A connected Lie group for which the exp map is not surjective

Asked Oct 27 '19 at 23:32

Active Oct 27 '19 at 23:56

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I know that if a Lie group $G$ is not compact so the map $\exp: \mathfrak{g} \to G$ is not necessarily surjective. But I am not absolutely sure if $\exp$ must be surjective if $G$ is connected.

I'm looking for an answer, maybe an example of connected Lie group for which the $\exp$ map wouldn't be surjective.

Many thanks, in advance.

edited Oct 27 '19 at 23:56

asked Oct 27 '19 at 23:32

Gabriel Palau

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https://math.stackexchange.com/questions/301504/on-surjectivity-of-exponential-map-for-lie-groups – Tsemo Aristide Oct 27 '19 at 23:37
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Your first sentence is incorrect. Certainly the exponential map can be surjective when $G$ is non-compact. – Ted Shifrin Oct 27 '19 at 23:50
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The simplest example of a noncompact group where exp isn't surjective is SL(2,R), and that's connected. – anon Oct 28 '19 at 00:08
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Voted as duplicate of On surjectivity of exponential map for Lie groups: this answer gives a connected example ($\mathrm{SL}_2(\mathbf{C})$) for failure of surjectivity. – YCor Oct 29 '19 at 19:47

A connected Lie group for which the exp map is not surjective

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