Proposition 18

Let $G$ be a group and $U$ and $V$ two subsets of $G$ containing $e$. Suppose that $U$ has an analytic manifold structure satisfying the following conditions: (i) $V = V^{-1}, V^2 \subset U, V$ is open in $U$; (ii) the mapping $(x, y) \mapsto xy^{-1}$ of $V \times V$ into $U$ is analytic; (iii) for all $g \in G$, there exists an open neighbourhood $V'$ of $e$ in $V$ such that $gV'g^{-1} \subset U$ and such that the mapping $x \mapsto gxg^{-1}$ of $V'$ into $U$ is analytic.

Then there exists one and only one analytic manifold structure on $G$ with the following properties: (α) $G$ with this structure is a Lie group; (β) $V$ is open in $G$; (γ) the manifold structures on $G$ and $U$ induce the same structure on $V$.

(a) Let $A$ be an open subset of $V$ and $v_0$ an element of $V$ such that $v_0A \subset V$. Then $v_0A$ is the set of $v \in V$ such that $v_0^{-1}v \in A$ and hence is an open subset of $V$ (taking account of (ii)). Moreover, (ii) implies that the mappings $v \mapsto v_0v$ of $A$ onto $v_0A$ and $v \mapsto v_0^{-1}v$ of $v_0A$ onto $A$ are inverse analytic bijections and hence analytic isomorphisms.

(b) We choose an open neighbourhood $W$ of $e$ in $V$ such that $W = W^{-1}$, $W^3 \subset V$ and there exists a chart $(W, \phi, E)$ of the manifold $U$ with domain $W$. For all $g \in G$, let $\phi_g$ be the mapping $h \mapsto \phi(g^{-1}h)$ of $gW$ into $E$. We show that the charts $\phi_g$ of $G$ are analytically compatible. Let $g_1, g_2$ be elements of $G$ such that $g_1W \cap g_2W \neq \emptyset$, so that $g_2^{-1}g_1$ and $g_1^{-1}g_2$ belong to $W^2$. By (a), $W \cap g_1^{-1}g_2W$ is an open subset of $W$ and hence

is an open subset $D$ of $E$. For $d \in D$, by (a) we see that $\phi_{g_2} \circ \phi_{g_1}^{-1}$ is analytic.

(c) By (b) there exists on $G$ an analytic manifold structure such that $(\phi_g)_{g \in G}$ is an atlas on $G$. For all $g_0 \in G$, the mapping $g \mapsto g_0g$ ($g \in G$) leaves this atlas invariant and hence is an automorphism of $G$ with this manifold structure. In particular condition (GL₁) is satisfied.

(d) Let $v_0 \in V$. By (ii) there exists an open neighbourhood $A$ of $e$ in $W$ such that $v_0A \subset V$. This proves first that $V$ is open in $G$. By (a) the mapping $v \mapsto v_0v$ of $A$ onto $v_0A$ is an analytic isomorphism with the structures induced by U. By (c) we see that the manifold structures on G and U induce the same structure on v_0A and hence finally on V.

(e) by (d), (ii) and (iii), we see that conditions (GL_2) and (GL_3) hold. Hence G is a Lie group.

(f) If a manifold structure on G is compatible with the group structure of G and is such that V is an open submanifold of G, then (\phi_g)_{g\in G} is an atlas on G. Hence the uniqueness assertion of the proposition.