Rectifiability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation [B,∇×B]=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[B,\nabla \times B] = 0$$\end{document}. Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution u:S→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u: S \rightarrow {\mathbb {R}}$$\end{document} to the cohomological equation B|S(u)=∂nB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\vert _S(u) = \partial _n B$$\end{document} on a toroidal surface S mutually invariant to B and ∇×B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \times B$$\end{document}. The right hand side ∂nB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _n B$$\end{document} is a normal surface derivative available to vector fields tangent to S. In this situation, we show that the field B on S is either identically zero or nowhere zero with B|S/‖B‖2|S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\vert _S/\Vert B\Vert ^2 \vert _S$$\end{document} being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality ‖B‖2|S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert B\Vert ^2 \vert _S$$\end{document}). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that B|S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\vert _S$$\end{document} itself is rectifiable. The rectifiability and semi-rectifiability of B|S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\vert _S$$\end{document} is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.

a diffeomorphism : S → R 2 /Z 2 such that * X is a constant vector field on R 2 lowered to R 2 /Z 2 . That is, * X = a A slightly weaker condition for X is the existence of a positive function f on S for which X / f is rectifiable. We will say that such an X is semi-rectifiable with proportionality f . This is equivalent to the existence of a coordinate system in which the field-lines of X are straight [1]. The field-lines of X are known as windings in such a coordinate system [3].
This paper mainly concerns these properties when X is the vector field induced on an invariant 2-torus of a divergence-free vector field in an oriented Riemannian 3-manifold M with (or without) boundary. The motivation lies in producing and understanding magnetohydrodynamics (MHD) equilibria [4][5][6][7], which are solutions to the system of equations where p is a function on M interpreted as pressure, B is the magnetic field in M, and n is the outward unit normal of ∂ M.
The conjecture of Grad [8,9] remains unsettled; that smooth solutions with p admitting toroidally nested level sets only exist if M has a continuous isometry. One way to better understand this conjecture is to prove necessary structural features of solutions if they exist. A contribution by Arnold in this regard is his structure theorems [3,10,11]. In particular, Arnold obtained the following rectifiability result.
Theorem 1 (Arnold [3]) Suppose that p has a closed (compact, without boundary) regular level set. Then, the connected components of this level set are invariant tori of B and ∇ × B and in some neighbourhood U ⊂ M of such a component, there exist a diffeomorphism : U → R 2 /Z 2 × I where I ⊂ R is an interval, such that * B| U = a(z) where a, b, c, d : I → R are smooth functions and z is the projection onto the factor I .
Coordinates in which both B and ∇ × B are linear are known as Hamada coordinates [5]. In magnetic confinement fusion, coordinates systems in which the field lines of B are straight (regardless of ∇ × B) are called magnetic coordinates [6,12]. Arnold also remarked [3] that in the case of ∇ p = 0, assuming B is nowhere zero, then necessarily (i) B is a Beltrami field, namely ∇ × B = λB for some function λ, (ii) λ is a first integral and the closed regular level sets of λ are unions of tori, and (iii) in a neighborhood U ⊂ M of such a 2-torus, there exists a diffeomorphism : U → R 2 /Z 2 × I and a positive function f : U → R such that * (B| U / f ) = a(z) where again a, b : I → R are smooth functions and z is the projection onto the factor I .
In this paper, we will bring some more attention to this remark. More specifically, we show that it is not particularly a consequence of B being Beltrami field or even a consequence of assuming B is nowhere zero. For instance, if one assumes a first integral (we will also state an "infinitesimal version" which features no first integral assumption), we have the following corollary of our methods.

Corollary 2 Let B be a vector field on M which satisfies, for some function ρ,
where a, b : I → R are smooth functions and z is the projection onto the factor I .
A strong Beltrami field B satisfying ∇ × B = λB where λ is a constant, can have more complicated topology [3,7] despite being MHD equilibria solutions [9,13]. Less has been said about the structure (in relation to rectifiability) of strong Beltrami fields on invariant tori. In particular, if ∂ M has a toroidal connected component and B · n = 0, this component is an invariant 2-torus of B and it is reasonable to ask of the rectifiability properties of B here. This question has been linked with the aforementioned conjecture of Grad. In the Euclidean context, Enciso et al. [14] have shown that under non-degeneracy assumptions of a toroidial domain M ⊂ R 3 , piece-wise smooth MHD equilibria with non-constant pressure exist. The non-degeneracy assumptions include assuming the existence of a strong Beltrami field B for which ∂ M is a Diophantine invariant 2-torus. That is, as in the context of the KAM Theorem, the vector field X (induced from B on ∂ M) can be written in the form of Eq. (1) where the vector (a, b), known as the frequency vector of X with respect to , is a Diophantine vector (a notion which will be defined in Sect. 2). Of course, the frequency being Diophantine does not depend on the diffeomorphism chosen (see, for instance, Propositions 23 and 24). The authors managed to show that the so-called thin toroidal domains are generically non-degenerate [14]. In view of Corollary 2, the existence of a first integral of B will ensure some structure of B. Although, in face of this complicated topology, there is no reason to expect that a first integral exists.
As mentioned, the "infinitesimal version" of Corollary 2 does not assume a first integral. We will now state this version in the case most relevant to strong Beltrami fields; namely when M is embedded in R 3 and S is a toroidal (connected) component of the boundary ∂ M. For instance, relevant to the Stepped Pressure Equilibrium Code [15] (SPEC, a program which numerically solves for MHD equilibria), S could be taken as either component of the boundary ∂ M when M is a hollow torus, that is, when M is diffeomorphic to R 2 /Z 2 × [0, 1].

The vector field B| S preserves a top-form μ on S. 3. Either B = 0, or B is nowhere zero on S and B| S is semi-rectifiable with proportionality
With such a solution u and closed curves C 1 , C 2 : [0, 1] → S ⊂ R 3 whose homology classes generate the first homology H 1 (S), form the integrals Our other results about vector fields on a 2-torus will mostly be stated in terms of their winding number (or frequency ratios, as in [3]). We have employed a homology-dependent means of defining winding numbers, so that by way of de Rham cohomology, we may compute them with integrals like those in Corollary 3. In the context of magnetic confinement [6,18], the winding number corresponds to what is known as the rotational transform. The rotational transform plays an integral role in stability of magnetically confined plasmas [19]. In a future paper, we will relate known rotational transform formulae to what is presented here. This paper is structured as follows. First, in Sect. 2, we state the main results with some preliminary definitions. In Sect. 3, we prove these results using Witten-deformed cohomology [20] and the elliptic PDE theory developed by Bers [21,22]. In Sect. 4, we give some applications and examples including proofs of Corollaries 2, 3 and 4. In Sect. 5, we discuss our use of the theory of Bers and how, in a certain sense, this generality is needed for the full result. In Appendices A and B.1, we establish correctness of the definitions. In Appendix B.2, we discuss some foundational properties of the winding number in relation to rectifiability.

Main results and definitions
In this section, we state the main results. For this, we will need to first define the normal surface derivative mentioned in the Introduction. For terminology with smooth manifold theory, we follow Lee's book [23]. Unless otherwise stated, everything is assumed to be smooth for convenience.

Definition 1
Let M be a Riemannian manifold with boundary with metric g. Let S be an orientable codimension 1 embedded submanifold with (or without) boundary. Let V : S → T M be a normal vector field to S, and B : M → T M be a vector field tangent to S. Define the function ∂ V B : S → R by where V : U → T M is a local vector field extending V| U ∩S . This function is called the normal derivative of B with respect to V along S.
We will show in Appendix A that Definition 1 is correct. With respect to this definition, the main part of our results giving Corollary 3 is the following.

Theorem 5 Let M be an oriented Riemannian 3-manifold with boundary. Let S be an embedded 2-torus in M. Let n : S → T M be a unit normal for S. Let B be a vector field on M which satisfies
Consider the ι-related vector field ı * B on S where i : S ⊂ M. If there exists a solution u ∈ C ∞ (S) to the cohomological equation then either B| S = 0 or B| S is nowhere zero and ı * B is semi-rectifiable with proportionality B 2 | S .
A particular application of Theorem 5 is granted in the special context of first integrals as follows.

Proposition 6
Let M be an oriented Riemannian 3-manifold with boundary. Let S be a codimension 1 embedded submanifold with boundary in M. Let B be a vector field on M which satisfies ∇ · B| S = 0. Assume that B has a first integral ρ ∈ C ∞ (M) which is constant and regular on S. Then, B| S · n = 0 and, setting u = − ln ∇ρ | S , Theorem 5 is proven by the fact that ı * B is a P-harmonic vector field on the 2-torus. We will now discuss the definitions and main results concerning these fields.
Let S be an oriented Riemannian manifold with differential d and codifferential δ = (−1) nk+n+1 d where is the Hodge star operator acting on k-forms and n = dim S. Let Accordingly, a vector field X on S is called P-harmonic if its flat (metric dual 1-form) X ∈ 1 (S) is a P-harmonic 1-form. Our result concerning P-harmonic 1-forms on the 2-torus is the following. Theorem 7 Let S be an oriented Riemannian 2-torus. Let H 1 P (S) denote the real vector space of P-harmonic 1-forms on S. Then, the following holds.

The map assigning a P-harmonic 1-form to its de Rham cohomology class
then ω is either identically zero or nowhere zero. The cohomology class of a closed 1-form dual to a vector field is not particularly telling of the field-line topology on a oriented Riemannian 2-torus. This will be illustrated by an example in Sect. 4. In particular, the strictly dual result to Theorem 7 does not address the field-line topology directly. Nevertheless, P-harmonic fields have a winding number, which is metric independent.
Instead of using fractions a/b to define the winding number, we will use elements [(a, b)] of the projective real line P(R) = (R 2 \{0})/∼ where vectors u, v ∈ R 2 \{0} are considered equivalent u ∼ v iff u and v are linearly dependent. This is to account for the case of b = 0 and to emphasise the connection with Diophantine vectors. Definition 2 Let S be a 2-torus. Then, a densely-nowhere zero vector field X is said be winding if there exists a nowhere zero closed 1-form ω on S such that ω(X ) = 0. Let γ 1 , γ 2 ∈ H 1 (S) be generating classes of the first homology H 1 (S). Let ω be a closed 1-form with non-trivial cohomology class [ω] = 0 such that ω(X ) = 0. Set Then, X is said to have Diophantine winding number if the vector (a, b) is Diophantine. The vector (a, b) ∈ R 2 is non-zero and hence defines a class [(a, b)] ∈ P(R). The class [(a, b)] is called the winding number of X with respect to the generators γ 1 and γ 2 .
Here, a vector u ∈ R 2 is said to be a Diophantine vector if there exists γ > 0 and τ > 1 such that, for all k ∈ Z 2 \{0}, there holds | u, k | ≥ γ k −τ .
The proof of correctness of Definition 2 is found in Appendix B.1 along with some compatibilities with other notions of winding number. Our result concerning P-harmonic vector fields on the 2-torus is the following. Theorem 8 Let S be an oriented Riemannian 2-torus. Let 0 < P ∈ C ∞ (S) and let H P V (S) denote the real vector space of P-harmonic vector fields on S. Then, the following holds.
Then either X is identically zero or X is nowhere zero. 3. If X is nowhere zero, then X is semi-rectifiable with proportionality X 2 . Moreover, X is winding and if X has Diophantine winding number, then X is rectifiable. 4. Let γ 1 , γ 2 be generating classes of the first homology of S. The winding number of X with respect to If γ 1 , γ 2 are represented by closed curves, C i : [0, 1] → S, i ∈ {1, 2}, then where μ is the area element on S. 5. For every τ ∈ P(R), there exists a unique Y ∈ H P V (S) (up to non-zero scalar multiplication) with winding number τ with respect to γ 1 , γ 2 .
In the next section, we will prove these results. It should be emphasised early on that the less trivial part of Theorem 8 is the nowhere zero behaviour of solutions and dimension of the solution space. The remaining properties follow directly from the convenient algebraic structure of the equations. If one is willing to make nowhere zero assumptions from the beginning, as known to celestial mechanics [1] the rectifiability properties in Theorem 8 hold for vector fields satisfying more general equations. We formulate the result here and prove it in Appendix B.2 using a covariant approach as an alternative to Sternberg's proof in [1].

Proof of the main results
Theorem 5 is proven in Sect. 3.2.3 via Theorems 7 and 8 and some basic computations. The proof of Theorem 7 relies on Witten-deformed cohomology theory as well as Bers' pseudoanalytic function theory. We will introduce these theories in the course of the proof. The corresponding result for constant P has a simplified proof which only relies on classical theory. We will discuss this in relation to the full result. After this, Theorem 8 follows from Theorem 7 using the properties of the winding numbers and rectifiability established in Appendix B. We will first present the basic computations required for the proof of Theorem 5.

Computations with codimension 1 submanifolds
The computations we do here will be of higher generality than needed for Theorem 5 since no additional difficulty is met. For the following, let M be an oriented Riemannian manifold with boundary, with metric g and top form μ. Let S be a codimension 1 oriented Riemannian submanifold with boundary. Let n be the outward unit vector field on S. Write ı : S ⊂ M for the inclusion and μ S for the inherited area form on S from M.
Proof We have dω = ı * db. Relating the Hodge stars on M and S on S (see Proposition 22 in Appendix A), we get, Proposition 11 Let N be a vector field on M such that N | S = n. Let B be a vector field on M. Set b = B and ω = ı * b. Let δ S be the codifferential on S. Then We also have the ı-relatedness Hence, With this, Now, for any vector field Y on M, since Y − g(X , Y )X is ι-related to a vector fieldỸ on S and ı * μ ∈ n (S) = {0}, we get, Hence, we arrive at If B is tangent to S, the final term g( [N , B], N )| S is the normal surface derivative along S in Definition 1. Moreover, since the operations are local, we may drop the assumption that S has a vector field N on M with N | S = n since this is always true locally (see Appendix A). Thus, in the tangential case, we can rewrite our formula intrinsically in terms of B and S as follows.

Corollary 12
Assume that B is tangent to S. Set b = B and ω = ı * b. Let ı * B denote the vector field on S ı-related to B. Then we get,

P-harmonic 1-forms and vector fields on an oriented Riemannian 2-torus
Here, we prove Theorems 7 and 8. Before specialising to tori, we will consider a general closed oriented Riemannian manifold S.
It is natural to "symmetrise" these equations by the linear automorphism on 1 (S) given by multiplication ω → P −1/2 ω, giving the isomorphism, where p = ln √ P and, following Witten [20] in his 1982 paper on supersymmetry and Morse theory, for h ∈ C ∞ (S), we set Although in [20], the case of primary interest is h being a non-degenerate Morse function, it is observed for general h that the operators d h , δ h are adjoints of one another with respect to the L 2 inner product on k (S), d 2 h = 0, δ 2 h = 0 and the relation d h e −h = e h d gives an isomorphism of the k th de Rham cohomology group and the cohomology group is now known in the literature as a Witten deformation of the cohomology group H k dR (S). In addition, Witten [20] notes that by standard arguments, the kernel of the associated Laplace operator has the same dimension as H k dR (S), that is, the k th Betti number B k of S. Standard arguments include those present in the proof of the standard Hodge decomposition Theorem for the de Rham complex. These arguments may be adapted to a very large class of complexes known as elliptic complexes [24] giving rise to a generalised Hodge decomposition theorem. Although, in our current position, we may already establish the following.

Proposition 13 The assignment ω → [ω] of a closed k-form on S to its de Rham cohomology class gives an isomorphism
Proof We first consider the Witten deformation to the cohomology with some h ∈ C ∞ (S). From the adjointness of d h and δ h , we have at once that every d h -exact k-form ω which is δ h -co-closed, is necessarily ω = 0. Hence, the restricted linear quotient map is an injection. Again by adjointness, So that, by the rank-nullity theorem, this linear map is an isomorphism. Setting h = p and recalling the isomorphism ω → P −1/2 ω, the result follows.
We will now draw our attention to tori. We record Proposition 13 in this instance.

Corollary 14 Let S be an oriented Riemannian 2-torus. Then the assignment ω → [ω] of a closed 1-form on S to its de Rham cohomology class gives an isomorphism
We will now address the nowhere zero property of the elements of H 1 P (S). For this, we will need to recall elements of Riemann surface theory and in particular some of its extensions due to Bers in the 1950s. Our strategy is to relate elements of H 1 P (S) to Bers' pseudo-analytic differentials on S as a Riemann surface. In this context, differentials play the role of holomorphic 1-forms in his generalised Riemann-Roch theorem. His theorem shows that differentials on S have the nowhere zero property when they are regular; that is, when they do not have poles. Lastly we translate this property back to H 1 P (S).

Elements of Riemann surface and pseudo-analytic function theory for the 2-torus
Here we will introduce the required results from Riemann surface theory and pseudo-analytic function theory. In the next section, we will apply these results and conclude with a proof of Theorem 7.
Recall that a Riemann surface, S, is a connected 2-manifold with a holomorphic atlas A = {(U α , z α )}; namely, the transition maps z α • z −1 β of A are holomorphic between the open subsets z β (U α ∩ U β ) and z α (U α ∩ U β ) of the complex plane (i.e. of R 2 ). Charts (U , z) on S with holomorphic transition maps z • z −1 β for (U β , z β ) ∈ A will be called holomorphic charts.
Following Bers [22], if F, G : S → C are functions such that Im(FG) > 0, (F, G) is called a generating pair if at every point p ∈ S, for any holomorphic chart (U , z) on S about p, the representatives of F, G in the (U , z) are Hölder continuous. The pair (F, G) also defines a second pair (F * , G * ) given by The pair (F, G) play similar roles to that of 1 and i play in the classical theory of holomorphic functions. Bers was able to conclude many similarities between holomorphic and pseudo-analytic functions; including definite orders of poles of meromorphic pseudo-analytic functions and differentials and a generalised Riemann-Roch theorem [22]. We are interested in Bers' conclusion about differentials on a Riemann surface. We will now define these.
Bers [21] defines a differential, W , on a domain D ⊂ S (connected and open subset of S) to be an assignment (U , z) → W /dz where (U , z) is a holomorphic chart with U ⊂ D and W /dz : U → C is a function such that, given two holomorphic charts (U , z), One may also multiply W by a function f : S → C obtaining a differential f W defined in the obvious way.
The differential W is said to be continuous (or have partial derivatives etc.
With this, given a continuous differential W on S and a closed continuously differentiable curve C : If the (F, G)-period vanishes over any C homologous to zero, then W is called a regular (F, G)-differential. The following is a direct consequence of the generalised Riemann-Roch Theorem [22,.
Theorem 15 A regular (F, G)-differential W on a closed Riemann surface S of genus g = 1 is either identically zero or nowhere zero.
We will now apply this result in the case of P-harmonic forms on an oriented Riemannian 2-torus to prove Theorem 7.

P-harmonic 1-forms
The following [25, Chapter 10] is a well-known means of giving any smooth surface a Riemann surface structure. We now will now prepare an application of Bers' differentials in the context of P-harmonic 1-forms. On any smooth manifold S, the complexified cotangent bundle T * C S is given by

Proposition 16 Let S be an oriented Riemannian 2-manifold. Then, there exists a maximal holomorphic atlas
where C is regarded with its vector space structure over R. Regularity and operations on sections of T * C S are defined component-wise. For example, for f ∈ C ∞ (S; C), the set of smooth functions S → C, writing f = u + iv for some unique u, v ∈ C ∞ (S), we set to the function f : U → C where ı * ω = f dz for any holomorphic chart (U , z), is a bijective equivalence between type (1, 0) sections of T * C S and Bers' differentials on M. If ω is a type (1, 0) section of T * C S and W is its differential equivalent, then ω is continuous if and only if W is. Moreover, for any continuously differentiable curve C : [0, 1] → S, we have where the latter integral is a C-linear extension of integrals of continuous 1-forms along the curve C. With this, we are ready to prove Theorem 7.
Proof of Theorem 7 Let S be an oriented Riemannian 2-torus and 0 < P ∈ C ∞ (S). Corollary 14 proves the first two statements of the Theorem. For the third, let ω be a P-harmonic 1-form. So, Then, as previously discussed, considering the 1-formω = P −1/2 ω, with p = ln √ P, we have that Now, give S the structure of a closed Riemann surface of genus g = 1 as per Proposition 16.
Since p is smooth, the pair (F, G) given by is a generating pair on S. Then, the pair (F * , G * ) are given by Consider the sectionω of T * C S given bŷ ω =ω + i ω.
Then, from ω = iω, it easily follows from Proposition 16 thatω is a T * C S section of type (1, 0). Hence, we may consider its Bers' differential equivalent W . Let C : [0, 1] → S be a continuously differentiable closed curve. Then, we get If C is homologous to zero, then by Stokes' Theorem for chains, since d p −1ω = 0 and d p ω = 0, we obtain that Thus, W is a regular (F, G) differential. Hence, from Theorem 15, we obtain that W is either identically zero or nowhere zero. Thus, ω is either identically zero or nowhere zero.
We will now move on to P-harmonic vector fields.

P-harmonic vector fields
We will now prove Theorem 8, which primarily involves reinterpreting 1-form data in terms of the corresponding vector field data.

Proof of Theorem 8
Let S be an oriented Riemannian 2-torus and 0 < P ∈ C ∞ (S). Denote by H P V (S) the real vector space of P-harmonic vector fields on S.

Proof of statements 1 and 2 Consider the map H
given by X → X . Then, from Theorem 7, we immediately obtain that dim H P V (S) = 2 and that every X ∈ H P V (S) is either identically zero or nowhere zero.
For the remaining statements, let X ∈ H P V (S) be nowhere zero and let γ 1 , γ 2 be classes generating the first homology of S.

Proof of statement 3 Set
is the perpendicular to X . The 1-forms ω = X and η = P ω, are closed 1-forms on S and Hence, [X / X 2 , Y ] = 0. Hence, from a well-known result in Lie group theory (see Appendix B.2, Proposition 25), X / X 2 is rectifiable. From, for example, η(X ) = 0, we see that X is winding and in turn, from a well-known result in celestial mechanics (see Appendix B.2, Theorem 9) if X has Diophantine winding number, then X is rectifiable.

Proof of statement 4
From the previous, we have a nowhere zero η = P ω with η(X ) = 0.
In particular (see Proposition 23 in Appendix B), [η] = 0 and the winding number of X with respect to γ 1 , γ 2 is given by [(a, b)] where Moreover, for a curve C : [0, 1] → S we have So, if γ 1 , γ 2 are represented by curves C 1 and C 2 , Proof of statement 5 We have the linear isomorphisms The final linear isomorphism is due to Theorem 7. Hence, we have the linear isomorphism Hence, by de Rham's Theorem, we have the linear isomorphism We will conclude this section with a proof of Theorem 5. From Corollary 12, taking P = e u , the codifferential δ S on S gives

Proof of Theorem 5 Let
Hence, ω is a P-harmonic 1-form on S. Considering the sharp S on S, ω S = ı * B. Hence, ı * B is a P-harmonic vector field on S. The conclusion now follows from Theorem 8.

Applications and examples
An application mentioned of Theorem 5 is Proposition 6. We will now prove this. Next, B(ρ) = 0 and N (ρ) = g(N , ∇ρ) =ũ so that,
On the topic of first integrals, we now provide a proof of Corollary 2.

Proof of Theorem 9
Let M be an oriented Riemannian 3-manifold with boundary. Let B be a vector field on M which satisfies, for some function ρ, First, observe the following. Let S be a closed connected component of a regular level set of ρ. For later, give S the orientation where the unit normal n = ∇ρ| S ∇ρ | S is outward. Setting u = − ln ∇ρ | S , and ı : S ⊂ M to be the inclusion, our assumptions together with Proposition 6 give, Hence, by Proposition 10 and Corollary 12, ı * B is a P = ∇ρ −1 | S -harmonic vector field on S . In particular, fix a closed connected component S of a regular level set of ρ. From the above, by Theorem 5, if B is not identically zero on S and S is a 2-torus, then B is nowhere zero on S. Conversely, if B is nowhere zero on S, then since S is closed and connected and oriented, the nowhere zero vector field ı * B on S implies that S is a 2-torus.
Assume B is nowhere zero on S and that S is a 2-torus. Consider the neighborhood V = {∇ρ = 0} of S and the local vector field, With this, let z ∈ I . Then, S = (S × {z}) is an embedded 2-torus in M for which X z = ı * B is a P = ∇ρ | S -harmonic vector field on S where ı : S ⊂ M is the inclusion. Hence, following the proof of statement 3 in Theorem 8, being the perpendicular to X z and P = ∇ρ | S . Thus, with the local vector fields where a, b : I → R are smooth functions and z is the projection onto the factor I .
We will now clarify the application of our results to strong Beltrami fields. Lastly, consider a solution u to du(ı * B) = ∂ n B on S. We have that ı * B is P = e uharmonic by Proposition 10 and Corollary 12. Take closed curves C 1 , C 2 : [0, 1] → S ⊂ R 3 whose homology classes generate the first homology H 1 (S) and form the integrals We will now address the example on the sphere from the introduction.

Proof of Corollary 4
Let B be a divergence-free vector field on an oriented Riemannian 3manifold M with boundary. Suppose that B and ∇ × B have a mutual first integral ρ on M with a connected component S of a regular level set diffeomorphic to S 2 .
Just as observed in [17], there is a neighbourhood U of S foliated by spheres for which ρ is constant and regular thereon. Then, on such a sphere S ⊂ U , by Proposition 6 and Corollary 12, setting b = B and ω = ı * b, we obtain that ω is P-harmonic on S with P = ∇ρ −1 | S . However, recall that from Proposition 13, that H 1 P (S ) and H 1 dR (S ) are isomorphic. In particular, we have H 1 dR (S ) = {0} so that ω = 0. Hence, B| S = 0. Thus, B| U = 0.
Our results also provide some easy bounds on the size of the space of divergence-free Beltrami fields for a special class of proportionality factors in a similar vein to [17,26].

Corollary 17 Let λ ∈ C ∞ (M) be a smooth function on an oriented connected Riemannian 3-manifold with boundary which is constant and regular on ∂ M, where ∂ M is diffeomorphic to a 2-torus. Then, the space of vector fields B satisfying
is at most two dimensional. We will now illustrate with a simple example that the cohomological assumption in Theorem 5 is not redundant.

Example 1
We will first consider on R 3 the vector field Eventually, we will lower this vector field into the manifold M = (R/2πZ) 2 × R along with some of its properties. To this end, writingŜ for the plane {z = 0}, one finds and denoting byî :Ŝ ⊂ R 3 the inclusion, for any u ∈ C ∞ (Ŝ), we get on the other hand, since ∂ ∂z is a geodesic vector field of unit length, Hence, du(ı * B ) = ∂nB has no solutions. Since f and h are 2π-periodic in x and y,B descends to a vector field B on M. Given the inherited oriented Riemannian structure from R 3 on M, we have where S is the 2-torus embedded in M lowered fromŜ in R 3 with outward unit normal n.
The other data shows that there does not exist a solution u ∈ C ∞ (S) to du(ı * B ) = ∂ n B. This is expected from Theorem 5 since we see that B has many zeros on S but is not identically zero on S.
We will now illustrate how the cohomology class of a nowhere zero closed 1-form ω on a 2-torus S does not explain the topology of the integral curves of ω .
Thus, in S, C(T ) + 2πZ 2 = C(0) + 2πZ 2 . Hence, η is not periodic. On the other hand, the perpendicular vector fields W = ( ω) and H = ( η) have the same winding number with respect to every pair of generators since [ω] = [η] and the flows of W and H are explicitly given by The curves t → ψ W (2πt, p + 2πZ 2 ) and t → ψ H (2πt, p + 2πZ 2 ), t ∈ [0, 1], are closed and easily seen to be homologous to the curve C 1 : This situation holds much more generally; as will be discussed in a future paper which will further explore the rotational transform in the context of magnetic confinement fusion.

Discussion
Theorem 7 in the case of constant P may be proven with the classical Riemann-Roch Theorem. In fact, together with the work of Calibi on intrinsically harmonic forms [27] we used to prove Theorem 9 by covariant means (see Theorem 27 in Appendix B.2), one can establish the following known result. It would be preferable to shorten the proof of Theorem 7 by a suitable reduction to the constant P case. For instance, from the conclusion of Theorem 7 and from Theorem 27 we get the following.

Corollary 19
Let S be an oriented 2-torus and 0 < P ∈ C ∞ (S). Let g be a metric on S and consider the P-harmonic 1-forms H 1 P (S, g) on (S, g). Let ω, η ∈ H 1 P (S, g) form a basis for H 1 P (S, g). Then, there exists a metricg with induced Hodge star˜ satisfying ω = η so that H 1 P (S, g) = H 1 (S,g), the harmonic 1-forms on (S,g).
However, the metrics in Corollary 19 are in a sense retrospective of the conclusion of Theorem 7 and are non-canonical. More precisely, the Witten-deformed co-differential δ p = P −1 δ P on (S, g) has a different kernel to the co-differentialδ on (S,g) when P is nonconstant in Corollary 19. This may be shown in higher generally as a comparison between different Witten-deformed co-differentials on S, without reference to Theorem 7, as follows.

Proposition 20 Let S be an oriented Riemannian 2-torus. Let g andg be Riemannian metrics on S. Let 0 < P, Q ∈ C ∞ (S) and consider the operators
Suppose that ker δ p ⊂ kerδ q .
Then g andg are conformally equivalent and P = cQ for some constant c > 0.
Proof Let R = P −1 Q, r = ln √ R and considerδ r = R −1δ R. Then, we have ker δ ⊂ kerδ r . Considering the bundle metrics , and , ∼ on 1 (S), we have for any f ∈ C ∞ (S) and ω ∈ 1 (S) that In particular, for any f ∈ C ∞ (S) and g-harmonic 1-form ζ , because also δ w ζ = 0, we get Take two g-harmonic 1-forms χ 1 , χ 2 which generate the first cohomololgy on S. Now, set ζ i = χ i . Then, for each point p ∈ S, (ζ 1 | p , ζ 2 | p ) forms a basis of 1 (T p S) . For (a, b) ∈ R 2 , form ζ (a,b) = aζ 1 + bζ 2 . We claim that, for all (a, b) ∈ R 2 such that (a, b) = cq for some c ∈ R and q ∈ Q 2 , for any point p ∈ S, there exists f ∈ C ∞ (S) such that d f | p = 0 and d f , ζ (a,b) = 0.
We have that, and that for any point p ∈ S, either d f 1 | p = 0 or d f 2 | p = 0. This proves the claim.
With this, let p ∈ S. Then, let (a, b) ∈ R 2 such that (a, b) = cq for some c ∈ R and q ∈ Q 2 . Then, consider the form η = aζ 1 | p + bζ 2 | p ∈ 1 (T p S). Then, take f ∈ C ∞ (S) such that d f | p = 0 and d f , ζ (a,b) = 0. Then, we have Hence, since Q 2 is dense in R 2 , with respect to the topology induced by the inner product , on 1 (T p S), there exists a dense subset S such that, for all η ∈ S, ˜ η, η = 0. Thus, since˜ is a linear operator, ˜ η, η = 0 for all η ∈ 1 (S). Now, considering on 1 (T p S), we see that, for all η ∈ 1 (T p S), there exists c ∈ R such that˜ η = c η. Since˜ and are linear operators and square to −Id, it follows that for η ∈ 1 (S),˜ η = η. In particular, g| p andg| p are conformally equivalent. Since p ∈ S was arbitrary, g andg are conformally equivalent.
On top-forms,˜ = κ and as in the above, on 1-forms,˜ = . Hencẽ Hence, ker δ ⊂ ker δ r . Now, we have for ω ∈ 1 (S) that Hence, for any g-harmonic 1-form ζ , we have Thus, r is constant. Hence, R is constant. Thus, P = cQ for some constant c > 0.
In this way, it is seen that P-harmonic 1-forms on surfaces have their differences with harmonic 1-forms. On the other hand, they have many similarities: as seen in Sect. 3.2, Theorem 7, and Corollary 19. the following. For any p ∈ S, there exists a neighbourhood U in M and a regular function f ∈ C ∞ (U ) which is constant on U ∩ S and for any h ∈ C ∞ (S), there exists a H ∈ C ∞ (U ) with H | U ∩S = h| U ∩S .
For the first part, since V is nowhere zero, h = V ∈ C ∞ (S). Moreover, for any p ∈ S, there exists a neighbourhood U in M and a regular function f ∈ C ∞ (U ) which is constant on U ∩ S and a H ∈ C ∞ (U ) with H | U ∩S = h| U ∩S . Then, since both V and ∇ f | S are normal-pointing, the local vector field Moreover, From this, we see that The following establishes the Hodge star formula used in Proposition 10. It suffices to consider the case of vector spaces because the Hodge star is defined point-wise. We will do this in arbitrary dimensions because no additional difficulty is met. O) be an oriented inner product space of dimension n ≥ 2. Let S be codimension 1 vector subspace with a unit normal n. Consider the induced oriented inner product space (S, i * g, O S ) where i : S ⊂ V is the inclusion. Consider the Hodge star on V and S on S. Then, for any k-form ω ∈ k (V ), Now, since T (n) = 0, we have i nη = 0 so that Hence, Let Then, for i ∈ {1, ..., n − 1} since f i ∈ S, T ( f i ) = f i . Hence, for 1 ≤ j 1 , ..., j k ≤ n − 1, we haveη Hence, .., j k ≤nη (e j 1 , ..., e j k )ω(e j 1 , ..., e j k ) One may easily check with the orthonormal bases introduced that μ S = i * (i n μ) so that, Thus, Hence,

Appendix B: The winding number
Besides proving correctness of definitions related to the winding number, we hope to highlight the elegance of cohomological and covariant approaches to basic properties of the winding number. Specifically, we will introduce the notions of intrinsically harmonic 1-forms and cohomologically rigid vector fields once they are needed.

B.1 Correctness and compatibility of definitions
A large part of the correctness of Definition 2 comes from the Poincaré Duality Theorem, which we emphasise in the proof.

Proof of statement 3 Consider first the vectors
Now, there exist integer matrices A,Ã ∈ Z 2×2 such that,γ i = A i j γ j and γ i =Ã i jγ j . From linear independence, we see that AÃ =Ã A = I , the identity 2 × 2 matrix. Hence, and R AR ∈ G L(2, Z). Now, in general, let L ∈ G L(2, Z) and u ∈ R 2 be a Diophantine vector. So that, there exists γ > 0 and τ > 1 such that, for all k ∈ Z 2 \{0}, Considerũ = Lu. Then, fix a constant C > 0 such that, L T u ≤ C u for all u ∈ R 2 and setγ = γ C −τ . Then, for k ∈ Z 2 \{0} non-zero, we have L T k ∈ Z 2 \{0} so that, Hence, Lu =ũ is Diophantine. With this, we see that We will now show that the winding number appearing in this paper is compatible with vector fields lying on straight lines.

B.2 Relationship with rectifiability
In this section, we will prove the Theorem 9 from celestial mechanics [1] along with Proposition 26, which was needed for the proof of Corollary 2. For completeness, we will first discuss the well-known rectifiability techniques employed by Arnold [3,11]. The first of which is focuses on a single 2-torus.

Proposition 25 Let S be a compact connected 2-manifold and let X and Y be vector fields which are point-wise independent and [X , Y ] = 0. Then, there exists a diffeomorphism
: S → R 2 /Z 2 and numbers a, b ∈ R such that R 2 /R 2 p is compact, we must have the latter. In particular, there exists an invertible matrix A ∈ GL(2, R) such that AZ 2 = R 2 p which induces a diffeomorphismÂ : R 2 /Z 2 → R 2 /R 2 p . By definition of F, one sees that, F * X = X andÂ * Ỹ = Y . The result then follows with the diffeomorphism = (Â • F) −1 .
Directly related to Arnold's structure Theorems is the following proposition which is a variant of Proposition 25 for multiple tori in three dimensions.
Proof Consider the map F : Fix a point p 0 ∈ S and consider the map G : R 2 × I → M given by G ((s, t), z) = F((s, t), ( p 0 , z)).
As seen in the proof of Proposition 25, we have for all z ∈ I that there exists a rank 2 lattice z ⊂ R 2 such that the map G z : R 2 / z → M given by G z ((s, t) is an embedding on to (S, z) in M satisfying where ∂ ∂ x and ∂ ∂ y are the constant vector fields lowered to the quotient R 2 / z . To turn the G z s into a diffeomorphism with domain R 2 /Z 2 × I , we must first check that the lattice z smoothly varies with z ∈ I . To this end, consider the vector field frame (X , Y , Z ) where Z = ∂ ∂z is the vector field on M induced by the factor I . Consider now the induced co-frame (α, β, γ ) of 1-forms to (X , Y , Z ) so that Note that γ = dz. Now, fix smooth curves C 1 , C 2 : [0, 1] → S which generate the first homology of S. For z ∈ I and i ∈ {1, 2} set We claim that (v 1,z , v 2,z ) forms a lattice basis for z for z ∈ J . Indeed, let z ∈ I . Then, we get curves c i : [0, 1] → R 2 / z induced by the embedding G z and curves C i (i ∈ {1, 2}). We also have that where dx, dy are the constant 1-forms lowered to the quotient R 2 / z . Now, write z = Zu 1 ⊕ Zu 2 for some u 1 , u 2 linearly independent in R 2 . Then, consider the curves D 1 , D 2 : [0, 1] → R 2 / z given by With this, we get that and similarly v 2 z,i = A i j u 2 j . In total, we have v z,i = A i j u j . Hence, since (u 1 , u 2 ) is a generator for z , so is (v z,1 , v z,2 ). Hence, our claim holds.
We will now use our v z,i (i ∈ {1, 2}) to make a diffeomorphism. To this end, for each z ∈ J , form the matrix Then, we have the smooth mapsÂ,B : R 2 × I → R 2 × I given bŷ wherebyÂ •B = Id =B •Â so that in particular,Â is a diffeomorphism. Moreover, we see that T G is everywhere invertible and that G(∂(R 2 × I )) = ∂ M. Similarly to the proof of Proposition 25, the tangent map T G is invertible everywhere. Hence, in the case of I = (− , ), the Inverse Function Theorem gives that G is a local diffeomorphism. In the case of I = [0, ), one may, for instance, globally extend the vector fields X and Y to vector fieldsX andỸ defined on a neighborhood of S × (− , ) tangent to the compact S × {z} for all z ∈ I , construct the suitableG, and apply the Inverse Function Theorem toG using the fact that S × {0} is left invariant by X and Y . Hence, in any case, we have the local diffeomorphism H = G •Â z : R 2 × I → M. Now, we have the product map = π × Id I : R 2 × I → R 2 /Z 2 × I where π : R 2 → R 2 /Z 2 is the quotient map so that is an onto local diffeomorphism. With this, since A z = (v z,1 , v z,2 ) is a matrix of generators of z for each z ∈ I , then there exists a unique map : R 2 /Z 2 × I → M such that Hence, is a local diffeomorphism. It is also clear that is bijective. Hence, is a diffeomorphism and the desired map is = −1 .
To continue with proving Theorem 9, we will also use a very special case of Calibi's theorem [27] on intrinsically harmonic 1-forms, which is the following.

Theorem 27
Let M be a compact oriented manifold. Let ω ∈ 1 (M) be closed and nowhere zero. Then ω is intrinsically harmonic; that is, there exists a metric g on M such that δω = 0.
The last result we need to prove Theorem 9 is the following solvability of the cohomological equation [2, Proposition 2.6].

Proposition 28
Let S be a 2-torus, : S → R 2 /Z 2 be a diffeomorphism and set X such where (a, b) is a Diophantine vector. Then, for any v ∈ C ∞ (S), there exists c ∈ R and a solution u ∈ C ∞ (S) to the cohomological equation That is, setting μ = * μ 0 , where μ 0 is the standard volume form on R 2 /Z 2 , for any v ∈ C ∞ (S), there exists a solution u ∈ C ∞ (S) to the cohomological equation if and only if S vμ = 0.
We will now prove Theorem 9.

Proof of Theorem 9
Let X be a nowhere zero vector field on a 2-torus S.

Statement 2 implies statement 3
Suppose that X is winding. Let ω be a closed nowhere zero 1-form such that ω(X ) = 0. Since ω is nowhere zero, by Theorem 18, there exists an Riemannian metric g on S for which ω is harmonic on (S, g). Then, consider η = ω. Then, η is closed and the top form ω ∧ η is nowhere zero. Thus, since X is nowhere zero, i X (ω ∧ η) is nowhere zero. On the other hand, i X (ω ∧ η) = ω(X )η − η(X )ω = −η(X )ω.
From this, consider the unique vector field Y on S such that ω(Y ) = 1, η(Y ) = 0.
Then, X and Y are point-wise independent and [X /η(X ), Y ] = 0. Hence, from Proposition 25, X /η(X ) is rectifiable and so X is semi-rectifiable.
Lastly, suppose that X has Diophantine winding number. Then, from the above, together with Propositions 23 and 24, we get for some 0 < f ∈ C ∞ (S), Diophantine vector (a, b) ∈ R 2 , and diffeomorphism : S → R 2 /Z 2 , that * (X / f ) = a With this, consider the vector fieldsX and Y such that * X = a Since (a, b) is Diophantine, so is (−b, a). With this, let u ∈ C ∞ (S) and consider the vector field