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