Suppose I have an isometric action by a finite group on the standard round sphere $S^n$ so that the quotient $S^n/\Gamma$ is a smooth manifold.
First, does this imply that the action is free? I know the converse is true, but it is not clear to me that the quotient being a smooth manifold implies that the action was free.
Second, what are the possible (smooth) quotients? Certainly one sees both $S^n$ (trivial action) and $\mathbb{RP}^n$ ($\mathbb{Z}_2$ action), but are these the only ones?