Yeah, about 10 years ago I was reading one of the earliest papers applying these ideas to computer vision and I was pretty confused, as I never had any exposition to Lie group/algebras in my engineering undergrad. So I tried to read mathematical texts on this subject and I was 100x more confused! Would have been useful to article like the GP's one back then.
From reading on mobile (so might have missed the clarification) it also seems to do the standard mistake in physics literature to assume the exponential map is surjective so the log map is defined on the whole group (which they call M?).
This is not always true, I think sl2(R) is an example of a connected, non compact group such that the exponential map is not surjective.
Thanks, I just realized I was also assuming exp to be surjective as well, I stand corrected.
The Wikipedia page[1] mentions the issue briefly but I was curious of the counter-examples. IIUC, matrices in SL(2) with trace < -2 have two distinct eigenvalues, one of which is negative[2], and such matrices cannot be reached by exponentiating elements of the Lie algebra sl(2) (traceless matrices).
[1] https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#S...
[2] https://en.wikipedia.org/wiki/SL2(R)#Classification_of_eleme...
I found this nice accessible proof for the interested: https://planetmath.org/slnrisconnected
This treatment is closer to the mathematician's than to the physicist's, but it still has some of the physics flavor about it—for example, it at least implicitly discusses Lie groups as if they come with a preferred action, which they need not—and, if you're not in this domain already, it's good to know what difficulties you can face later if you try to move between domains.