# Examples of applications of real-valued closed 1-forms in physics [closed]

Closed 1-forms are well-studied in foliation topology, algebraic geometry, and theory of manifolds. What are examples of their most typical or most interesting applications in physics?

I do not mean exact 1-forms (roughly speaking, functions -- not interesting). I am interested in examples of applications of real-valued closed 1-forms that that are not exact.

My motivation is to mention several good examples in an introductory section of a mathematical paper on closed 1-forms to show their importance to physics, both classical and modern. So several good (typical, or interesting) examples suitable to be mentioned in such a section would do.

• This question is too broad - many areas of physics may be formulated with differential forms, and almost all of them will consequently deal with closed forms in particular. – ACuriousMind Jan 3 '15 at 23:52
• why exactly are you interested in just closed 1-forms? – Phoenix87 Jan 4 '15 at 0:01
• @Phoenix87 I am a mathematician, I study closed 1-forms, because they have many specific mathematical properties. In the Preface to my papers I want to show their importance for physics, classical (mechanics, electrodynamics, crystallography?) and modern (cosmology and gravitation?). But I am not a physicist and I'm not sure which applications are most important to mention. – Irina Jan 4 '15 at 0:06
• @ ACuriousMind I don't think the question is too broad: I mean precisely closed 1-forms --- not differential forms in general. And at the moment there is no answer :( – Irina Jan 4 '15 at 1:21
• I edited the question to ask for good/best examples of applications and not for all applications. This allows for a reasonably short and specific answer. – Irina Jan 4 '15 at 4:41

## 2 Answers

Most notably, part of Maxwell's equations states that the Faraday 2-form is closed: $$dF=0$$ From this we can infer from Poincare's lemma that there exists a 1-form $A$ such that $dA=F$. In some elementary treatments $F$ is considered to be an exact form. But when considering magnetic monopoles is it important to treat it as a closed form because of the "locally" clause in the Poincare lemma.

A really trivial example is the following: let $g$ be an orthonormal metric. Then it is a closed 0-form $$dg=0$$ This is merely the equation for the antisymmetry of the spin connection on a Riemannian manifold with orthonormal metric.

Cohomology is used quite extensively in a little sector of physics called String Theory. I'm sure you know how important closed forms are for that. A really important closed form is the Kahler form: $$dJ=0$$

EDIT: Those weren't 1-forms. The curl operator is $\star d$. Thus a closed one-form is isomorphic to a vector that has zero curl! Some examples I can think of off the top of my head:

Take Faraday's law $\nabla\times\mathbf{E}+\dot{\mathbf{B}}=0$. Suppose the fields are static. Then $\dot{\mathbf{B}}=0$ and $\nabla\times\mathbf{E}=0$. If $\mathcal{E}=\mathbf{E}^\flat$ $$d\mathcal{E}=0$$

The same works for the Maxwell-Ampere law in a vacuum. Then the magnetic 1-form $\mathcal{B}=\mathbf{B}^\flat$ is closed $$d\mathcal{B}=0$$

Suppose the integral of some force $\mathbf{F}$ is path-independent. Work is defined by $$W_P=\int_P\mathbf{F}\cdot d\mathbf{x}$$ If $\mathcal{F}=\mathbf{F}^\flat$ then $$W_P=\int_P\mathcal{F}$$ The difference of work along two different paths vanishes ($P'-P$ is a closed curve which is the boundary of a surface $S$) $$W_{P'}-W_P=\int_{P'-P}\mathcal{F}=\int_S d\mathcal{F}=0$$ by Stokes' theorem. This implies for any conservative force $$d\mathcal{F}=0$$

• Thank you! But Faraday form, Kahler form, even metrics are 2-forms. And I ask about a closed 1-form. – Irina Jan 4 '15 at 1:07
• Oh snap, didn't notice that. I'll rack my brain! – Ryan Unger Jan 4 '15 at 1:09
• Oh, so any irrotational vector field (in particular, electric or magnetic irrotational field) on a 3-manifold corresponds to a closed 1-form! Thank you. As for a conservative force -- it corresponds to an exact form, which is trivially closed, and thus is not very interesting. Except for irrotational vector fields -- they are so classic-- is there something relevant in modern physics? – Irina Jan 4 '15 at 3:26
• I'm actually not sure if a conservative force is by definition exact. It really depends on definitions! A conservative force is path-independent. As I showed above, it implies that F is closed. By the converse of the gradient theorem (and certain properties of $\mathbb{R}^n$) we can find a potential $V$ such that $F=-\nabla V$ globally. It's a special case of the Poincare lemma! – Ryan Unger Jan 4 '15 at 3:35
• Here's a really good one! In the study of axisymmetric spacetimes it is convenient to introduce the complex-valued Ernst form on a two-dimensional Riemannian submanifold. On the vacuum part of the manifold (i.e. $\operatorname{Ric}=0$), this form is closed $d\mathcal{E}=0$. If you need more details, google first but ask if you don't find anything good. It's quite the complicated object actually. – Ryan Unger Jan 4 '15 at 3:57

Many. Classical mechanics is essentially geometry. In the Hamiltonian formulation, the dynamics takes place on a cotangent bundle to a manifold, the configuration space $\Gamma$, known as the phase space $T^*\Gamma$. The tautological, or Poincaré 1-form $\theta$, leads through exterior derivative to the natural symplectic 2-form $\omega$ on the cotangent bundle $T^*\Gamma$, that is $\omega = \text d\theta$.

In Electrodynamics, the 4-potential $A$ can be viewed as a 1-form, and its exterior derivative $\text dA$ is the Faraday, or electromagnetic, tensor $F$, which describes both electric and magnetic fields and is linked to the 4-current 1-form $J$ through Maxwell's equations. For more on this subject see this answer.

For some other ideas in General Relativity see this other answer.

• Thank you! The $\theta$ is not a closed form, otherwise $\omega=\mathrm{d}\theta$ were zero ($\omega$ is a closed form iff $\mathrm{d}\omega=0$). Your $\omega$ is closed but not a 1-form :-( It seems that the same holds for $A$ and $F$. – Irina Jan 4 '15 at 0:12
• Yep, these are examples of forms rather than 1-forms, so this is why i was asking you for you interests in just closed ones. – Phoenix87 Jan 4 '15 at 0:15
• Perhaps fluid dynamics is another important example. Closed 1-forms describe irrotational flows under some circumstances, but i don't remember much at the moment – Phoenix87 Jan 4 '15 at 0:21