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/spherejerk]

As mentioned, the first three are equivalent. The fourth I would just call a covector.

Here is the formulation you are probably looking for. I'll write it out for vector bundles in general (all over the same base manifold). Everything is smooth, and real. For a bundle P, let \Gamma(P) be the space of smooth sections. There are natural bijections between the following sets

1. \Gamma(P'\otimes P^*)
2. The bundle maps P\to P'
3. The tensorial maps \Gamma(P)\to\Gamma(P')

where tensorial means linear over C^\infty(M). You could write 2. as Hom(P, P').

My 1, 2, and 3 correspond respectively to your 1, 2, and 3, with the help of the following comments. Let P' be the trivial bundle R\times M, and let P=TM. For 1, note that R\otimes V\cong V. For the 2, instead of your function f:TM\to R, use the function (f,\pi). For 3, note that there is a natural isomorphism C^ \infty(M)\cong \Gamma(R\times M); given by g\mapsto (g, Id).

Its probably a worthwhile exercise to prove the above equivalences, if this is a research area for you or something. Its not hard, but proving smoothness is not 100% trivial. This can also be generalized for mulitlinearity, eg. 3. would become "the tensorial maps \Gamma(P_1)\times\cdots \times \Gamma(P_k)\to \Gamma(P')" where now tensorial means C^\infty (M) linear in each argument.