Is the exponential map $\exp: \mathfrak{so}(3) \rightarrow \textrm{SO}(3)$ injective? How about the case $n>3$?
Is the exponential map $\exp: \mathfrak{so}(3) \rightarrow \textrm{SO}(3)$ injective?
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lie-groups
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1Exponential maps to compact groups are rarely injective. Surely you now that if you continue rotating around an axis, you get back to the identity at some point (when the angle is $2\pi$). – 2017-02-18
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0Okay, thank you for your comment, I modified the question a little, do you know the answer to the modified one as well? – 2017-02-18
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1Well what I said is true for _any_ rotation. So for taking any nonzero $X,Y$ there are $s,t\in\Bbb R$ such that $\exp(sX)=I=\exp(tY)$, but this does not imply $sX=tY$. – 2017-02-18
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0You completely changed the question after accepting an answer. Notice that the result of that was that the currently accepted answer does not answer anything close to what is being asked! **Please** do not do that. I will revert your edit, and if you want to ask a *new* question please do so. – 2017-02-18
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0You are right, sorry! and thanks for your reply. – 2017-02-18
1 Answers
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A compact group contains a closed subgroup isomorphic to $S^1$, and the exponential of the big group restricts to the exponential of the subgroup, which is not injective.