The definition from my notes says that a differential $k$-form is a section of $\bigwedge^k T^*X \rightarrow X$, so $\omega \in \Omega^k(X)$ would be a map $\omega : X \rightarrow \bigwedge^k T^*X$ satisfying the condition for a section.
John Lee's text defines a $k$-form as a section of $\coprod_{p \in X} \bigwedge^k (T_pX) \rightarrow X$. Are these two definitions equivalent? I know that the cotangent bundle $T^*X$ is defined as $$T^*X = \{ (p, \alpha) \mid p \in X, \alpha \in T_p^*X \}$$ so that $\alpha$ is a map $\alpha : T_pX \rightarrow \mathbb R$ (the tangent space in this case is a real vector space). I know that we can also write the cotangent bundle as $T^*X = \coprod_{p \in X} T^*_pX$. But then how are the above two definitions equivalent? Substituting what I just wrote for the cotangent bundle into $\bigwedge^k T^*X$ from the first definition gives $\bigwedge^k \coprod_{p \in X} T^*_pX = \coprod_{p \in X} \bigwedge^k T^*_pX$ (I have assumed that the disjoint union and exterior product commute here) whereas we should have $T_pX$ in this equation rather than $T^*_pX$ according to the second definition. Could anyone please shed some light on where I am going wrong here?
On a related note, I would like to ask what a differential form evaluated at a point (denote it $\omega(p) =: \omega_p$) takes as its argument? I am an unsure of this because I am unsure of the above discussion. I think it would either take a coefficient times a wedge of things in $T_pX$ or $T_p^*X$, depending on where I've gone wrong above.
Edit: I suppose that $T_pX \cong T_p^*X$ since in general a (finite dimensional, which we have here) vector space is isomorphic to its dual, but I'm still not sure if this solves my questions, i.e. what should I be taking as the arguments for $\omega_p$, and can what I asked really be resolved as simply as writing the standard isomorphism between a vector space and its dual?
Any help is much appreciated. Many thanks.
Best Answer
It's just a question of notation. You didn't say which of my books you're looking at, but I gather it's either Riemannian Manifolds or the first edition of Introduction to Smooth Manifolds. I originally learned most of my differential geometry from Spivak, who uses the notations $\mathcal T^k(V)$ and $\Omega^k(V)$ for the spaces of covariant $k$-tensors and alternating $k$-tensors on a vector space $V$, and correspondingly $\mathcal T^k(TX)$ and $\Omega^k(TX)$ for the bundles of $k$-tensors and $k$-forms on a manifold. When I wrote those books, I decided to change the notations a bit, to $T^k(V)$ and $\Lambda^k(V)$, but I kept Spivak's convention of using the notation $V$ to reflect the fact that covariant tensors are functions on $V$.
After those books were published, I realized that Spivak was an outlier in this notation, and almost everyone else in the world uses something like $T^k(V^*)$ and $\Lambda^k(V^*)$ for the spaces of covariant $k$-tensors and alternating $k$-tensors on $V$, with analogous notations for tensors and forms on a manifold. (The reason for this is that $T^k(V)$ is naturally isomorphic to $V^*\otimes \cdots \otimes V^*$, the $k$-th tensor product of $V^*$ with itself.) So in the second edition of ISM (and the second edition of Riemannian Manifolds, which should appear sometime in January or February), I've switched to the more common notations.
The upshot is that the definition of $\Lambda^k(T_pX)$ in my older books is exactly the same as the definition of $\Lambda^k(T_p^*X)$ in my newer ones and in your notes: It's the space of real-valued alternating multilinear functions on $T_pX$.
While it's true that $T_pX$ and $T_p^*X$ are isomorphic vector spaces (simply because they have the same dimension), there is no "standard isomorphism" between them. So it definitely makes a difference that we think of differential forms as multilinear functions on the tangent space, not on its dual.