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.
Isometric actions
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riemannian-geometry
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2The co$n$jugation action of a Lie group equipped with an invariant Riemannian metric on itself...? – 2011-08-03