The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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

6 Comments »

  1. [...] 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 [...]

    Pingback by Pulling Back Forms « The Unapologetic Mathematician | July 13, 2011 | Reply

  2. [...] 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 [...]

    Pingback by The Exterior Derivative « The Unapologetic Mathematician | July 15, 2011 | Reply

  3. [...] 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, [...]

    Pingback by De Rham Cohomology « The Unapologetic Mathematician | July 20, 2011 | Reply

  4. [...] 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 [...]

    Pingback by Integration on the Standard Cube « The Unapologetic Mathematician | August 2, 2011 | Reply

  5. [...] 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 [...]

    Pingback by The Poincaré Lemma (setup) « The Unapologetic Mathematician | December 2, 2011 | Reply

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

    Pingback by Tenth Linkfest | August 21, 2012 | Reply


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