I'm interested in the special Euclidean group, especially in the case $n=3$. We know $\exp \colon se(3) \rightarrow SE(3)$ is surjective but not injective, however $\exp$ is a local diffeomorphism between a nbhd of $0 \in se(3)$ to a nhbd of $e \in SE(3)$.
Is it possible to explicitly write the $\log$ part of the diffeomorphism then transport it around any point of $SE(3)$ by left or right translation ?
Thanks!