Every smooth manifold is assumed to be Hausdorff and second-countable.
Suppose $G$ is a Lie group, $H$ is a Lie subgroup of $G$, $N$ is a closed Lie subgroup of $G$ such that $N$ is normal, $N\cap H=\{e\}$, and $NH=G$, where $NH=\{ab:a\in N, b\in H\}$.
Is $H$ closed in $G$?
Lee《光滑流形导论》第171页Problem 7-2(b)之解答
定义$G\to G\times G$ by
$g\mapsto (g,g^{-1})$
定义$G\times G\to G$ by
$(g,h)\mapsto gh$
将上面两个光滑映射复合，则有
$G\to G\times G \to G$
$g\mapsto (g,g^{-1})\mapsto e$
相对应的，我们有切空间的如下复合映射(其中用到7-2(a)与第59页Proposition 3.14结论).
$T_eG\to T_eG\oplus T_eG\to T_eG$
$X\mapsto (X,di_e(X))\mapsto X+di_e(X)=0$
故$di_e(X)=-X$.
――― by Brother Jin
