If we denote $X, Y \in se(3)$, and they have this relationship
$$e^X = e^Y$$
is it safe to assume that $X = Y$ for every element? If it is not, may I know the case when it is not?
Intuitively, the rotation mapping is not injective because if an object is rotated around certain axis for $\theta$ and $\theta + 2\pi$, they will result in the same pose. But is there another case?
Thanks.