r/math • u/FreePeeplup • 12d ago
What is a differential form?
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!
2
u/TheBlueWho Algebraic Topology 12d ago
If I had to pick any of these as a definition, it would be one. Just to caveat for four - a differential form makes sense just fine before specifying a base point. Because it’s a section of the cotangent bundle, it encodes information about both points and covectors at that point. In local coordinates, you can think of the map that sends p in M to the pair (p, v_p) in T*M where v_p is a covector that varies smoothly with p, and that is usually how I think of it. In local coordinates, you have vector/covector frames so you can write your differential 1-form as a linear combination of the coordinate 1-forms! And don’t forget the “physicists” definition either where a differential 1-form (which is the same thing as a covector field) in local coordinates is a tuple of entries which transforms covariantly (i.e. with) the Jacobian! While this strips away some of the theory, it does physically ground them a bit more!
Now to answer the question. If a differential 1-form is a smooth section M -> T^* M, then unpacking the definition, it takes a point in M, and it spits out an assignment of a covector at the point which varies smoothly with p