I want to know the existence of global section of $\pi : M\rightarrow M/G$, where
$M$ is a Riemannian manifold with $G$-action.
For instance in case of $M=S^2$ and $G={\bf Z}_2$ there exists no global section.
This case is easy by considering continuity.
(1) But I cannot show the noexistence of global section of $\pi : S^3 \rightarrow S^3/S^1=S^2$.
(2) And if $G$-action on $M$ has a fixed point, then there exists a global section.
How can we show ?
Thank you in advance.
Hint for #1: A principal bundle has a global section if and only if it is a trivial bundle.