suppose G and H connected Lie groups. Is there $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is bijective but $\phi$ is not an isomorphism of Lie groups?
I know that $d\phi$ surjective and H connected $\Rightarrow \phi$ surjective. And I know that $d\phi$ injective $\Rightarrow Ker \phi$ is discrete.
So to come up with a potential example, I need to prove that $Ker\phi$ could be non trivial. Is that a possibility?
Many thanks for your hints or help.