Problem 14 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Prove that cvery closed 1-form on \(S^{2}\) is exact. Show that this statement does not, extend to 2 -forms by showing that the 2 -form $$ \alpha=r^{-3 / 2}(x \mathrm{~d} y \wedge \mathrm{d} z+y \mathrm{~d} z \wedge \mathrm{d} x+z \mathrm{~d} x \wedge d y) $$ is closed, but has non-vanishing integral on \(S^{2} .\)

Step-by-Step Solution

Answer

We have proven that every closed 1-form on \(S^{2}\) is exact by contradiction. In addition, we showed that this statement does not extend to 2-forms by proving the specific 2-form, \(\alpha\), is closed but has a non-vanishing integral over \(S^{2}\), thus is not exact.

Step 1: Understanding the definitions

First, we need to understand what the terms 'closed' and 'exact' mean in the context of differential forms. A differential form is closed if its exterior derivative is equal to zero. A form is exact if there exists a form of one degree less such that it's exterior derivative is our original form. Our first task is to prove that every closed 1-form on \(S^2\) is exact. Considering the Poincare lemma which states that every closed form is exact in contractible domains, and since \(S^2\) is not contractible, its application is not straightforward. So we need a specific treatment for \(S^2\).

Step 2: Prove Closed 1-forms are exact on \(S^2\)

We can prove that all closed 1-forms on \(S^2\) are exact by contradiction. Assume there is a closed but not exact 1-form \(\omega\) on \(S^2\), this would mean that the integral of \(\omega\) over some non-contractible cycle (like the equator of \(S^2\)) is non-zero. We can cut \(S^2\) along this equator and get a topological disk, which is contractible, and over this region, being closed, \(\omega\) must be exact by the Poincare lemma. So its integral over the boundary (our equator) must be zero, which contradicts our initial assumption.

Step 3: Prove the 2-form not extending the property

For the 2-form, \(\alpha\), first we need to prove it is closed by showing that its exterior derivative \(d\alpha\) is zero. Then to show it is not exact, we need to recognize that \(\alpha\) can be viewed as the volume form of the standard metric on \(S^{2}\). It is known that its integral over the whole \(S^{2}\) is not zero, specifically, it equals \(4\pi\). So \(\alpha\) is not exact as its integral over \(S^{2}\) is not zero unlike exact forms.

Key Concepts

Closed Form

In differential geometry, a *closed form* is an important concept related to differential forms. To call a form closed means that its exterior derivative is zero. In simple terms, if you take the exterior derivative of a closed form, you end up with zero. This is akin to having a function with a zero derivative—it's in a state of equilibrium. Closed forms often represent conserved quantities in physics, like in the context of vector fields where they correspond to irrotational fields.

• For a form \( \omega \) to be closed: \( d\omega = 0 \)

For instance, when working with 1-forms on the sphere \( S^2 \), if a 1-form is closed, the integral around any closed loop on the sphere would ideally measure this "conservation" nature effectively, unless the form is also *exact*.

Exact Form

An *exact form* is a step further than a closed form. If a differential form is exact, it means that there exists another form of one degree less such that applying the exterior derivative to this form results in the original form. Essentially, all exact forms are closed, but not all closed forms are exact.In practical terms:

• If \( \omega \) is exact, \( \omega = d\eta \) for some form \( \eta \).

When considering 1-forms on a manifold such as \( S^2 \), proving a form is exact involves demonstrating the existence of a specific "potential" form \( \eta \) whose exterior derivative produces \( \omega \). The sphere's complexity stems from its topology, wherein not all closed forms can be proven to be exact without deeper examination, such as using the Poincare lemma.

Poincare Lemma

The *Poincare Lemma* is a significant theoretical tool in differential geometry. It states that on a contractible domain—a space that can be continuously "shrunk" to a point—every closed form is also exact.However, this becomes intriguing when dealing with spaces that are not contractible, like the 2-sphere \( S^2 \). The lemma itself cannot be directly applied here, so it requires alternate reasoning. For a closed form on such non-contractible spaces, proving exactness can involve clever topological arguments.

• Poincare Lemma: "Every closed form is exact on a contractible domain."

This lemma highlights the beauties and complexities of topology when dealing with differential forms.

Integral on Manifolds

In the world of differential geometry, integrating a form on a manifold explores quite literally "summing" the form's effects over that space. When discussing whether a form is exact or not, examining the integral of the form over a manifold can be revealing.For the 2-form \( \alpha \) as discussed on the sphere \( S^2 \), it was shown to be closed, but not exact because its integral over the sphere yielded a non-zero value. This breaks the condition set by the definition of exact forms, as exact forms would integrate to zero over closed manifolds.

• If \( \alpha \) is exact, the integral \( \int_{S^2} \alpha = 0 \)

This concept encapsulates the subtle interaction between forms, their topology, and the manifold they reside on, showcasing why such integral calculations are vital in understanding the structure and properties of forms.