Lie Subgroups of $\mathrm{GL}(n, \mathbb{C})$

Abstract

If U is an open subset of ${\mathbb{R}}^{n}$, we say that a map $\phi: U\longrightarrow {\mathbb{R}}^{m}$ is smooth if it has continuous partial derivatives of all orders. More generally, if $X \subset {\mathbb{R}}^{n}$ is not necessarily open, we say that a map $\phi: X\longrightarrow {\mathbb{R}}^{n}$ is smooth if for each x ∈ X there exists an open set U of ${\mathbb{R}}^{n}$ containing $x$ such that ϕ can be extended to a smooth map on U. A diffeomorphism of $X \subseteq {\mathbb{R}}^{n}$ with $Y \subseteq {\mathbb{R}}^{m}$ is a homeomorphism $F: X\longrightarrow Y$ such that both F and F ⁻¹ are smooth. We will assume as known the following useful criterion.