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Is the exponential map $\exp: \mathfrak{so}(3) \rightarrow \textrm{SO}(3)$ injective? How about the case $n>3$?

Mariano Suárez-Álvarez
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Mohammadreza Bidar
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    Exponential 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 $2\pi$). – Marc van Leeuwen Feb 18 '17 at 05:28
  • Okay, thank you for your comment, I modified the question a little, do you know the answer to the modified one as well? – Mohammadreza Bidar Feb 18 '17 at 07:31
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    Well 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 $sX=tY$. – Marc van Leeuwen Feb 18 '17 at 08:39
  • You 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. – Mariano Suárez-Álvarez Feb 18 '17 at 17:15
  • You are right, sorry! and thanks for your reply. – Mohammadreza Bidar Feb 18 '17 at 21:39

1 Answers1

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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.

Mariano Suárez-Álvarez
  • 135,076