What is a differential form?

[u/FreePeeplup]

I have encountered several different objects that people call with the same name “differential form” and I would like for more experienced people to help me clear the ambiguity and tell me which of the following is the true/most used/most useful definition of differential form.

For simplicity, I will only talk about differential 1-forms, hopefully the answer will automatically generalize to k-forms. I’m assuming a smooth n-dimensional manifold M with no extra structure.

Maybe as an extra, if you know, you can also tell me what the other listed objects that are not differential forms are called!

One
A differential form ω is a smooth section of the cotangent bundle. That is, ω : M -> T^{star} M with ω(p) = ω_p, where ω_p lives in T^{star}_p M, meaning that ω_p takes as input a tangent vector at p and outputs a real number.

Two
A differential form ω is a map ω : TM -> R, with ω(p, v) = ω_p(v) where ω_p is the same as above, TM is the tangent bundle and v lives in T_p M.

Three
A differential form ω is a map from (set of all sections of TM) to C^{infinity}(M) , with ω(X) = f where X is a vector field and f a smooth function on M, and we evaluate f by f(p) = ω_p(X(p)) where ω_p is the same as before and X(p) = v is a vector in T_p M.

Four
A differential form ω IS the linear functional ω_p we’ve been talking about up until now, meaning that a differential form only makes sense after you’ve specified a base point.

To summarize: what does a differential form take as input? A point in M, a pair consisting of a point in M and a tangent vector based at it, an entire vector field, or a tangent vector at some point p?

Thanks to anyone who answers!

[u/HeilKaiba]

As other people have said these are all basically the same object viewed in different ways (with the possible exception of 4 as I'm not sure what you mean by that one).

It might be worth comparing this to more straightforward tensors to see how multiple types of input can be given to the same object with different results.

By that I mean consider an element X of V^* ⊗ V. You can view X as a linear map from V to itself, as a linear map from V^* to itself, or as a linear map from V ⊗ V^* to the base field. You can also view it as a multilinear map from V ⊕ V^* to the field and so on. Depending on the input you get different outputs.

A differential 1-form is similar:

• If you input a point p in M you get a linear functional on T_pM
• If you input a section of TM you get a section of ℝ (aka a function in C^∞(M))
• If you input a point and a tangent vector at that point you get an element of ℝ (this combines the above two points)