist an open neighbourhood U of $x$ in M and a mapping $m \mapsto \phi(m) = (m, f(m))$ of class $C^{r-1}$ of U into $M \times G$ such that $f(x) = g$ and $\phi^*(0) = 0$. Then

Let $f'$ be a mapping of class $C^{r-1}$ of U into G such that $f'(x) = g$ and ${f'}^{-1}df' = \alpha$. By § 3, 18.9, ${f'}^{-1}$ is locally constant and hence $f' = f$ in a neighbourhood of $x$.