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.

July 12, 2011 Posted by | Differential Topology, Topology | 6 Comments

   

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