Closed subgroup theorem

Let $G$ be a Lie group and let $H\le G$ be a closed subgroup .

Theorem (Closed Subgroup Theorem).

1. There is a unique smooth manifold structure on $H$ making it a Lie group such that the inclusion $\iota:H\hookrightarrow G$ is a smooth injective immersion and a homeomorphism onto its image. In particular, $H$ is an embedded Lie subgroup of $G$.
2. The Lie algebra of $H$ is the subalgebra $\mathrm{Lie}(H)=\{X\in \mathfrak{g} : \exp(tX)\in H \text{ for all } t\in \mathbb{R}\},$ matching the description in the Lie algebra of a subgroup lemma .
3. The coset space $G/H$ admits a unique smooth manifold structure such that the projection $\pi:G\to G/H$ is a smooth submersion, making $G/H$ into a basic example of a homogeneous space .

Context. This theorem is the bridge between “topological subgroup” and “geometric submanifold.” It is also what makes quotients by closed normal subgroups into Lie groups (compare quotient Lie groups ).