Exercise: Show that a principal bundle admits a (global) section of and only if it is trivializable. In more detail: show that if $\sigma$ is a section of the principal G-bundle $\pi : P \rightarrow M$ then
$$ F_\sigma : M \times G \rightarrow P, F_\sigma (x,g) = \sigma(x).g $$
Is an isomorphism of pricinpal G-bundles; and then show that $\sigma \longleftrightarrow F_\sigma $ defines a 1-1 correspondence between sections of P and isomorphisms between the trivial principal G-bundle and P.
I know how to do the first part of this exercise, What it remains is how to construct a global section $\sigma $of P out of an isomorphism of trivial bundles : $ F: M \times G \rightarrow P $, such that $F= F_\sigma $.
Thanks!
Best Answer
Let $F:M\times G\rightarrow P$ be an isomorphism and $e$ the neutral element of $G$. For every $x\in E$, write $\sigma(x)=F(x,e)$. The map $\sigma$ is a section of $\pi:P\rightarrow M$ since $F$ is a morphism of bundle.
To $\sigma$ is associated the isomorphism $F_\sigma: M\times G\rightarrow P$ defined by $F_\sigma(x,g)=\sigma(x).g=F(x,e).g=F(x,g)$.