The Algebra of Differential Forms

We’ve defined the exterior bundle $\Lambda^*_k(M)$ over a manifold $M$ . Given any open $U\subseteq M$ we’ve also defined a $k$ -form over $U$ to be a section of this bundle: a function $\alpha:U\to\Lambda^*_k(U)$ such that $\pi\circ\alpha=I_U:U\to U$ . We write $\Omega^k(U)=\Omega_M^k(U)$ for the collection of all such $k$ -forms over $U$ . It’s straightforward to see that this defines a sheaf on $M$ .

This isn’t just a sheaf of sets; it’s a sheaf of modules over the structure sheaf $\mathcal{O}_M$ of smooth functions on $M$ . We define the necessary operations pointwise:

$\displaystyle\begin{aligned}\left[\alpha+\beta\right](p)&=\alpha(p)+\beta(p)\\\left[f\alpha\right](p)&=f(p)\alpha(p)\end{aligned}$

where the right hand sides are defined by the vector space structures on the respective $\mathcal{T}_pM$ .

We can go even further and define the sheaf of differential forms

$\displaystyle\Omega_M(U)=\Omega(U)=\bigoplus\limits_{k=1}^n\Omega^k(U)$

This sheaf $\Omega_M$ is not just a sheaf of modules over $\mathcal{O}_M$ , it’s a sheaf of algebras. For an $\alpha\in\Omega^k(U)$ and a $\beta\in\Omega^l(U)$ , we define their exterior product pointwise:

$\displaystyle\left[\alpha\wedge\beta\right](p)=\alpha(p)\wedge\beta(p)$

In fact, this is a graded algebra, and the multiplication has degree zero:

$\displaystyle\wedge:\Omega^k(U)\otimes\Omega^l(U)\to\Omega^{k+l}(U)$

Even better, this is a unital algebra. We see this by considering the zero grade, since the unit must live in the zero grade. Indeed, $\Lambda_0^*(U)\cong\mathbb{R}$ , so sections of $\Lambda_0^*(U)$ are simply functions on $U$ . That is, $\Omega^0(U)\cong\mathcal{O}(U)$ . Given a function $f\in\mathcal{O}(U)$ we will just write $f\alpha$ instead of $f\wedge\alpha$ .

July 12, 2011 - Posted by John Armstrong | Differential Topology, Topology

6 Comments »

[…] just seen that smooth real-valued functions are differential forms with grade zero. We also know that functions pull back along smooth maps; if is a smooth function […]

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[…] looks sort of familiar as a derivative, but we have another sort of derivative on the algebra of differential forms: the “exterior derivative”. But this one doesn’t really look like a derivative at […]

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[…] really important thing about the exterior derivative is that it makes the algebra of differential forms into a “differential graded algebra”. We had the structure of a graded algebra before, […]

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[…] forms” — top because is the highest degree possible for a differential form on a differential form — has rank over the algebra of smooth functions. That is, if is a top form then we can […]

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[…] smooth maps, and homotopies form a 2-category, but it’s not the only 2-category around. The algebra of differential forms — together with the exterior derivative — gives us a chain complex. Since pullbacks of […]

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[…] Armstrong: The algebra of differential forms, Pulling back forms, The Lie derivative on forms, The exterior derivative is a derivative, The […]

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