Smooth 1-forms and smooth sections

Question

The following statement is from the book introduction to manifolds by Loring Tu.

" We define a 1-form $\omega$ on a manifold M to be smooth if $\omega : M \longrightarrow T^*M $ is smooth as a section of the cotangent bundle $\pi : T^*M \longrightarrow M $ "

Well I know what is the definition of a smooth map between manifolds, but never knew about a map which is smooth "as a section" of a fiber bundle! Could you please explain to me this definition.

I'm just a begginer in differential geometry and while trying to teach my self more about it, I've notice the large use of the expression "smooth sections" but I'm not aware of its importance and why do people emphasis on the property of smootheness of sections ?

Thanks a lot!

Answer 1

Smooth sections are defined to be maps $\sigma : M \to E$ between the base space of a smooth fiber bundle (which is a smooth manifold) and the total space of the bundle (which is another smooth manifold) which are smooth as maps in the usual sense, and satisfy some particular properties. In other contexts you might want to restrict to bundles and manifolds of lower regularity, maybe $C^k$ or even $C^0$, and in that case the sections will be $C^k$ or $C^0$ as well. So their regularity is really just a sanity check (in the world of smooth manifolds, everything had better be smooth!) – what really matters is the fundamental property of "preserving fibers", $$\pi \circ \sigma = \operatorname{id}_M $$ which distinguishes sections from more general maps $M \to E$.

Answer 2

You might not be familiar with the concept of vector bundles but this is related to it. So when you have the cotagent bundle or the tangent bundle we have a canonical smooth map $\pi : T^*M\rightarrow M$ and a smooth section will be a smooth map $s:M\rightarrow T^*M$ such that $\pi\circ s=id$, which is exactly saying that at each point we get an element in $T_p^*M$.