I'm working on problem 20-11 from Lee's "Introduction to Smooth Manifolds", which asks us to prove:
- Every continuous homomorphism $\gamma : \mathbb R \to G$ is smooth ($G$ a Lie group).
- Every continuous homomorphism $F : G \to H$ of Lie groups is smooth.
The first part comes with a hint: let $V \subseteq \mathrm{Lie}(G) = \mathfrak g$ be a neighborhood of $0$ such that $\exp: 2V \to \exp(2V)$ is a diffeomorphism (with $2V = \{2X : X \in V\}$). Choose $t_0$ small enough that $\gamma(t) \in \exp(V)$ whenever $|t| \leq t_0$, and let $X_0$ be the element of $V$ such that $\gamma(t_0) = \exp X_0$. Then one can show $\gamma(qt_0) = \exp(qX_0)$ whenever $q = m/2^n$ for some $m,n$.
I've been able to show all of this in the hint, but I'm not sure why that implies $\gamma$ is smooth. Is it because it now depends smoothly on $X_0$, which is in one-to-one correspondence with $t_0$? But why should that be true? And why do we care about the dyadic rational $q$?
Part 2 also comes with a hint: show that there's a map $\phi : \mathfrak g \to \mathfrak h$ so that the following diagram commutes: $\require{AMScd}$ \begin{CD} \mathfrak g @>\phi>> \mathfrak h\\ @V \exp V V @VV \exp V\\ G @>>F> H \end{CD} and then show $\phi$ is linear. But without knowing whether we can talk about $dF_e$, how could we construct such a $\phi$?
Any help with either of these problems would be greatly appreciated (or even a good resource on why continuous homomorphisms of Lie groups are automatically smooth).