Example

Proposition 6 applies in particular when G is a connected Lie group over $\mathbf{R}$ or $\mathbf{C}$. There then exists on $\widetilde{G}$ a unique structure of analytic manifold such that the projection $p:\widetilde{G}\rightarrow G$ is an étale morphism of analytic manifolds (VAR R, §1, 5.8.2, p. 48). For this manifold structure, $\widetilde{G}$ is a Lie group (LIE, III, p. 113, corollary).