Just trying to understand the exponential map in Lie groups. How do I answer this question:
Show that the Lie algebra of a Lie subgroup $H \subset G$ can be identified with $$\mathfrak{h}=\{X\in \mathfrak{g}:\exp(tX)\in H\text{ for all sufficiently small }t\in \mathbb{R}\}$$
?
I understand that there is a correspondence between Lie subalgebras $\mathfrak{h}\subset \mathfrak{g}$ and connected Lie subgroups $H \in G$, but what's the relationship with $\exp$ here?
Let $X\in \mathfrak{g}$, $X$ defines on $G$ a vector field invariant by left multiplication $\hat X(g)=dL_g.X$ where $L_g(u)=gu$. $exp(X)$ is the value of the flow $\phi_X$ of $\hat X$ at $1$. $dexp_0(\hat X)=X$, thus if $exp(tX)\in H$, ${d\over{dt}}exp(tX)=X\in T_IH=\mathfrak{h}$.