Proposition 9

Let $G$ be a finite-dimensional Lie group and $H$ a subgroup of $G$.

(i) There exists on $H$ one and only one analytic manifold structure with the following property: for all $r$ between 1 and $\omega$, for every manifold $V$ of class $C^r$ and for every mapping $f$ of $V$ into $H$, $f$ is of class $C^r$ as a mapping of $V$ into $H$ if and only if $f$ is of class $C^r$ as a mapping of $V$ into $G$.

(ii) With this structure, $H$ is a Lie group, the canonical injection $i$ of $H$ into $G$ is an immersion and $L(i)(L(H))$ is the Lie subalgebra tangent at $e$ to $H$.

In (i) the uniqueness is obvious. We prove the existence. Let $\mathfrak{h}$ be the Lie algebra tangent at $e$ to $H$. Let $H'$ be a Lie subgroup germ of $G$ with Lie algebra $\mathfrak{h}$. By replacing $H'$ by an open subgroup germ of $H'$, it can be assumed that $H' \subset H$ (Lemma 4 (iv)). For all $x \in H$, $xH'x^{-1}$ is a Lie subgroup germ of $G$ with Lie algebra $x\mathfrak{h}x^{-1} = \mathfrak{h}$. Hence $H' \cap (xH'x^{-1})$ is open in $H'$ (no. 2, Theorem 3) and the mapping $y \mapsto xyx^{-1}$ is an isomorphism of $H' \cap x^{-1}H'x$ onto $xH'x^{-1} \cap H'$. Using Proposition 18 of § 1, no. 9, there exist an open Lie subgroup germ $W$ of $H'$ and a Lie group structure on $H$ with the following properties: $W$ is open in $H$ and the manifold structures on $H$ and $H'$ induce the same structure on $W$. It then follows that the canonical injection $i$ of $H$ into $G$ is an immersion and that $L(i)(L(H)) = L(H') = \mathfrak{h}$. Moreover, let $V$ and $f$ be as in (i). If $f : V \to H$ is of class $C^r$, $i \circ f : V \to G$ is of class $C^r$. Suppose that $i \circ f : V \to G$ is of class $C^r$; then we prove that $f : V \to H$ is of class $C^r$. By translation, it suffices to consider the case where there exists $v_0 \in V$ such that $f(v_0) = e$ and to prove that $f : V \to H$ is of class $C^r$ in an open neighbourhood of $v_0$. Now, by Lemma 4 (iii), $f(v) \in H'$ for $v$ sufficiently close to $v_0$, whence our assertion. Thus we have proved (i) and (ii) has been obtained on the way.