What it means for a differential form to be on a manifold and how to "find" it

Question

In most of the cases I have studied, I haven't seen what it means for a form "to be" on the manifold. We usually let $M$ to be a manifold, and we let $\omega\in \bigwedge ^kT^*_pM$ in the exterior algebra of the dual tangent space, we then say $\omega$ is of the form $\sum_{I}f_Idx^I$ (I is a multiindex) and we don't give too much meaning. But all this is abstract and much general.

How we can make sense of a form been on a manifold. Apart from the obvious, to say that $\omega$ "takes values" from the tangent space of the manifold.

Can we say anything about the coefficients (functions) of the form ?

I have been trying to answer this by looking the example of the sphere $S^2$, trying to deduce some information about the functions of the form by looking some evaluations of vector fields on the sphere.

So I think a nice answer to this question would be, to give an example of a $2$-form on the sphere.

Any example of your choice that answers this will do. But I always take the sphere as a "natural choice".

Answer 1

A quote from Lee's Smooth Manifolds second edition page 360.

In any smooth chart, a k-form can be written locally as

$$\omega = \sum_I\omega_I dx^{i_1}\land \dots \land dx^{i_k}$$

where the coefficients $\omega_I$ are continuous functions defined on the coordinate domain. Proposition $10.22$ shows that $\omega$ is smooth on $U$ if and only if the coordinate functions $\omega_I$ are smooth.

So the answer to your question is that the coordinate functions must be continuous to be a k-form, and smooth to be a smooth k-form.

Here is an example in $\mathbb{R}^3$:

• A 0-form is a continuous real valued function.

• A 1-form is a co-vector field.

• Some examples of 2-forms are $$\omega = (\sin xy)dy\land dx$$ which is smooth because $\sin xy$ is smooth and $$\eta = dx\land dy + dx \land dz + dy \land dz$$

• Every 3-form on $\mathbb{R}^3$ is a continuous real-valued function times $dx\land dy\land dz$

In conclusion the thing that we can say about the coefficients is that they are (smooth) continuous real valued functions of the coordinate domain to be a (smooth) k-form.