Viewing a differential form on a manifold as a map $\mathfrak{X}(M) \times \cdots \mathfrak{X}(M) \rightarrow C^\infty(M)$

Question

I am used to viewing a differential form as a map $M \rightarrow \bigwedge^k(T_p^*M)$, where $\bigwedge^k(T_p^*M)$ is the set of all alternating multilinear maps on contangent space of $M$ at $p \in M$. In other words, we assign to each point in the manifold an alternating (covariant) tensor. This definition makes sense to me and is found in books such as Tu's Introduction to Manifolds.

I recently came across another definition in the book Mathematical Gauge Theory by Hamilton, where a differential form is defined as the map $\mathfrak{X}(M) \times \cdots \mathfrak{X}(M) \rightarrow C^\infty(M)$ that is $C^\infty(M)$-linear in each entry. $\mathfrak{X}(M)$ denotes the set of all vector fields on $M$. I do not fully understand this definition and have been having some trouble seeing how the two definitions are equivalent. Why is the range taken to be as general as $C^\infty(M)$?

Answer 1

Consider the case of vector fields (it is completely analogous for covector fields i.e. one-forms).

A smooth vector field $X$ on some smooth manifold $M$ is a smooth section of the tangent bundle $TM$, i.e. to every $p\in M$ we assign a vector $X_p\in T_pM$ varying smoothly with $p$. The vectors $X_p$ are operators on smooth functions on $M$. Given a smooth function $f\in C^\infty(M)$ we can apply the vector $X_p$ to $f$ to yield a real number $X_p(f)\in \mathbb R$.

In this way one can define the action of a vector field on a smooth function: $(X(f))_p:=X_p(f)$. So given a smooth vector field $X$ and a smooth function $f$, we get a new smooth function $X(f)\in C^\infty(M)$. In differential geometry it is now customary to identify the induced mapping $$ C^\infty(M)\to C^\infty(M), \ f\mapsto X(f) $$ with the vector field $X$ itself. Alternatively one has the induced mapping $$ \Gamma(T^*M)\to C^\infty(M), \ \omega\mapsto X(\omega), $$ where $X(\omega)$ is defined by $(X(\omega))(p):=\omega_p(X_p)$. The key fact about this mapping is (as noted in comments) that it is $C^\infty(M)$-linear.

For differential forms (and indeed general tensor fields) one can proceed in a very similar fashion. Let $\omega$ be a smooth $k$-form on $M$. Given smooth vector fields $X_1,\dots,X_k$ on $M$, we can define a smooth function $$ \omega(X_1,\dots,X_k)\colon M\to \mathbb R, \ p\mapsto \omega_p(X_1(p),\dots,X_k(p)). $$ Similarily to vector fields, this induces a mapping $$ \Gamma(TM)\times \dots\times \Gamma(TM)\to C^\infty(M), \ (X_1,\dots,X_k)\mapsto \omega(X_1,\dots,X_k). $$ It is again customary to identify the form $\omega$ with this mapping. This induced mapping will be $C^\infty(M)$-multilinear.

Answer 2

This is based on my limited knowledge. Let $\omega$ be a differential $k$-form. $\omega_p(v_1,v_2,...,v_k)$ is a real number where $p \in M$, $v_i \in T_pM$ so this is a map from $M$ to $k$ - fold product of $T_pM^*$.

Now view this as $\omega_p(X_1(p),X_2(p),...,X_k(p))$ where $p \in M$ and $X_i$ is a vector field on $M$ and $X_i(p) \in T_pM$ and this quantity $\omega_p(X_1(p),X_2(p),...,X_k(p))$ is a smooth function on $M$ to $\mathbb{R}$.

This is similar to how tangent space has 2 equivalent definitions. One definition based on vector space and the other based on derivative of smooth functions w.r.t curves on $M$.

Both are consistent definitions but whether they are equivalent, needs to be verified.