Proposition 15

*Let G be a Lie group, 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 $d\alpha + [\alpha]^2 = 0$. Suppose that M is simply connected. For all $x \in M$ and all $s \in G$, there exists one and only one mapping f of class $C^{r-1}$ of M into G such that $f(x) = s$ and $f^{-1} \cdot df = \alpha$.

The uniqueness of f follows from § 3, no. 17, Corollary 2 to Proposition 59 and the fact that M is connected. We prove the existence of f. There exist an open covering $(U_i)_{i \in I}$ of M and, for all $i \in I$, a mapping $g_i : U_i \to G$ of class $C^{r-1}$ such that $g_i^{-1} \cdot dg = \alpha$ on $U_i$ (§ 4, no. 6, Theorem 5). By § 3, no. 17, Corollary 2 to Proposition 59, $g_i g_j^{-1}$ is locally constant on $U_i \cap U_j$. Let $g_i g_j^{-1} = g_{ij}$. Let $G_d$ be the group G with the discrete topology. The $g_{ij} : U_i \cap U_j \to G_d$ are continuous and $g_{ij} g_{jk} = g_{ik}$ on $U_i \cap U_j \cap U_k$. Since M is simply connected, there exist continuous mappings $\lambda_i : U_i \to G_d$ such that $g_i g_j^{-1} = \lambda_i \lambda_j^{-1}$ on $U_i \cap U_j$. Let g be the mapping of M into G whose restriction to $U_i$ is $\lambda_i^{-1} g_i$ for all $i \in I$. This mapping is of class $C^{r-1}$ and $g^{-1} dg = \alpha$. The mapping f of M into G defined by $f = s(g(x))^{-1} g$ satisfies the conditions $f^{-1} \cdot df = \alpha$ and $f(x) = s$.