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I am looking for a reference to the following claims:

1. Any compact group (connected or not) acting on S^2 is differentiably conjugate to a linear action. This must be classical.

2. A circle S^1 acting on RP^3 (and supposedly any spherical space form) is differentiably conjugate to a linear action. This is probably true for every compact group acting on a 3-dimensional spherical space form?

Wolfgang Ziller

I am looking for a reference to the following claims:

1. Any compact group (connected or not) acting on $S^2$ is differentiably conjugate to a linear action. This must be classical.
2. A circle $S^1$ acting on $RP^3$ (and supposedly any spherical space form) is differentiably conjugate to a linear action. This is probably true for every compact group acting on a $3$-dimensional spherical space form?

Wolfgang Ziller