The condition for the exponential map of Lie group is surjective or injective

Asked Dec 28 '15 at 15:37

Active Dec 28 '15 at 15:37

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$G$ is a connected Lie group, $g$ is its Lie algebra.

1) What is the necessary and sufficient condition for the exponential map from $g$ to $G$ is surjective?

2) What is the necessary and sufficient condition for the exponential map from $g$ to $G$ is one-to-one?

asked Dec 28 '15 at 15:37

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This has been asked many times; check the links, e.g. here, here, here, here, etc. – Dietrich Burde Dec 28 '15 at 15:53
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There isn't a nice criterion for either (1) or (2). You can read about it here: http://www.heldermann-verlag.de/jlt/jlt07/DOKHOFPL.PDF. – levap Dec 28 '15 at 15:54
@levap Yes, that is a very good link (I was just thinking of it). But nice or not nice, there are enough results. – Dietrich Burde Dec 28 '15 at 15:58
@DietrichBurde, levap: I disagree with both of you about injectivity (I agree that surjectivity is more messy): for injectivity, a reasonable criterion can be formulated. See my answer to http://math.stackexchange.com/questions/475385/under-what-conditions-is-the-exponential-map-on-a-lie-algebra-injective/1592257#1592257 – YCor Dec 29 '15 at 01:34
@YCor: I never said anything to disagree about injectivity. I agree with you. – Dietrich Burde Dec 29 '15 at 11:53
@DietrichBurde OK no problem :) – YCor Dec 29 '15 at 12:10

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