Chapter I of Kobayashi's Transformation Groups in Differential Geometry contains a very general theorem on transformation groups, due to Palais. I have some questions about its proof (which I attach below, highlighting the unclear points):

  • Is the local action that the author is referring to with $f$ really an action? I.e. do we have $f(e^X,f(e^Y,p))=f(e^X e^Y,p)$ for small $X,Y$? If this is assumed in the rest of the proof, how can one prove it? This seems to be a local version of Lie-Palais theorem and is not trivial to me.
  • How do we deduce, in Lemma 1, that $(\exp tZ)p=(\exp X)(\exp tY)(\exp -X)p$ out of $e^{tZ}=e^X e^{tY} e^{-X}$? An affirmative answer to the preceding question would clear up this doubt (at least for small $X$). A similar question arises for Lemma 3.
  • Why is $\mathfrak{G}^*$ a Lie group? It is not a subgroup of the abstract, simply-connected auxiliary group, but rather a subgroup of the original one (the one in the statement). Is the author implicitly extending the local action $f$ to a global action of the abstract, simply-connected group? (Again, if this is the case, I do not see how this is achieved.)




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