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

$G$ 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?

• This has been asked many times; check the links, e.g. here, here, here, here, etc. – Dietrich Burde Dec 28 '15 at 15:53
• There isn't a nice criterion for either (1) or (2). You can read about it here: 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 math.stackexchange.com/questions/475385/… – YCor Dec 29 '15 at 1:34
• @YCor: I never said anything to disagree about injectivity. I agree with you. – Dietrich Burde Dec 29 '15 at 11:53