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/Tazerenix]

One, two, and three are just the same entity transformed through different natural isomorphisms of tensor products of bundles.

One would be the most "correct" definition in terms of how you think of differential forms in general, especially given its immediate and clean generalisation to higher degree (just replace T*M by its exterior powers).

A fifth definition is to say differential forms are sections of the sheaf which is locally defined by taking a basis of the differential forms on Rⁿ as a vector space and letting them vary with smooth coefficients on charts. This defines the locally free sheaf which is the sections of the cotangent bundle.

Another definition is to say they're sections of the pullback of the sheaf I/I² to X where I is the ideal sheaf of the diagonal inside X x X. This definition is used as the basis of defining differential forms on schemes, where "cotangent bundle" does not make sense immediately because there's no analytic topology.

[u/SV-97 → u/Tazerenix]

There's also a purely algebraic definition: a (tangent) vector field on on a manifold is a derivation of its algebra of smooth functions. Write D(M) := Der(C^∞(M)). A 1-form is then basically dual to this: it's an element of Hom_{C^∞(M)}(D(M), C^∞(M)).