implication of a locally trivial bundle

Question

I am following video lectures by Prof. Fredric Schuller titled "lectures on the geometrical anatomy of theoretical physics".In the following video Manifold bundles at 1:07:25 he says that locally trivial bundle has an immediate implication that any section can be represented as a map from the base space to the fibre.What does that mean? I have been trying to dissect this statement for a long time but can't get my head around it.Thanks.

Answer 1

A word is missing in the statement: "Any section of a locally trivial fibre bundle can be locally represented as a map from the base to the fibre."

Let $M\times N\overset{\pi}{\rightarrow}M$ be the trivial bundle over $M$ with fiber $N$, then a section of $\pi$ is $M\overset{\sigma}{\rightarrow} M\times N$ such that $$\pi\circ\sigma=\operatorname{id}_M,$$ so that there is a bijective correspondence between sections of $\pi$ and maps from $M$ to $N$, given by: $$f\mapsto(\operatorname{id}_M,f).$$ For a locally trivial bundle, this correspondence holds locally, since the bundle is locally isomorphic to $\pi$. Whence the affirmation made by F. Schuller.

I hope this discussion cleared things up for you, otherwise feel free to let me know!