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What is a differential form?

There are all sorts of answers to that question in the title, such as

• it’s an alternating tensor
• it’s a section of an exterior power of the cotangent bundle
(in particular, differential 1-forms are the dual objects to vector fields)
• it’s something that can be integrated

but none of these seem very satisfying to me in and of themselves—they give you the technical specifications to formally define things, but leave the motivation for defining them somewhat of a mystery (although the last answer kind of hints at some of this motivation.) Imagine you were trying to figure out what a vacuum cleaner was, and someone told you “they’re air pumps which are set up to create partial vacuums; you can roll them along your carpet.”

With that in mind, let me supplement the above answers with an additional, rather vague but hopefully slightly more motivational one: in the words of an (unidentified) analysis postdoc at Chicago via Alan Chang, differential (2-)forms are “pretentious parallelograms” (and I suppose more general k-forms would be pretentious parallelopipeds.)

In other words, they are compact book-keeping devices which help keep track of the data needed to make sense of things like integration, or solutions to differential equations, across coordinate charts, in a coordinate-free way—meaning, for the most part, that the coordinates are there waiting to be unrolled if you need them, but stay hidden, out the way and unobtrusive, as long as you don’t.

For example

Consider integrating a 3-form $\omega$ over a 3-manifold. In order for such an integration to be globally well-defined, we would want overlap transformations between coordinate charts U, V to satisfy $\omega_V = (\varphi_{UV})_*(\omega_U)$, where $(\varphi_{UV})_*$ denotes the pullback operator.

We may verify (i.e. it is a computation to verify) that choosing $\omega$ to be a (smooth) section of the exterior product vector bundle $\bigwedge^3 T^*M$ satisfies this.

A longer discussion of why exterior products might come into play may be found in Section 5.1 of Simon Donaldson’s excellent Riemann Surfaces.

To go back to and slightly overstretch the analogy with the vacuum cleaner—now that we’ve been told that the vacuum cleaner is meant to help clean your carpet, this computation would be analogous to figuring out why using an air pump to create a partial vacuum and then rolling that vacuum around your carpet might achieve that objective.

Looking at differential equations of various shapes, and/or complex structures and various regularity conditions gives rise to varying transformation rules, which leads to the minting of such related exoticisms as (-1,1)-forms—also known as (after imposing a few more technical conditions) Beltrami differentials—and holomorphic quadratic differentials, both of which play an important role in Teichmüller theory.

These varying exoticisms may also be viewed as sections of suitable vector bundles: e.g. a (-1,1)-form is a section of the tensor product $T\Sigma \otimes \overline{T^*\Sigma}$, where $T\Sigma$ denotes the (holomorphic) tangent bundle (canonically identified with its double dual) and $T^*\Sigma$ its dual.

Why would I ever need a (-1,1)-form?

Beltrami differentials are naturally obtained as (essentially bounded) solutions $\mu$ to the differential equation $\dfrac{\partial w}{\partial \overline{z}} = \mu \dfrac{\partial w}{\partial z}$ where w is a given complex distribution (i.e. generalized function) of z with locally $L^2$ derivative.

Such an equation (known as the Beltrami equation) comes up e.g. when one seeks to find isothermal coordinates on a Riemannian surface, i.e. coordinates in which the given metric is conformal to a Euclidean one: if the given metric is given by $ds^{2}=E\,dx^{2}+2F\,dxdy+G\,dy^{2}$, then we may rewrite it, using the complex coordinate $z = x + iy$, as $ds^2 = \lambda|dz + \mu d\bar{z}|^2$, for suitable $\lambda$ and $\mu$, which turn out to be smooth and satisfy $\lambda > 0$ and $|\mu| < 1$ (explicit formulae may be found here.)

In isothermal coordinates (u, v) we would have $ds^{2}= \rho (du^{2}+dv^{2})$ with $\rho > 0$ smooth; note the complex coordinate $w = u + iv$ satisfies $|dw|^2 |w_z|^2|\,dz + \frac{w_{\bar{z}}}{w_z} \,d\bar{z}|^2$.

Since $ds^2 = \rho |dw|^2 = |dw|^2 |w_{z}|^2 |\,dz + \frac{w_{\bar{z}}}{w_z} \,d{\bar{z}}|^2 = \lambda|dz + \mu d\bar{z}|^2$, we will be done if $\rho|w_z|^2 = \lambda$ and $\mu = \dfrac{w_{\bar{z}}}{w_z}$. The former is no constraint, since we are free to choose $\latex rho$ to be any positive smooth function; hence we are left with the latter, which is exactly the Beltrami equation with $\mu$ given and where we wish to solve for w.

In other words, there is a bijective correspondence between conformal structures on a Riemann surface and Beltrami differentials.

Beltrami differentials are also closely related to quasiconformal homeomorphisms via the measurable Riemann mapping theorem, and thus appear in Teichmüller theory, as e.g. tangent vectors to Teichmüller space.