Flat connection over a simply connectied manifold I am looking for a proof of the following result.


  Let $p:P\to M$ be a $G$-principal bundle over a connected and simply connected manifold.
  Suppose there is a flat connection $A\in\Omega^1(P,\mathfrak{g})$, that is $F^A\equiv 0$.
  Then $P$ is trivial and there exists a global section $s:M\to P$ such that $s^*A \equiv 0$.


That a global section on $P$ implies that it is trivial is well known, so the first task is showing that $P$ is trivial.
It seems that this is a "well known fact", but I could find no concise proofs.
Wikipedia suggests that this is closely related to the holonomy of $A$.

Even then, I don't know how do we hit the $s^*A$ condition.
A connection in a principal $G$-bundle $P$ is said to be flat at $u\in P$ if there is a local trivialization of the bundle in a neighborhood of u, $(U_{\alpha},\psi_{\alpha})$,  under which the given horizontal subspaces map to the horizontal subspaces of the trivial connection on the corresponding trivial bundle.
Consider the local section $s(m)=\psi_{\alpha}(m,e)$, being $e$ the identity element in $G$. Since the horizontal space at $s(m)$ consists of those $V\in T_{s(m)}P$ such that $A_{s(m)}(V)=0$  it is clear that
$$
s^*(A)_{m}(X)=A_{s(m)}(ds_m(X))=0
$$
and so $s^*(A)=0$.
On the other hand, I don't know how to solve the part that $P$ being trivial, that is, that $s$ is global. Sorry.
A connection on a principal bundle is defined by a distribution ${\cal H}$ transverse to the fibers. Saying that the connection is flat is equivalent to saying that ${\cal H}$ is integrable and define a foliation transverse to the fibres this is equivalent to saying that $P$ is a suspension bundle, that is a quotient  $(x,y)\rightarrow (\gamma(x),h(\gamma)(y))$ where $h:\pi_1(M)\rightarrow Diff(G)$ is a representation of $\pi_1(M)$ and $\hat M$ the universal cover of $M$, if $\pi_1(M)=1$, then $P$ is isomorphic to $M\times G$.