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Let $φ : G → H$ be a continous morphism of linear Lie groups with H connected.

Prove that $φ$ is surjective iff $dφ$ surjective

Using the expression : $exp\circ dφ = φ \circ exp$ we can see that

$φ$ is surjective $\Rightarrow φ \circ exp = exp\circ dφ$ is surjective but how to prove $dφ$ is surjective?

I have no idea on how to tackle the other implication.

Edit

Thanks to comments I see now that

$dφ$ is surjective $\Rightarrow dφ(Lie(g))=Lie(h) \Rightarrow exp\circ dφ(Lie(g))=\exp\circ Lie(h) \Rightarrow φ\circ exp(Lie(g))=H$ (because H is connected) $ \Rightarrow φ$ is surjective.

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    $\begingroup$ do you know that a connected Lie group is generated by any neighborhood of the identity? $\endgroup$ – Tim kinsella Apr 9 at 12:07
  • $\begingroup$ Thanks @Timkinsella I edited the question with your suggestion $\endgroup$ – PerelMan Apr 9 at 12:31

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