A necessary condition for the exponential map $\mathfrak{g}\to G$ to be a diffeomorphism is that $G$ is contractible. Is this also sufficient? I can't think of a proof or any counterexamples.
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@Jap88: the circle isn't contractible. – Qiaochu Yuan Sep 14 '22 at 01:09
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1Completely answered here: https://math.stackexchange.com/questions/475385/under-what-conditions-is-the-exponential-map-on-a-lie-algebra-injective In particular it is not sufficient. – Qiaochu Yuan Sep 14 '22 at 01:11