Definition of a $n$ - form on a manifold

Question

I am confused about the definition of a differential form on a manifold.

The definition I have comes from Bott and Tu and is as follows:

A differential form, $\omega$, on a manifold $M$ is a collection of forms $\omega_U$ for $U$ in the atlas defining $M$, which are compatible in the following sense:

$i^*\omega_U=j^*\omega_V$ where $i,j$ are the inclusion maps.

I am confused as to what exactly $\omega_u$ is. Is it the pull back by a chart of a form on Euclidean space?

Moreover, how can I fail the compatibility criterion? It seems to me like it should always be true.

I think I simple example would really help me understand but I can't find one.

Thanks for your help,

Answer 1

A differential form $\omega$ of degree $k$ on a real manifold $M$ of dimension $n$ is, formally, a section of the $p$-th exterior power of the cotangent bundle $T^{*}M$ over $M$, in symbols:

$$\omega\in \Gamma(\wedge^k T^{*}M).$$

This means that $\omega$ is a smooth map $M\rightarrow \wedge^k T^{*}M$ which, at any point $p\in M$ is given by

$$\omega_U(p):=\omega_{U,i_1,\dots,i_k}(p)dx^{i_1}\wedge\dots\wedge dx^{i_k}, $$

where $(x^{1},\dots, x^{n})$ denote local coordinates in the open set $U$ centered at $p\in M$. In other words, the differential form $\omega$ can be seen as a collection of "local" forms like above; any coordinate transformation $f: U\rightarrow V$ induces a transformation $\omega_V=f^{*}\circ\omega_U$ as pointed out by @John. As often happens in differential geometry, one starts with local data and gives a rule to glue them into a global structure (when possible).

$f^{*}\omega$ is the pull-back of $\omega$ along $f$: you can find its definition on every textbook. Essentially, the coordinate transformation formula reduces to an identity involving the Jacobian matrix of $f$.

Answer 2

This is just an intuitive opinion. I believe this works the same way as the differentiable structure definition works for a manifold. We want a maximal atlas that is the collection of all compatible atlases given any representative atlas. This probably tries to do something similar for a form. In as many words, as Avitus notes, this is an attempt to relate the local and global existences.