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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.

Irina
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    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 03 '15 at 23:52
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    why exactly are you interested in just closed 1-forms? – Phoenix87 Jan 04 '15 at 00:01
  • @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 04 '15 at 00:06
  • @ 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 04 '15 at 01:21
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    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 04 '15 at 04:41

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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 $$d\mathcal{F}=0$$

Ryan Unger
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  • Thank you! But Faraday form, Kahler form, even metrics are 2-forms. And I ask about a closed 1-form. – Irina Jan 04 '15 at 01:07
  • Oh snap, didn't notice that. I'll rack my brain! – Ryan Unger Jan 04 '15 at 01:09
  • 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 04 '15 at 03:26
  • 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 04 '15 at 03:35
  • Modern physics...That's hard. I can think of so many closed forms that are not 1-forms (Ricci form, NS-NS field strength, etc.) and tensor-valued forms... In fact, there are some tensor valued one-forms that are closed! That might be cheating, because we have to use the absolute exterior differential $D$. But here goes... The torsion 1-form is closed: $D\Theta^i=0$. Let $T_{\mu}=T_{\mu\nu}\theta^\nu$ be the energy-momentum 1-form. Then $DT_\mu=0$. Let $G_\mu=G_{\mu\nu}\theta^\nu$ be the Einstein 1-form. Then $DG_\mu=0$. I am looking furiously! More to come! – Ryan Unger Jan 04 '15 at 03:47
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    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 04 '15 at 03:57
  • "It really depends on definitions!": OK, I call a form exact if it is $\mathrm{d}f$ for some globally defined $f$. With this definition, it is a tautology that if the integral is path-independent, then the form is exact -- exactly as you show: $W(x)=\int_{x_0}^x{\mathcal{F}}$. (But see my next comment.) – Irina Jan 04 '15 at 04:28
  • However, if you change "Suppose the integral of some force $\mathbf{F}$ is path-independent" to "locally path-independent", that is, path-independent in any simply connected neighborhood, then we can find an interesting example. For example, I think the work of a magnet around a coil current is locally path-independent, but not globally: if you move the magnet through the coil, the work will be non-zero; does this give a closed but not exact form? A similar example could be the Kerr black hole, but I am not a specialist in this. Those are examples of what I am looking for. – Irina Jan 04 '15 at 04:29
  • Perhaps it makes sense to add new examples to the answer and not only to the comments. This will make your answer even better! – Irina Jan 04 '15 at 04:31
  • I'm not sure about the specific magnetic example you mentioned, but the magnetic field does not produce a conservative force. This is because you cannot put B in the form of a gradient. Alternatively, observe that the magnetic field couples to velocity in the Lorentz force. – Ryan Unger Jan 04 '15 at 04:39
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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.

Community
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Phoenix87
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  • 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 04 '15 at 00:12
  • 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 04 '15 at 00:15
  • 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 04 '15 at 00:21