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I am trying reach an understanding of precisely how the space of differential forms is related to the space of vector fields. These are the definitions that I understand and am using for these objects: Let $X$ be an open subset of $\mathbb{R}^n$. A vector field $v$ on $X$ is a mapping that assigns to each point $p \in X$ to an element $v(p) \in T_pX$ of the (geometric) tangent space $T_pX = \{(p, x) | x \in \mathbb{R}^n\}$. A one form $\omega$ on $X$ is a mapping that assigns to each point $p \in X$ an element $\omega(p) \in T^*_pX$, the cotangent space of $X$ at $p$ which is defined as the algebraic dual of the tangent space at $p$. If $C^1(X)$ denotes the set of all continuously differentiable functions on $X$ we can form the ring of $C^1$ functions using the standard pointwise addition and multiplication of functions. We can then form the two $C^1(X)$-modules $\mathcal{V}^1(X)$ and $\Omega^1(X)$ whose underlying groups are the set of all $C^1$ vector fields on $X$ and all differential forms on $X$, respectively.

Finally, now, I can ask my question: what is the precise relation between the modules $\mathcal{V}^1(X)$ and $\Omega^1(X)$?

My thoughts: I notice that a differential form $\omega$ carries a point $p$ to the cotangent space at $p$ while a vector field $v$ will carry the same point $p$ to the tangent space. We can't really form the composition, per-se of $\omega$ and $v$ since the both have the same domain $X$ but for $v(p) \in T_pX$ we can evaluate $\omega(p)$ at $v(p)$ since the domain of $\omega(p)$ is $T_pX$. This looks a little like a dual pairing; can it be that $\Omega^1(X)$ and $\mathcal{V}^1(X)$ are dual modules in some sense?

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In general, it seems to me that the common linear-algebraic operations that one performs on vector bundles correspond to performing those same operations on the spaces of global sections (or better, the sheaf of sections). One thing that might confuse: the linear algebra on the bundle seems to be "done over $\mathbf R$", whereas with the sections we have pairings and tensors into and over $C(X, \mathbf R)$. –  Dylan Moreland Apr 2 '12 at 20:29
    
One book, by the way, that does spell these things out (but which I found quite confusing otherwise) is Madsen-Tornehave's From Calculus to Cohomology. –  Dylan Moreland Apr 2 '12 at 20:34
    
@DylanMoreland Although I have on more than one occasion considered burning, shredding or otherwise mutilating my copy of the Madsen-Tornehave text that you reference, I will check it out for this topic. –  ItsNotObvious Apr 2 '12 at 20:42

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(Below, by "vector space" I always mean "finite-dimensional real vector space.")

For $X$ any topological space there is a notion of a vector bundle over $X$ which formalizes the idea of a family of vector spaces parameterized by $X$. Vector bundles can be organized into a category similar to the category of vector spaces (which one recovers by taking $X$ to be a point); in particular one can define direct sums, duals, and tensor products of vector bundles.

An important notion here is that of a section of a vector bundle, which is roughly speaking a continuous choice of vector in each vector space in the family.

If $X$ is a smooth manifold (in particular if $X$ is an open subset of $\mathbb{R}^n$), then we can define a distinguished (smooth) bundle on $X$, the tangent bundle $T(X)$, coming from the tangent spaces at each point. The (smooth) sections of the tangent bundle are precisely vector fields on $X$. There is a dual bundle $T^{\ast}(X)$, the cotangent bundle, whose (smooth) sections are precisely differential forms on $X$.

So the tangent and cotangent bundles are in fact dual bundles, which means they have a dual pairing, and taking sections gives a dual pairing between vector fields and differential forms.

If in addition $X$ is compact, then we can make use of the Serre-Swan theorem, which identifies vector bundles over $X$ with finitely-generated projective modules over the ring $C^{\infty}(X)$ of smooth functions $X \to \mathbb{R}$. In this context I believe it's still true that the module of vector fields and the module of differential forms are dual, but I haven't worked out the details.

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If you need to calculate in the local coordinates with a metric, then you need to use the musical isomorphism. You can treat it purely algebraically if you like, but if you want to do calculation in an actual manifold this is needed.

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