What are differential forms?

Question

For a manifold $M$, if we want to speak of "tangent vectors," we often say the tangent bundle $TM$ is the space of tangent vectors. This is sort of an abuse of terminology, I guess you could say, because $TM$ isn't even a vector space. I'm having trouble with this because differential $k$-forms (at least one formulation of them anyway) are multilinear maps from $\Pi^k TM$ to $\mathbb{R}$. But how does multilinearity work if we can't even define addition on $TM$? We can't define diff. forms as maps from $\Pi^k T_pM$ to $\mathbb{R}$ because this is specific of a point $p$ and when $k=1$, this is just a covector. Some people say diff. forms as "covector fields" that map $M$ to $T^*M$, but we usually need forms to "eat" vectors.

Traditionally, "forms" in general are multilinear maps from $V^k$ to $\mathbb{R}$ for a vector space $V$. This is true for every notion of "form" I've come across, so this should be true for differential forms as well.

My question is, can we rigorously define differential forms as multilinear maps from tangent vectors to $\mathbb{R}$?

Answer 1

A covariant tensor field is not a map $TM^{\times k}\to C^\infty(M)$, but instead a map $TM^{\oplus k}\to C^\infty(M)$. The difference is that $$TM^{\times k} =\{((p_1,v_1),\ldots,(p_k,v_k))\mid p_i\in M \mbox{ and } v_i \in T_{p_i}M\mbox{ for all }i\},$$while $$TM^{\oplus k} =\{(p,v_1,\ldots,v_k)\mid p\in M\mbox{ and } v_1,\ldots, v_k\in T_pM\}.$$Multilinearity does not make sense for the product but it makes sense for the direct sum.

Answer 2

The unifying structure you need is that of a vector bundle. The idea here is that each point $p \in M$ has a vector space attached to it, namely its tangent space $T_pM$, and the tangent bundle is the manifold formed by taking the disjoint union of all of these tangent spaces:

$$ TM \;\; =\;\; \bigsqcup_{p\in M} T_pM. $$

While we can't talk about linearity with respect to the tangent bundle globally, we can impose linearity pointwise. For instance, a smooth vector field on $M$ can now be described as a smooth section into the tangent bundle, i.e. a smooth mapping $X:M\to TM$ where which satisfies $\pi(X_p) = p$, and $\pi:TM \to M$ is the canonical projection $\pi(p,v) = p$. Linearity can now take place pointwise where for two vector fields $X,Y$ we define $X+Y$ to be defined pointwise by $(X+Y)_p = X_p + Y_p$.

This is the general notion of a vector bundle where given a base space $B$ and a vector space $F$ we can define a vector bundle $E$ to be a topological space $E$ with a surjective mapping $\pi:E\to B$ and such that $B$ is covered by neighborhoods $U$ satisfying a local triviality condition: each point has a neighborhood $U$ such that $\pi^{-1}(U)$ is homeomorphic to $U\times F$ via a mapping $\Phi:\pi^{-1}(U) \to U\times F$ such that $\Phi$ is an isomorphism between each fiber $\pi^{-1}(x)$ and the vector space $F$.

From this, we can define many different vector bundles in differential geometry including the cotangent bundle $T^*M$ and the space of differential $k$-forms $\Lambda^k(M)$ which is the disjoint union

$$ \Lambda^k(M) \;\; =\;\; \bigsqcup_{p\in M} \Lambda^k(T_pM). $$

Differential forms are therefore defined as smooth sections $\omega:M\to \Lambda^k(M)$ where linearity holds pointwise.