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Let $G$ be a compact connected Lie group acting smoothly on a smooth connected manifold $M$. Say that $p,q\in M$ have the same orbit type if their stabilizer subgroups $G_p$ and $G_q$ are conjugate in $G$.

Can there be infinitely many orbit types?

In other words, if $[G_p]$ denotes the conjugacy class of $G_p$ in $G$, is it possible that the set $$\{[G_p]:p\in M\}$$ is infinite?

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  • $\begingroup$ Is M connected? compact?.. $\endgroup$ – Tsemo Aristide Sep 20 '16 at 15:44
  • $\begingroup$ @TsemoAristide As stated in question, $M$ is connected. I don't assume that $M$ is compact, but if you have an answer in that situation, I would still be interested. $\endgroup$ – Simon Parker Sep 20 '16 at 15:45
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There exits finitely orbit type if $M$ is compact. See p. 15 here

https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf

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