# Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces".

Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:(

It starts with the definition of holomorphic 1-forms which is as follows:

Definition: A holomorphic 1-form on an open set $V\subset \mathbb C$ is an expression $\omega$ of the form

$$\omega=f(z)\text{dz}$$

where $f$ is a holomorphic function on $V$. We say that $\omega$ is a holomorphic 1-form in the coordinate z.

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The main problem for me is now that i can't really work with this expression. What is $\omega=f(z)\text{dz}$? Why should we define something like this/ what is the benefit from that?

We already have a notion for the integral of a function on submanifolds. Why we should not apply this here?

Because of this open question I am not able to understand the definitions which come later for example the following two:

Definition: A $C^{\infty}$ 1-form on an open set $V\subset \mathbb C$ is an expression $\omega$ of the form

$$\omega=f(z,\overline{z})\text{dz}+g(z,\overline{z})\text{d}\overline{\text{z}}$$

where $f$ and $g$ are $C^{\infty}$ function on $V$. We say that $\omega$ is a $C^{\infty}$ 1-form in the coordinate $z$.

I think my problems with this definition are the same as above.

Definition: A $C^{\infty}$ 2-form on an open set $V\subset \mathbb C$ is an expression $\eta$ of the form

$$\eta=f(z,\overline{z})\text{dz}\wedge\text{d}\overline{\text{z}}$$ where $f$ is a $C^{\infty}$ function on $V$. We say that $\eta$ is a $C^{\infty}$ 2-form in the coordinate $z$.

My additional problem/question here is: What is $\wedge?$. At this point its the first time that the author introduces this symbol.

I would be very very glad if someone can make this clear for me. I appreciate any kind of help.

• It might help to add to your question your notion of integration of a function on a submanifold. (Chances are, you're already integrating either a smooth differential form or its absolute value.) As to the importance of holomorphic forms, presumably you've taken complex analysis, and have encountered the pervasive tools of line integration: Cauchy's and Morera's theorems, the Cauchy integral formula, the residue theorem...? :) – Andrew D. Hwang Jun 24 '16 at 11:22

## 2 Answers

I will give you only idea and motivation why differential forms(because that's what you are asking for) are so important and show you some of their applications. You can learn about them in detail in classical Spivak's book Calculus on manifolds. The formalism behind machinery is rather technical, but nevertheless easy, so I strongly recommend to you consulting this reference.

In my opinion differential forms were invented because they can be nicely integrated over a differentiable manifold. So if you have a $k$-form $\omega$ on your compact manifold $M$, then roughly speeking you can integrate $\omega$ over any closed oriented submanifold of dimension $k$. For example $1$-forms can be integrated over oriented embedded curves, $2$-forms over oriented embedded surfaces and so on. You can also differentiate differential forms by taking exterior derivative. Moreover, there is a famous Stokes theorem, which generalizes fundamental theorem of calculus. Let me state it for bounded open subset $\Omega\in \mathbb{R}^n$ with piecewise smooth boundary $\partial\Omega$ and orientation inherited from the standard orientation on $\mathbb{R}^n$. If $\omega$ is $n-1$-form on $\mathbb{R}^n$, then Stokes formula gives you: $$\int_\Omega d\omega=\int_{\partial \Omega}\omega$$ Here $d\omega$ denotes exterior derivative for the form $\omega$ and $\partial \Omega$ has orientation given by so called an outward pointing vector field.

Let me give you some examples. First let me say that $0$-forms are differentiable functions. Now if your open subset is interval $(a,b)\subseteq \mathbb{R}$ and you have a differential $0$-form $\omega$, which is by the remark above just a function $F$, then $dF=F'(x)dx$ and applying Stokes theorem you derive: $$\int^b_aF'(x)dx=\int_{(a,b)}F'(x)dx=\int_{(a,b)}dF=\int_{\partial (a,b)}F=\int_{\{b\}-\{a\}}F=F(b)-F(a)$$ If $\Omega$ is open set inside $\mathbb{C}$ and $\omega=f(z)dz$ is $1$-form with $f(z)$ being holomorphic then: $$d\omega=0$$ which is just consequence of Cauchy's-Riemann equations and hence: $$0=\int_{\Omega}d\omega=\int_{\partial \Omega}\omega=\int_{\partial \Omega}f(z)dz$$ which is a fundamental result in one complex variable(and even several complex variables by application of Fubini's theorem to derive multivariate Cauchy's formula).

• Did you by any chance mean Munkres' Analysis on Manifolds or Spivak's Calculus on Manifolds as your reference text? – Blake Jun 24 '16 at 11:47
• Oh, I'm sorry in my native language it is rather like Analysis on Manifolds than Calculus on Manifolds. Of course I meant Spivak's Calculus on Manifolds. I will edit my answer according to your comment. – Slup Jun 24 '16 at 14:32

My understanding is when we do integration, we are not integrating functions but forms instead. What eats vectors from the tangent bundle of surface/manifold is things living in cotangent bundle. So it's $dz$ part that deals with vectors coming from $\Omega$ and yield number out to reflect orientation and infinitesimal area.

As to the mysterious wedge product, what helps us understand it may be the simple version relation between volume of space spanned by vectors and derterminant of corresponding matrix. The antisymmetric property of derterminant may give a hint why we also need antisymmetic property for forms.