TITLE: group actions in dimension 2 and 3
QUESTION [9 upvotes]: I am looking for a reference to the following claims:

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

REPLY [7 votes]: For (1), the usual trick is to average a Riemannian metric (using
Haar measure on the group) to get a fixed metric. Hence the group
will be a subgroup of $SO(3)$ (e.g., by uniformization, the metric
on $S^2$ is conformal to the round $S^2$, so the group must act
by conformal transformations, i.e. $PSL_2(\mathbb{C})$. The
maximal compact subgroup of $PSL_2(\mathbb{C})$ is $SO(3)$). 
For (2),  the quotient space is a 2-dimensional orbifold (maybe
with boundary corresponding to the fixed locus). Away from the
boundary, it will be a Seifert fibration, and one may use
the classification of Seifert fibrations to see that the action
is conjugate to a linear action.