I am looking for a good reference (textbook, lecture notes, etc) for the theory on smooth actions of compact Lie groups on manifolds. In particular, I am interested in the Slice Theorem and its consequences. Ideally (but not necessary), it should be self-contained and focus on the simpler case of compact Lie groups rather than the more general case.
There have been several other questions already posted here asking for references for the Slice Theorem, however most of them are unanswered. The one recommendation that I have found is "The Topology of Torus Actions on Symplectic Manifolds" by Audin. However, it only covers Lie group actions in the first chapter and it is not very self-contained. I was wondering if anyone had a recommendation for a more in-depth and self-contained treatment of this topic.
Best Answer
The standard reference I know is:
Bredon, Glen E., Introduction to compact transformation groups, Pure and Applied Mathematics, 46. New York-London: Academic Press. XIII,459 p. (1972). ZBL0246.57017.
Another reference is
Koszul, J. L., Lectures on groups of transformations. Notes by R. R. Simha and R. Sridharan, Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics. 32. Bombay: Tata Institute of Fundamental Research. 97 p. (1965). ZBL0195.04605.
freely available here. See the proof of the slice theorem for proper actions (which is more general than actions of compact groups) in Theorem 1 in in Chapter II.4. Actually, I like Koszul's lectures quite a bit since he covers useful material on group actions which is hard to find elsewhere.