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I'd like to know an example of a concrete riemannian isometric action $\mu: G\times M \rightarrow M$ such that the fixed point set is easy to calculate. If anyone could point me in the right direction for any nice references where they actually make the calculations or could give me an example easy enough to do it for myself I'd really appreciate it.

Sak
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    Rotations of $\mathbb{R}^n$... – anon Aug 03 '11 at 03:45
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    Yep, I'm aware of those :) I meant for something a bit more convoluted. – Sak Aug 03 '11 at 03:47
  • What kind of $G$ do you want to have? And what kind of $M$ do you like? I mean $G$ discrete, compact, finite. Or $M$ of constant curvature, negative/positive curvature, you name it, but if you say what examples of manifolds you know well it may be easier to give you good examples that are more convoluted but not too convoluted. Also, what kind of calculations do you want to make? – t.b. Aug 03 '11 at 03:55
  • @Chu: It might be useful if you explain why you want such an example? – Zhen Lin Aug 03 '11 at 03:57
  • I want to get a good understanding on what types of fixed point sets can one have. I know the basic facts such as that the connected components are totally geodesic submanifolds and such. But I want an example a bit more difficult than the trivial ones. I'd preffer G not discrete for example. – Sak Aug 03 '11 at 04:09
  • I forgot to say that I just want to calculate the Fixed point set. – Sak Aug 03 '11 at 04:21
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    The conjugation action of a Lie group equipped with an invariant Riemannian metric on itself...? – Qiaochu Yuan Aug 03 '11 at 05:18

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