Most textbooks define a vector field on a smooth manifold $M$ as a section of the tangent bundle of $M$. My question is: why is it even necessary to talk about bundles when defining vector fields on $M$?
To be clear, when I refer to the tangent bundle, I refer to the triple $(M, TM, \pi)$, where $\pi: TM \to M$ is a smooth surjection. The problem is, vector fields can be defined without ever referring to $\pi$ at all. We can just define a smooth vector field as a smooth map $f: M \to TM$ satisfying $f (p) \in T_pM$ for all points $p \in M$ (that is, once we've given $TM$ a topology and a smooth atlas).
I'm likely missing something about the importance of bundles. Any comments would be most welcome. Thanks!
Best Answer
For each $p$, $f(p)\in T_pM$ happens if and only if $\pi(f(p))=p$. This is the same as saying "$f$ is section for $\pi$", so indeed you said the same without ever using the word "section". Why use it? Well, it's just convenient because the same concept applies for maps which are not neccesarilly bundles. E.g., sheaf theory, algebraic topology, category theory.