Problem
Problem 5.15 Let $\pi: P \rightarrow M$ be a surjective submersion and let $P \times G \rightarrow P$ be an action of $G$ on $P$ satisfying the following two properties:
(i) $G$ acts freely, i.e. $u \cdot g=u, u \in P, g \in G$, implies $g=e$.
(ii) The fibres of $\pi$ coincide with the orbits of $G$, i.e.
$$
\pi^{-1}(x)=u \cdot G, \quad u \in \pi^{-1}(x), x \in M
$$
Prove:
1. If $\pi$ admits a global section (i.e. there exists a smooth map $s: M \rightarrow P$ such that $\pi \circ s=\operatorname{id}_{M}$ ), then $\pi$ is trivialisable, i.e. there exists a $G$-equivariant diffeomor$\operatorname{phism} \Phi: M \times G \rightarrow P$. 2. Every $x \in M$ admits an open neighbourhood $U$ such that $\pi^{-1}(U)$ is $G$ equivariantly diffeomorphic to $U \times G$.