Nature of a differential form and pullbacks

Question

I've been trying to study differential geometry on the context of physics but somethings are really cloudy and I can't figure out a correct and fluid interpretation on some things from the textbooks alone. So, my questions:

1) How can I understand what a cotangent bundle is intuitively? That is, to be convinced that it is the space of linear functionals on the tangent bundle by some geometric insight.

2) How can I see that a differential form (say, a 2-form $\omega$) is a map from some manifold to its cotangent bundle? Or that it can be viewed as a section of it?

3) Also, here is how I see understand a pullback: let $\varphi:M \to N$ be a map between manifolds and $\omega$ (which I don't know how to define since my second question is exactly regarding if $\omega(Y)\in T^{ *}N$ for $Y\in N$) a differential r-form on $N$. The pullback $\varphi^{*} \omega$ is a map which takes elements of $M$ to the image of the r-form correspondent to the image (regarding $\varphi$) of the original element? That is, takes an element $X$ of $M$ to $\omega(\varphi (X))$. Is it correct?

Please correct me if any of the statements is not correct. Thanks!

Answer 1

1. The cotangent bundle is not the "space of linear functionals on $TM$" rather than it is a bundle over $M$ whose fiber at a point $p \in M$ consists of linear functionals on the tangent space $T_p M$. This is usually just the definition of the cotangent bundle so I don't see how one should be convinced this is the case unless the definition of the cotangent bundle given is not what I refer to.
2. A differential $k$-form on $M$ can be thought of as an object $\omega$ that for each $p \in M$ gives you a alternating multilinear map $\omega|_{p} \colon T_p M \times \dots \times T_p M \rightarrow \mathbb{R}$ (where there are $k$ copies of $T_p M$ on the domain) and satisfies some regularity conditions. In other words, for each $p \in M$ we want to choose a $k$-multilinear alternating map on $T_pM$ (note how at each $p \in M$ this is a different vector space!) "which varies smoothly" with $p$. One way to define such a form rigorously is to construct a bundle $\Lambda^k(T^{*}M) \rightarrow M$ (which includes specifying the topology, the smooth structure and the vector bundle structure) whose fiber at each $p \in M$ is the space of $k$-multilinear alternating maps on $T_p M$. Then a $k$-form is just a smooth section $\omega \colon M \rightarrow \Lambda^k(T^{*}M)$ of the bundle. For each $p \in M$, $\omega|_{p}$ (which more properly should be written as $\omega(p)$ because it is the value of a function at a point $p$ but the $\omega|_{p}$ notation makes things appear less cluttered later) "chooses" a $k$-multilinear alternating map on $T_pM$ and the "smooth" part makes such that the choices vary smoothly with $p$.
3. I don't understand your sentence but let me describe the pullback operation rigorously. Given a $k$-form $\omega$ on $N$ and a smooth map $f \colon M \rightarrow N$, we want to define a $k$-form on $M$. That is, for each $p \in M$, we need to specify a $k$-multilinear alternating map on $T_p M$. This is done by the formula: $$ (f^{*} \omega)|_{p}(X_1,\dots,X_k) := \omega|_{f(p)}(df|_p(X_1), \dots, df|_p(X_k)). $$ Namely, we evaluate the $k$-multilinear map $\omega|_{f(p)}$ defined on $T_{f(p)}M$ with the pushforwards under the differential $df|_p$ of the tangent vectors $X_1,\dots,X_k$.