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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
by andi999 4y ago
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.
- chombier 4y agoThanks, 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)#Surjectivity_of_the_exponential https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#S... [2] https://en.wikipedia.org/wiki/SL2(R)#Classification_of_elements https://en.wikipedia.org/wiki/SL2(R)#Classification_of_eleme...
- chombier 4y agoI found this nice accessible proof for the interested: https://planetmath.org/slnrisconnected https://planetmath.org/slnrisconnected
- andi999 4y agoIs this part of the proposition: x=exp X, and x having a double eigenvalue implying that X has a double eigenvalue somehow clear? How do you prove it?