1 Introduction

The rectifiability of a vector field on a 2-torus is a well understood dynamical problem [1, 2]. A vector field X on a 2-torus S is said to be rectifiable if there exists numbers \(a,b \in {\mathbb {R}}\) and a diffeomorphism \(\Phi : S \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\) such that \(\Phi _* X\) is a constant vector field on \({\mathbb {R}}^2\) lowered to \({\mathbb {R}}^2/{\mathbb {Z}}^2\). That is,

$$\begin{aligned} \Phi _* X = a \frac{\partial }{\partial x} + b \frac{\partial }{\partial y}. \end{aligned}$$
(1)

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

$$\begin{aligned} \nabla \cdot B&= 0,&(\nabla \times B) \times B&= \nabla p,&B \cdot n&= 0 \text { on } \partial M, \end{aligned}$$

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 \(\partial M\).

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 \(\nabla \times B\) and in some neighbourhood \(U\subset M\) of such a component, there exist a diffeomorphism \(\Phi : U \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2 \times I\) where \(I \subset {\mathbb {R}}\) is an interval, such that

$$\begin{aligned} \Phi _* B\vert _U&= a(z) \frac{\partial }{\partial x} + b(z) \frac{\partial }{\partial y},&\Phi _* \nabla \times B \vert _U&= c(z) \frac{\partial }{\partial x} + d(z) \frac{\partial }{\partial y}, \end{aligned}$$

where \(a,b,c,d: I \rightarrow {\mathbb {R}}\) are smooth functions and z is the projection onto the factor I.

Coordinates in which both B and \(\nabla \times B\) 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 \(\nabla \times B\)) are called magnetic coordinates [6, 12].

Arnold also remarked [3] that in the case of \(\nabla p = 0\), assuming B is nowhere zero, then necessarily

  1. (i)

    B is a Beltrami field, namely \(\nabla \times B = \lambda B\) for some function \(\lambda \),

  2. (ii)

    \(\lambda \) is a first integral and the closed regular level sets of \(\lambda \) are unions of tori, and

  3. (iii)

    in a neighborhood \(U\subset M\) of such a 2-torus, there exists a diffeomorphism \(\Phi : U \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2 \times I\) and a positive function \(f: U \rightarrow {\mathbb {R}}\) such that

    $$\begin{aligned} \Phi _* (B\vert _U/f)&= a(z) \frac{\partial }{\partial x} + b(z) \frac{\partial }{\partial y} \end{aligned}$$

    where again \(a,b: I \rightarrow {\mathbb {R}}\) 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 \(\rho \),

$$\begin{aligned} \nabla \cdot B&= 0,&B \cdot \nabla \rho&= 0,&(\nabla \times B) \cdot \nabla \rho&= 0. \end{aligned}$$

Let S be a closed connected component of a regular level set of \(\rho \). Then, the following are equivalent

  1. 1.

    B is not identically zero on S and S is a 2-torus

  2. 2.

    B is nowhere zero on S

and when this is the case, additionally assuming \(S \cap \partial M = \emptyset \) (or \(S \subset \partial M\)), there exists a neighbourhood \(U\subset M\) of S and a diffeomorphism \(\Phi : U \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2 \times I\) where I is an open (or half-open) interval in \({\mathbb {R}}\) such that \(B\vert _U\) is nowhere zero and

$$\begin{aligned} \Phi _* \left( \frac{B\vert _U}{\Vert B\Vert ^2 \vert _U}\right) = a(z) \frac{\partial }{\partial x} + b(z) \frac{\partial }{\partial y} \end{aligned}$$

where \(a,b: I \rightarrow {\mathbb {R}}\) are smooth functions and z is the projection onto the factor I.

A strong Beltrami field B satisfying \(\nabla \times B = \lambda B\) where \(\lambda \) 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 \(\partial M\) has a toroidal connected component and \(B \cdot n = 0\), 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 \(M \subset {\mathbb {R}}^3\), 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 \(\partial M\) is a Diophantine invariant 2-torus. That is, as in the context of the KAM Theorem, the vector field X (induced from B on \(\partial M\)) can be written in the form of Eq. (1) where the vector (a, b), known as the frequency vector of X with respect to \(\Phi \), 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 \(\Phi \) 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 \({\mathbb {R}}^3\) and S is a toroidal (connected) component of the boundary \(\partial M\). 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 \(\partial M\) when M is a hollow torus, that is, when M is diffeomorphic to \({\mathbb {R}}^2/{\mathbb {Z}}^2 \times [0,1]\).

Corollary 3

Let M be embedded in \({\mathbb {R}}^3\) with the inherited Euclidean structure. Let S with a toroidal boundary component of M. Let \(n: \partial M \rightarrow {\mathbb {R}}^3\) be the outward unit normal on M. Let B be a vector field on M satisfying, for some \(\lambda \in {\mathbb {R}}\),

$$\begin{aligned} \nabla \cdot B&= 0,&\nabla \times B&= \lambda B,&B\vert _{\partial M} \cdot n&= 0. \end{aligned}$$

Consider \(B\vert _{S}\), the vector field B along S. Then, the following are equivalent.

  1. 1.

    There exists a solution \(u \in C^{\infty }(S)\) to the cohomological equation

    $$\begin{aligned} B\vert _{S}(u) = \partial _n B. \end{aligned}$$
  2. 2.

    The vector field \(B\vert _{S}\) preserves a top-form \(\mu \) on S.

  3. 3.

    Either \(B = 0\), or B is nowhere zero on S and \(B\vert _{S}\) is semi-rectifiable with proportionality \(\Vert B\Vert ^2\vert _{S}\).

With such a solution u and closed curves \(C_1,C_2: [0,1] \rightarrow S \subset {\mathbb {R}}^3\) whose homology classes generate the first homology \(H_1(S)\), form the integrals

$$\begin{aligned} I(C_i,u) = \int _{0}^1 \exp {(u(C_i(t)))} \det (C_i'(t),B(C_i(t)),n(C_i(t)))\mathrm{d}t,~i \in \{1,2 \}. \end{aligned}$$

If \((I(C_1,u),I(C_2,u))\) is Diophantine, then S is a Diophantine invariant 2-torus for \(B\vert _{S}\). The vector \((I(C_1,u),I(C_2,u))\) being Diophantine is independent of the solution u and curves \(C_1,C_2\) chosen.

Concerning Corollary 3 we have a few remarks.

  1. 1.

    The implication 2. \(\Rightarrow \) 3. is well-known in the case that B is non-vanishing and does not rerquire that B is a strong Beltrami field (see for instance Theorem 9 in Sect. 2).

  2. 2.

    If the field is zero on the boundary component S, \(B\vert _{S} = 0\), then from Gerner’s result [16,  Lemma 2.1] we must have \(B = 0\) on the entire domain M (see also Proposition 17).

  3. 3.

    The function \(\partial _n B\) in Corollary 3 and elsewhere will be defined in Sect. 2 of the main text. It represents a derivative in the normal direction which may be evaluated using any local extension of n. In particular, if a first integral \(\rho \) of B is constant and regular on S, then \(u = -\ln \Vert \nabla \rho \Vert \vert _{S}\) satisfies the cohomological equation in the corollary (this is Proposition 6 in Sect. 2).

Aside from the 2-torus, the techniques used to show the main results may also be easily applied to the 2-sphere. This gives a similar result to that obtained by Enciso et al. [17] for divergence-free vector fields which are not necessarily Beltrami.

Corollary 4

Let B be a divergence-free vector field on an oriented Riemannian 3-manifold M with boundary. Suppose that B and \(\nabla \times B\) have a mutual first integral \(\rho \) on M with a connected component S of a regular level set diffeomorphic to \({\mathbb {S}}^2\). Then, B vanishes entirely in a neighbourhood of S.

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.

2 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 \({\mathcal {V}}: S \rightarrow TM\) be a normal vector field to S, and \(B:M\rightarrow TM\) be a vector field tangent to S. Define the function \(\partial _{{\mathcal {V}}}B: S \rightarrow {\mathbb {R}}\) by

$$\begin{aligned} \partial _{{\mathcal {V}}}B\vert _p = g\vert _p([V,B]\vert _p,V\vert _p) \end{aligned}$$

where \(V: U \rightarrow TM\) is a local vector field extending \({\mathcal {V}}\vert _{U \cap S}\). This function is called the normal derivative of B with respect to \({\mathcal {V}}\) 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 \(n: S \rightarrow TM\) be a unit normal for S. Let B be a vector field on M which satisfies

$$\begin{aligned} \nabla \cdot B\vert _S&= 0,&\nabla \times B\vert _S \cdot n&= 0,&B\vert _S \cdot n&= 0. \end{aligned}$$

Consider the \(\iota \)-related vector field \(\imath ^*B\) on S where \(i: S \subset M\). If there exists a solution \(u \in C^{\infty }(S)\) to the cohomological equation

$$\begin{aligned} \mathrm{d}u(\imath ^*B) = \partial _n B, \end{aligned}$$

then either \(B\vert _S = 0\) or \(B\vert _S\) is nowhere zero and \(\imath ^*B\) is semi-rectifiable with proportionality \(\Vert B\Vert ^2\vert _S\).

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 \(\nabla \cdot B\vert _S = 0\). Assume that B has a first integral \(\rho \in C^{\infty }(M)\) which is constant and regular on S. Then, \(B\vert _S \cdot n = 0\) and, setting \(u = -\ln \Vert \nabla \rho \Vert \vert _S\),

$$\begin{aligned} \mathrm{d}u(\imath ^*B) = \partial _n B. \end{aligned}$$

Theorem 5 is proven by the fact that \(\imath ^*B\) 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 \(\delta = (-1)^{nk+n+1}\star \mathrm{d} \star \) where \(\star \) is the Hodge star operator acting on k-forms and \(n = \dim S\). Let \(0 < P \in C^{\infty }(S)\). A k-form \(\omega \in \Omega ^k(S)\) is called P-harmonic if

$$\begin{aligned} \mathrm{d}\omega = 0,\quad \delta P \omega = 0. \end{aligned}$$

Accordingly, a vector field X on S is called P-harmonic if its flat (metric dual 1-form) \(X^{\flat }\in \Omega ^1(S)\) 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 \({\mathcal {H}}_P^1(S)\) denote the real vector space of P-harmonic 1-forms on S. Then, the following holds.

  1. 1.

    \(\dim {\mathcal {H}}_P^1(S) = 2\).

  2. 2.

    The map assigning a P-harmonic 1-form to its de Rham cohomology class

    $$\begin{aligned} {\mathcal {H}}_P^1(S) \ni \omega \mapsto [\omega ] \in H^1_{\text {dR}}(S), \end{aligned}$$

    is a linear isomorphism.

  3. 3.

    If \(\omega \in {\mathcal {H}}_P^1(S)\), then \(\omega \) 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 \({\mathbb {P}}({\mathbb {R}}) = ({\mathbb {R}}^2\backslash \{0\})/{\sim }\) where vectors \(u,v \in {\mathbb {R}}^2\backslash \{0\}\) are considered equivalent \(u \sim v\) 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 \(\omega \) on S such that \(\omega (X) = 0\). Let \(\gamma _1,\gamma _2 \in H_1(S)\) be generating classes of the first homology \(H_1(S)\). Let \(\omega \) be a closed 1-form with non-trivial cohomology class \([\omega ] \ne 0\) such that \(\omega (X) = 0\). Set

$$\begin{aligned} a&= -\int _{\gamma _2}\omega ,&b&= \int _{\gamma _1}\omega . \end{aligned}$$

Then, X is said to have Diophantine winding number if the vector (a, b) is Diophantine. The vector \((a,b) \in {\mathbb {R}}^2\) is non-zero and hence defines a class \([(a,b)] \in {\mathbb {P}}({\mathbb {R}})\). The class [(a, b)] is called the winding number of X with respect to the generators \(\gamma _1\) and \(\gamma _2\).

Here, a vector \(u \in {\mathbb {R}}^2\) is said to be a Diophantine vector if there exists \(\gamma > 0\) and \(\tau > 1\) such that, for all \(k \in {\mathbb {Z}}^2\backslash \{0\}\), there holds \(\vert \langle u,k \rangle \vert \ge \gamma \Vert k\Vert ^{-\tau }\).

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 \(0 < P \in C^{\infty }(S)\) and let \({\mathcal {H}}_P V(S)\) denote the real vector space of P-harmonic vector fields on S. Then, the following holds.

  1. 1.

    \(\dim {\mathcal {H}}_P V(S) = 2\).

  2. 2.

    Let \(X \in {\mathcal {H}}_P V(S)\). Then either X is identically zero or X is nowhere zero.

  3. 3.

    If X is nowhere zero, then X is semi-rectifiable with proportionality \(\Vert X\Vert ^2\). Moreover, X is winding and if X has Diophantine winding number, then X is rectifiable.

  4. 4.

    Let \(\gamma _1,\gamma _2\) be generating classes of the first homology of S. The winding number of X with respect to \(\gamma _1,\gamma _2\) is [(a, b)] where

    $$\begin{aligned} a&= - \int _{\gamma _2}P\star X^{\flat },&b&= \int _{\gamma _1}P\star X^{\flat }. \end{aligned}$$

    If \(\gamma _1,\gamma _2\) are represented by closed curves, \(C_i: [0,1] \rightarrow S\), \(i \in \{1,2\}\), then

    $$\begin{aligned} \int _{\gamma _i}P\star X^{\flat } = \int _{0}^1 P(C_i(t)) \mu (X(C_i(t)),C_i'(t))\mathrm{d}t, \end{aligned}$$

    where \(\mu \) is the area element on S.

  5. 5.

    For every \(\tau \in {\mathbb {P}}({\mathbb {R}})\), there exists a unique \(Y \in {\mathcal {H}}_P V(S)\) (up to non-zero scalar multiplication) with winding number \(\tau \) with respect to \(\gamma _1,\gamma _2\).

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].

Theorem 9

Let X be a nowhere zero vector field on a 2-torus S. Then, the following are equivalent.

  1. 1.

    X preserves a top-form \(\mu \) on S.

  2. 2.

    X is winding.

  3. 3.

    X is semi-rectifiable.

Moreover, if X has Diophantine winding number, then X is rectifiable.

3 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’ pseudo-analytic 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.

3.1 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 \(\mu \). Let S be a codimension 1 oriented Riemannian submanifold with boundary. Let n be the outward unit vector field on S. Write \(\imath : S \subset M\) for the inclusion and \(\mu _S\) for the inherited area form on S from M.

Proposition 10

Assume \(\dim M = 3\). Let B be a vector field on M. Set \(b = B^{\flat }\) and \(\omega = \imath ^*b\). Then

$$\begin{aligned} \mathrm{d}\omega = g\vert _S(\nabla \times B\vert _{S},n)\mu _S. \end{aligned}$$

Proof

We have \(\mathrm{d}\omega = \imath ^*\mathrm{d}b\). Relating the Hodge stars \(\star \) on M and \(\star _S\) on S (see Proposition 22 in Appendix A), we get,

$$\begin{aligned} \star _S \imath ^*\mathrm{d}b = \imath ^*(i_n \star \mathrm{d}b\vert _S) = \imath ^*(i_{n} (\nabla \times B\vert _S)^{\sharp }) = \imath ^*(g\vert _S(n,\nabla \times B\vert _S)) = g\vert _S(n,\nabla \times B\vert _S). \end{aligned}$$

\(\square \)

Proposition 11

Let N be a vector field on M such that \(N\vert _S = n\). Let B be a vector field on M. Set \(b = B^{\flat }\) and \(\omega = \imath ^*b\). Let \(\delta _S\) be the codifferential on S. Then

$$\begin{aligned} \delta _S \omega = -\nabla \cdot B +g(B,N)\nabla \cdot N + g([N,B],N)~\vert _S. \end{aligned}$$

Proof

We may write

$$\begin{aligned} \delta _S \omega = -\star _S {\mathcal {L}}_{\omega ^{\sharp }}\mu _S. \end{aligned}$$

We also have the \(\imath \)-relatedness

$$\begin{aligned} T\imath \circ \omega ^{\sharp } = (B-g(B,N)N) \circ \imath \\ \mu _S = \imath ^*(i_N\mu ). \end{aligned}$$

Hence,

$$\begin{aligned} {\mathcal {L}}_{\omega ^{\sharp }}\mu _S = \imath ^* {\mathcal {L}}_{B-g(B,N)N}i_N\mu . \end{aligned}$$

With this,

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_{B-g(B,N)N}i_N\mu&= {\mathcal {L}}_{B}i_N\mu - {\mathcal {L}}_{g(B,N)N}i_N\mu \\&= ([{\mathcal {L}}_{B},i_N]+i_N{\mathcal {L}}_{B})\mu -(i_{g(B,N)N}di_X\mu + di_{g(B,N)N}i_N\mu )\\&= (i_{[B,N]}+i_N{\mathcal {L}}_{B})\mu -g(B,N)i_{N}di_N\mu \\&= i_{[B,N]}\mu +i_N{\mathcal {L}}_{B}\mu -g(B,N)i_{N}(i_Nd\mu + di_N\mu )\\&= i_{[B,N]}\mu +i_N{\mathcal {L}}_{B}\mu -g(B,N)i_{N}{\mathcal {L}}_X\mu \\&= i_{[B,N]}\mu +i_N(\nabla \cdot B)\mu -g(B,N)i_{N}(\nabla \cdot N)\mu \\&= i_{[B,N]}\mu +(\nabla \cdot B -g(B,N)\nabla \cdot N)i_{N}\mu . \end{aligned} \end{aligned}$$

Now, for any vector field Y on M, since \(Y-g(X, Y) X\) is \(\iota \)-related to a vector field \({\tilde{Y}}\) on S and \(\imath ^*\mu \in \Omega ^n(S) = \{0\}\), we get,

$$\begin{aligned} \begin{aligned} \imath ^*(i_Y\mu )&= \imath ^*(i_{Y-g(Y,N)N +g(Y,N)N}\mu )\\&= \imath ^*(i_{Y-g(Y, N)N}\mu ) + g(Y,N)\vert _S\imath ^*(i_N\mu )\\&= i_{{\tilde{Y}}}\imath ^*\mu + g(Y, N)\vert _S\mu _S\\&= g(Y, N)\vert _S\mu _S. \end{aligned} \end{aligned}$$

In particular, \(\imath ^*i_{[B,X]}\mu = g([B,X],X)\vert _S\mu _S\). Thus,

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_{\omega ^{\sharp }}\mu _S&= \imath ^* {\mathcal {L}}_{B-g(B,N)N}i_N\mu \\&= \imath ^*(i_{[B,N]}\mu +(\nabla \cdot B -g(B,N)\nabla \cdot N)i_{N}\mu )\\&= g([B,N],N)\vert _S\mu _S+(\nabla \cdot B -g(B,N)\nabla \cdot N)\vert _S \mu _S\\&= (g([B,N],N)+\nabla \cdot B -g(B,N)\nabla \cdot N)\vert _S \mu _S. \end{aligned} \end{aligned}$$

Hence, we arrive at

$$\begin{aligned} \begin{aligned} \delta _S \omega&= -\star _S {\mathcal {L}}_{\omega ^{\sharp }}\mu _S\\&= -g([B,N],N)-\nabla \cdot B +g(B,N)\nabla \cdot N~\vert _S\\&= -\nabla \cdot B +g(B,N)\nabla \cdot N + g([N,B],N)~\vert _S. \end{aligned} \end{aligned}$$

\(\square \)

If B is tangent to S, the final term \(g([N,B],N)\vert _S\) 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 \(N\vert _S = n\) 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 \(b = B^{\flat }\) and \(\omega = \imath ^*b\). Let \(\imath ^*B\) denote the vector field on S \(\imath \)-related to B. Then we get,

$$\begin{aligned} \delta _S \omega = -\nabla \cdot B\vert _S + \partial _n B. \end{aligned}$$

Moreover, if \(\nabla \cdot B\vert _S = 0\), then if \(P,u \in C^{\infty }(S)\) are with \(P = e^{u}\), then,

$$\begin{aligned} \delta _S P \omega = 0 \Leftrightarrow \mathrm{d}u(\imath ^*B) = \partial _n B. \end{aligned}$$

3.2 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. Let \(0 < P \in C^{\infty }(S)\). Consider the space of P-harmonic k-forms

$$\begin{aligned} {\mathcal {H}}^k_P(S) = \{\omega \in \Omega ^k(S): \mathrm{d}\omega = 0,~\delta P \omega = 0\}. \end{aligned}$$

It is natural to “symmetrise" these equations by the linear automorphism on \(\Omega ^1(S)\) given by multiplication \(\omega \mapsto P^{-1/2} \omega \), giving the isomorphism,

$$\begin{aligned} {\mathcal {H}}^1_P(S) \cong \{\omega \in \Omega ^1(S): \mathrm{d}_{p} \omega = 0,~\delta _{p} \omega = 0\}, \end{aligned}$$

where \(p = \ln \sqrt{P}\) and, following Witten [20] in his 1982 paper on supersymmetry and Morse theory, for \(h \in C^{\infty }(S)\), we set

$$\begin{aligned} \mathrm{d}_h = e^{h}\mathrm{d}e^{-h} : \Omega ^k(S) \rightarrow \Omega ^{k+1}(S),\\ \delta _h = e^{-h}e^{h} : \Omega ^k(S) \rightarrow \Omega ^{k-1}(S). \end{aligned}$$

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 \(\mathrm{d}_h\), \(\delta _h\) are adjoints of one another with respect to the \(L^2\) inner product on \(\Omega ^k(S)\), \(\mathrm{d}_h^2 = 0\), \(\delta _h^2 = 0\) and the relation \(\mathrm{d}_h e^{-h} = e^h \mathrm{d}\) gives an isomorphism of the \(k^{\text {th}}\) de Rham cohomology group and the cohomology group

$$\begin{aligned} H^k_{\text {dR}}(S) \cong H^k_{\text {W-dR}}(S,h) = \ker (\mathrm{d}_h: \Omega ^k(S) \rightarrow \Omega ^{k+1}(S))/\text {im}(\mathrm{d}_h: \Omega ^{k-1}(S) \rightarrow \Omega ^{k}(S)). \end{aligned}$$

The cohomology group \(H^k_{\text {W-dR}}(S,h)\) is now known in the literature as a Witten deformation of the cohomology group \(H^k_{\text {dR}}(S)\). In addition, Witten [20] notes that by standard arguments, the kernel of the associated Laplace operator

$$\begin{aligned} \Delta _h = \mathrm{d}_h\delta _h + \delta _h \mathrm{d}_h: \Omega ^k(S) \rightarrow \Omega ^k(S) \end{aligned}$$

has the same dimension as \(H^k_{\text {dR}}(S)\), that is, the \(k^{\text {th}}\) 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 \(\omega \mapsto [\omega ]\) of a closed k-form on S to its de Rham cohomology class gives an isomorphism \({\mathcal {H}}^k_P(S) \cong H^k_{\text {dR}}(S)\).

Proof

We first consider the Witten deformation to the cohomology with some \(h\in C^{\infty }(S)\). From the adjointness of \(\mathrm{d}_h\) and \(\delta _h\), we have at once that every \(\mathrm{d}_h\)-exact k-form \(\omega \) which is \(\delta _h\)-co-closed, is necessarily \(\omega = 0\). Hence, the restricted linear quotient map

$$\begin{aligned} \ker \mathrm{d}_h \cap \ker \delta _h \ni \omega \mapsto [\omega ]_{\text {W}} \in H^k_{\text {W-dR}}(S,h), \end{aligned}$$

is an injection. Again by adjointness, \(\ker \mathrm{d}_h \cap \ker \delta _h = \ker \Delta _h\). Hence, \(\dim \ker \mathrm{d}_h \cap \ker \delta _h = B^k = \dim H^k_{\text {W-dR}}(S,h)\). So that, by the rank-nullity theorem, this linear map is an isomorphism. Setting \(h = p\) and recalling the isomorphism \(\omega \mapsto P^{-1/2}\omega \), the result follows. \(\square \)

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 \(\omega \mapsto [\omega ]\) of a closed 1-form on S to its de Rham cohomology class gives an isomorphism \({\mathcal {H}}^1_P(S) \cong H^1_{\text {dR}}(S)\). In particular, \(\dim {\mathcal {H}}^1_P(S) = 2\).

We will now address the nowhere zero property of the elements of \({\mathcal {H}}^1_P(S)\). 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 \({\mathcal {H}}^1_P(S)\) 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 \({\mathcal {H}}^1_P(S)\).

3.2.1 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 \({\mathcal {A}} = \{(U_{\alpha },z_{\alpha })\}\); namely, the transition maps \(z_{\alpha } \circ z_{\beta }^{-1}\) of \({\mathcal {A}}\) are holomorphic between the open subsets \(z_{\beta }(U_{\alpha } \cap U_{\beta })\) and \(z_{\alpha }(U_{\alpha } \cap U_{\beta })\) of the complex plane (i.e. of \({\mathbb {R}}^2\)). Charts (U, z) on S with holomorphic transition maps \(z \circ z_{\beta }^{-1}\) for \((U_{\beta },z_\beta ) \in {\mathcal {A}}\) will be called holomorphic charts.

Following Bers [22], if \(F,G: S \rightarrow {\mathbb {C}}\) are functions such that \(\text {Im}({\overline{F}}G)>0\), (F, G) is called a generating pair if at every point \(p \in S\), 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

$$\begin{aligned} F^*&= \frac{2{\overline{G}}}{F{\overline{G}}-{\overline{F}}G},&G^*&= \frac{2{\overline{F}}}{F{\overline{G}}-{\overline{F}}G}. \end{aligned}$$

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 \(D \subset S\) (connected and open subset of S) to be an assignment \((U,z) \mapsto W/\mathrm{d}z\) where (U, z) is a holomorphic chart with \(U\subset D\) and \(W/\mathrm{d}z: U \rightarrow {\mathbb {C}}\) is a function such that, given two holomorphic charts (U, z), \((V,{\tilde{z}})\) with \(U,V \subset D\), for \(p \in U \cap V\), we have

$$\begin{aligned} \frac{W}{\mathrm{d}z}\bigg \vert _p = \frac{W}{\mathrm{d}{\tilde{z}}} \bigg \vert _p \frac{\mathrm{d}{\tilde{z}}}{\mathrm{d}z}\bigg \vert _p \end{aligned}$$

where \(\frac{\mathrm{d}{\tilde{z}}}{\mathrm{d}z}\vert _p = ({\tilde{z}} \circ z^{-1})'(z(p))\) denotes the complex derivative of the holomorphic transition map \({\tilde{z}} \circ z^{-1}\). Setting \(f = \frac{W}{\mathrm{d}z}\) and \({\tilde{f}} = \frac{W}{\mathrm{d}{\tilde{z}}}\) this relation is also denoted \(f\mathrm{d}z = {\tilde{f}}\mathrm{d}{\tilde{z}}\). One may also multiply W by a function \(f: S \rightarrow {\mathbb {C}}\) obtaining a differential fW defined in the obvious way.

The differential W is said to be continuous (or have partial derivatives etc.) if \(W/\mathrm{d}z\) is continuous for every holomorphic chart (U, z) with \(U \subset D\). Similarly, the integral \(\int _{\Gamma } W\) along a continuously differentiable curve \(C: [0,1] \rightarrow D \subset S\) may be defined. That is, \(\int _{\Gamma } W = \int _{0}^1 k(t)dt\) where \(k: [0,1] \rightarrow {\mathbb {C}}\) is a function such that, for \(t \in [0,1]\), if (U, z) is a holomorphic chart with \(U \subset D\), where \(C(t) \in U\), then \((\varphi \circ C)'(t) \in {\mathbb {C}}\) is defined and we set

$$\begin{aligned} k(t) = \frac{W}{\mathrm{d}z}\bigg \vert _{C(t)} (\varphi \circ C)'(t). \end{aligned}$$

With this, given a continuous differential W on S and a closed continuously differentiable curve \(C: [0,1] \rightarrow S\), the number

$$\begin{aligned} \text {Re}\int _C F^*W - i \text {Re}\int _C G^*W \end{aligned}$$

is called the (F, G)-period of W over 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,  Page 163–164].

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.

3.2.2 P-harmonic 1-forms

The following [25,  Chapter 10] is a well-known means of giving any smooth surface a Riemann surface structure.

Proposition 16

Let S be an oriented Riemannian 2-manifold. Then, there exists a maximal holomorphic atlas \({\mathcal {A}}\) compatible with the smooth structure on S. The component functions x, y of holomorphic charts satisfy \(\star \mathrm{d}x = \mathrm{d}y\) where \(\star \) is the Hodge star of S in U.

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 \(T_{{\mathbb {C}}}^*S\) is given by

$$\begin{aligned} T_{{\mathbb {C}}}^*S = \sqcup _{p \in S} T_p^*S \otimes _{{\mathbb {R}}} {\mathbb {C}}, \end{aligned}$$

where \({\mathbb {C}}\) is regarded with its vector space structure over \({\mathbb {R}}\). Regularity and operations on sections of \(T_{{\mathbb {C}}}^*S\) are defined component-wise. For example, for \(f \in C^{\infty }(S;{\mathbb {C}})\), the set of smooth functions \(S \rightarrow {\mathbb {C}}\), writing \(f = u+iv\) for some unique \(u,v \in C^{\infty }(S)\), we set

$$\begin{aligned} \mathrm{d}f = \mathrm{d}u+i\mathrm{d}v, \end{aligned}$$

where \(\mathrm{d}u+i\mathrm{d}v\) is a smooth section of the complexified cotangent bundle \(T_{{\mathbb {C}}}^*S\). If S is a Riemann surface, a section \(\omega \) of \(T_{{\mathbb {C}}}^*S\) is called of type (1, 0) if in any holomorphic chart (U, z), \(\imath ^*\omega = f\mathrm{d}z\) for some function \(f: U \rightarrow {\mathbb {C}}\), where \(\imath : U \subset S\) is the inclusion. The assignment of a type (1, 0) section of \(T_{{\mathbb {C}}}^*S\) to the function \(f: U \rightarrow {\mathbb {C}}\) where \(\imath ^*\omega = f\mathrm{d}z\) for any holomorphic chart (U, z), is a bijective equivalence between type (1, 0) sections of \(T_{{\mathbb {C}}}^*S\) and Bers’ differentials on M. If \(\omega \) is a type (1, 0) section of \(T_{{\mathbb {C}}}^*S\) and W is its differential equivalent, then \(\omega \) is continuous if and only if W is. Moreover, for any continuously differentiable curve \(C: [0,1] \rightarrow S\), we have

$$\begin{aligned} \int _C W = \int _C \omega , \end{aligned}$$

where the latter integral is a \({\mathbb {C}}\)-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 \(0<P \in C^{\infty }(S)\). Corollary 14 proves the first two statements of the Theorem. For the third, let \(\omega \) be a P-harmonic 1-form. So,

$$\begin{aligned} \mathrm{d}\omega&= 0,&\delta P\omega&= 0. \end{aligned}$$

Then, as previously discussed, considering the 1-form \({\tilde{\omega }} = P^{-1/2}\omega \), with \(p = \ln \sqrt{P}\), we have that

$$\begin{aligned} \mathrm{d} p {\tilde{\omega }}&= 0,&\mathrm{d}\star p^{-1} {\tilde{\omega }}&= 0. \end{aligned}$$

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

$$\begin{aligned} F&= p,&G&= i/p, \end{aligned}$$

is a generating pair on S. Then, the pair \((F^*,G^*)\) are given by

$$\begin{aligned} F^*&= 1/p,&G^*&= ip. \end{aligned}$$

Consider the section \({\hat{\omega }}\) of \(T_{{\mathbb {C}}}^*S\) given by

$$\begin{aligned} {\hat{\omega }} = {\tilde{\omega }} + i \star {\tilde{\omega }}. \end{aligned}$$

Then, from \(\star {\hat{\omega }} = i{\hat{\omega }}\), it easily follows from Proposition 16 that \({\hat{\omega }}\) is a \(T_{{\mathbb {C}}}^*S\) section of type (1, 0). Hence, we may consider its Bers’ differential equivalent W. Let \(C: [0,1] \rightarrow S\) be a continuously differentiable closed curve. Then, we get

$$\begin{aligned} \text {Re}\int _C {F^*~}W - i \text {Re}\int _C {G^*~} W&= \text {Re}\int _C {F^*~}{\hat{\omega }} - i \text {Re}\int _C {G^*~} {\hat{\omega }}\\&= \int _C p^{-1}{\tilde{\omega }}+i\int _C p \star {\tilde{\omega }}. \end{aligned}$$

If C is homologous to zero, then by Stokes’ Theorem for chains, since \(\mathrm{d}p^{-1}{\tilde{\omega }} = 0\) and \(\mathrm{d} p \star {\tilde{\omega }} = 0\), we obtain that

$$\begin{aligned} \text {Re}\int _C {F^*~}W - i \text {Re}\int _C {G^*~} W = 0. \end{aligned}$$

Thus, W is a regular (F, G) differential. Hence, from Theorem 15, we obtain that W is either identically zero or nowhere zero. Thus, \(\omega \) is either identically zero or nowhere zero. \(\square \)

We will now move on to P-harmonic vector fields.

3.2.3 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 \(0 < P \in C^{\infty }(S)\). Denote by \({\mathcal {H}}_P V(S)\) the real vector space of P-harmonic vector fields on S.

Proof of statements 1 and 2

Consider the map \({\mathcal {H}}_P V(S) \rightarrow {\mathcal {H}}_P^1(S)\) given by \(X \mapsto X^{\flat }\). Then, from Theorem 7, we immediately obtain that \(\dim {\mathcal {H}}_P V(S) = 2\) and that every \(X \in {\mathcal {H}}_P V(S)\) is either identically zero or nowhere zero. \(\square \)

For the remaining statements, let \(X \in {\mathcal {H}}_P V(S)\) be nowhere zero and let \(\gamma _1,\gamma _2\) be classes generating the first homology of S.

Proof of statement 3

Set \(Y = X^{\perp }/P\Vert X\Vert ^2\), where \(X^{\perp } = (\star X^{\flat })^{\sharp }\) is the perpendicular to X. The 1-forms \(\omega = X^{\flat }\) and \(\eta = {P \star \omega }\), are closed 1-forms on S and

$$\begin{aligned} \omega (X/\Vert X\Vert ^2)&= 1&\eta (X/\Vert X\Vert ^2)&= 0\\ \omega (Y)&= 0&\eta (Y)&= 1. \end{aligned}$$

Hence, \([X/\Vert X\Vert ^2,Y] = 0\). Hence, from a well-known result in Lie group theory (see Appendix B.2, Proposition 25), \(X/\Vert X\Vert ^2\) is rectifiable. From, for example, \(\eta (X) = 0\), 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. \(\square \)

Proof of statement 4

From the previous, we have a nowhere zero \(\eta = {P \star \omega }\) with \(\eta (X) = 0\). In particular (see Proposition 23 in Appendix B), \([\eta ] \ne 0\) and the winding number of X with respect to \(\gamma _1,\gamma _2\) is given by [(a, b)] where

$$\begin{aligned} a&= -\int _{\gamma _2} \eta = -\int _{\gamma _2} P \star X^{\flat },&b&= \int _{\gamma _1} \eta = \int _{\gamma _1} P \star X^{\flat }. \end{aligned}$$

Moreover, for a curve \(C: [0,1] \rightarrow S\) we have

$$\begin{aligned} C^*(P \star X^{\flat }) = P(C(t)) (\star X^{\flat })(C'(t))\mathrm{d}t = P(C(t)) (i_X\mu )(C'(t))\mathrm{d}t = \mu (X(C(t)),C'(t))\mathrm{d}t. \end{aligned}$$

So, if \(\gamma _1,\gamma _2\) are represented by curves \(C_1\) and \(C_2\),

$$\begin{aligned} a= & {} -\int _{\gamma _2} P \star X^{\flat } = -\int _{C_2} P \star X^{\flat } = \int _0^1 \mu (X(\gamma (t)),C'(t))\mathrm{d}t,\\ b= & {} \int _{\gamma _1} P \star X^{\flat } = \int _{C_1} P \star X^{\flat } = \int _0^1 \mu (X(C(t)),C'(t))\mathrm{d}t. \end{aligned}$$

\(\square \)

Proof of statement 5

We have the linear isomorphisms

$$\begin{aligned} {\mathcal {H}}_PV(S)&\rightarrow {\mathcal {H}}_P^1(S),&X&\mapsto X^{\flat },\\ {\mathcal {H}}_P^1(S)&\rightarrow {\mathcal {H}}_{1/P}^1(S),&\omega&\mapsto {P\star \omega },\\ {\mathcal {H}}_{1/P}^1(S)&\rightarrow H^1_{\text {dR}}(S),&\omega&\mapsto [\omega ]. \end{aligned}$$

The final linear isomorphism is due to Theorem 7. Hence, we have the linear isomorphism

$$\begin{aligned} {\mathcal {H}}_PV(S)&\rightarrow H^1_{\text {dR}}(S),&X&\mapsto [{P \star X^{\flat }}]. \end{aligned}$$

Hence, by de Rham’s Theorem, we have the linear isomorphism

$$\begin{aligned}&L: {\mathcal {H}}_PV(S) \rightarrow {\mathbb {R}}^2,\\&L(X) = \left( -\int _{\gamma _2}P\star X^{\flat },\int {\gamma _1} P \star X^{\flat } \right) . \end{aligned}$$

Set \({\mathbb {H}}_PV(S) = ({\mathcal {H}}_PV(S) \backslash \{0\}) / \sim \) to be the projective quotient space where \(X,Y \in {\mathcal {H}}_PV(S)\) are equivalent \(X \sim Y\) if and only if X and Y are linearly independent. Then, by injectivity, L descends to an injection, \({\mathbb {L}}: {\mathbb {H}}_PV(S) \rightarrow {\mathbb {P}}({\mathbb {R}})\). This map is also surjective since L is. Lastly, for \(X \in {\mathcal {H}}_PV(S)\), from statement 4, we see that L([X]) gives the winding number of X with respect to \(\gamma _1\) and \(\gamma _2\). The result now follows from the definition of \({\mathbb {H}}_PV(S)\).

\(\square \)

We will conclude this section with a proof of Theorem 5.

Proof of Theorem 5

Let M be an oriented Riemannian 3-manifold with boundary. Let S be an embedded 2-torus in M. Let \(n: S \rightarrow TM\) be a unit normal for S. Let B be a vector field on M which satisfies

$$\begin{aligned} \nabla \cdot B\vert _S&= 0,&\nabla \times B\vert _S \cdot n&= 0,&B\vert _S \cdot n&= 0. \end{aligned}$$

Consider the \(\iota \)-related vector field \(\imath ^*B\) on S where \(i: S \subset M\). Assume there exists a solution \(u \in C^{\infty }(S)\) to the cohomological equation

$$\begin{aligned} \mathrm{d}u(\imath ^*B) = \partial _n B. \end{aligned}$$

Now, let \(b = B^{\flat }\) and \(\omega = \imath ^*b\). Then, from Proposition 10, denonting by \(\mu _S\) the area element on S,

$$\begin{aligned} \mathrm{d}\omega = g\vert _S(\nabla \times B\vert _S,n)\mu _S = 0. \end{aligned}$$

From Corollary 12, taking \(P = e^u\), the codifferential \(\delta _S\) on S gives

$$\begin{aligned} \delta _S P\omega = 0. \end{aligned}$$

Hence, \(\omega \) is a P-harmonic 1-form on S. Considering the sharp \(\sharp _S\) on S, \(\omega ^{\sharp _S} = \imath ^*B\). Hence, \(\imath ^*B\) is a P-harmonic vector field on S. The conclusion now follows from Theorem 8. \(\square \)

4 Applications and examples

An application mentioned of Theorem 5 is Proposition 6. We will now prove this.

Proof of Proposition 6

Let M be an oriented Riemannian 3-manifold with boundary. Let S be a codimension 1 embedded submanifold in M. Let B be a vector field on M which satisfies \(\nabla \cdot B\vert _S = 0\). Assume that B has a first integral \(\rho \in C^{\infty }(M)\) which is constant and regular on S.

Consider now the neighbourhood \(V = \{\nabla \rho \ne 0\}\) of S in M. Set \({\tilde{u}} = \Vert \nabla \rho \Vert \) and \(N = \nabla \rho \vert _V /{\tilde{u}} \vert _V: V \rightarrow TM\). We have that \(n = N\vert _S\) so that \(B\vert _S \cdot n = 0\) and,

$$\begin{aligned} \partial _n B = g\vert _S([N,B],N) = \frac{[N,B](\rho )}{{\tilde{u}}}~\bigg \vert _S. \end{aligned}$$

Next, \(B(\rho ) = 0\) and \(N(\rho ) = g(N,\nabla \rho ) = {\tilde{u}}\) so that,

$$\begin{aligned} \frac{[N,B](\rho )}{{\tilde{u}}} = \frac{N(B(\rho ))-B(N(\rho ))}{{\tilde{u}}} = -B({\tilde{u}})/{\tilde{u}} = B(-\ln {\tilde{u}}). \end{aligned}$$

Hence, since B is tangential, setting \(u = -\ln \Vert \nabla \rho \Vert \vert _S = -\ln {\tilde{u}}\vert _S\), we have

$$\begin{aligned} \frac{[N,B](\rho )}{{\tilde{u}}}\bigg \vert _S = \mathrm{d}u(\imath ^*B). \end{aligned}$$

So that \(\mathrm{d}u(\imath ^*B) = \partial _n B\). \(\square \)

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 \(\rho \),

$$\begin{aligned} \nabla \cdot B&= 0,&B \cdot \nabla \rho&= 0,&(\nabla \times B) \cdot \nabla \rho&= 0. \end{aligned}$$

First, observe the following. Let \(S'\) be a closed connected component of a regular level set of \(\rho \). For later, give \(S'\) the orientation where the unit normal \(n = \frac{\nabla \rho \vert _{S'}}{\Vert \nabla \rho \Vert \vert _{S'}}\) is outward. Setting \(u = -\ln \Vert \nabla \rho \Vert \vert _{S'}\), and \(\imath : S' \subset M\) to be the inclusion, our assumptions together with Proposition 6 give,

$$\begin{aligned} \nabla \cdot B\vert _{S'}&= 0,&\nabla \times B\vert _{S'} \cdot n&= 0,&B\vert _{S'} \cdot n&= 0,&\mathrm{d}u(\imath ^*B)&= \partial _n B. \end{aligned}$$

Hence, by Proposition 10 and Corollary 12, \(\imath ^*B\) is a \(P = \Vert \nabla \rho \Vert ^{-1}\vert _{S'}\)-harmonic vector field on \(S'\).

In particular, fix a closed connected component S of a regular level set of \(\rho \). 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 \(\imath ^*B\) 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 \(V = \{\nabla \rho \ne 0\}\) of S and the local vector field, \(N = \frac{\nabla \rho \vert _V}{\Vert \nabla \rho \Vert \vert _V}\). Recall that S is compact. In particular, if \(S \subset \partial M\), since S is then embedded in \(\partial M\), S is a connected component of \(\partial M\). Thus, if \(S \subset \partial M = \emptyset \) (or \(S \subset \partial M\)), the Flowout Theorem [23,  Theorem 9.20] (the Boundary Flowout Theorem [23,  Theorem 9.24]) with V and N gives an \(\epsilon > 0\) and a diffeomorphism \(\Sigma : S \times I \rightarrow U\) where \(I = (-\epsilon ,\epsilon )\) (\(I = [0,\epsilon )\)) and U is open subset in M such that the projection \(z: S \times I \rightarrow I\) satisfies \(z = \rho \circ \Sigma + c\) for some constant \(c \in {\mathbb {R}}\).

With this, let \(z \in I\). Then, \(S' = \Sigma (S \times \{z\})\) is an embedded 2-torus in M for which \(X_z = \imath ^*B\) is a \(P = \Vert \nabla \rho \Vert \vert _{S'}\)-harmonic vector field on \(S'\) where \(\imath : S' \subset M\) is the inclusion. Hence, following the proof of statement 3 in Theorem 8, \([X/\Vert X\Vert ^2,Y] = 0\) where \(Y_z = X_z^{\perp }/P\Vert X_z\Vert ^2\), \(X_z^{\perp } = (\star X_z^{\flat })^{\sharp }\) being the perpendicular to \(X_z\) and \(P = \Vert \nabla \rho \Vert \vert _{S'}\). Thus, with the local vector fields

$$\begin{aligned} X&= \frac{B\vert _U}{\Vert B\Vert ^2\vert _U},&Y&= \frac{\nabla \rho \times B\vert _U}{\Vert \nabla \rho \Vert ^2 \Vert B\Vert ^2\vert _U} = \frac{\nabla \rho \times B\vert _U}{\Vert \nabla \rho \times B\Vert ^2\vert _U}, \end{aligned}$$

we have that \([X,Y] = 0\) on U. Hence, using the diffeomorphism \(\Sigma \) with Proposition 26 in Appendix B.2, there exists a diffeomorphism \(\Phi : U \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2 \times I\) where \(I \subset {\mathbb {R}}\) is an interval, such that \(B\vert _U\) is nowhere zero and,

$$\begin{aligned} \Phi _* \left( \frac{B\vert _U}{\Vert B\Vert ^2 \vert _U}\right) = a(z) \frac{\partial }{\partial x} + b(z) \frac{\partial }{\partial y} \end{aligned}$$

where \(a,b: I \rightarrow {\mathbb {R}}\) are smooth functions and z is the projection onto the factor I. \(\square \)

We will now clarify the application of our results to strong Beltrami fields.

Proof of Corollary 3

Let M be a manifold with boundary embedded in \({\mathbb {R}}^3\) with the inherited Euclidean structure. Let S be a toroidal connected component of \(\partial M\). Let \(n: \partial M \rightarrow {\mathbb {R}}^3\) be the outward unit normal on M. Let B be a vector field on M satisfying, for some \(\lambda \in {\mathbb {R}}\),

$$\begin{aligned} \nabla \cdot B&= 0,&\nabla \times B&= \lambda B,&B\vert _{\partial M} \cdot n&= 0. \end{aligned}$$

Let \(\imath : S \subset M\) denote the inclusion and consider \(\imath ^*B\). Corollary 12, Theorem 9 and [16,  Lemma 2.1] easily show that the three statements in Corollary 3 are indeed equivalent.

Lastly, consider a solution u to \(\mathrm{d}u(\imath ^*B) = \partial _n B\) on S. We have that \(\imath ^*B\) is \(P = e^{u}\)-harmonic by Proposition 10 and Corollary 12. Take closed curves \(C_1,C_2: [0,1] \rightarrow S \subset {\mathbb {R}}^3\) whose homology classes generate the first homology \(H_1(S)\) and form the integrals

$$\begin{aligned} I(C_i,u) = \int _{0}^1 \exp {(u(C_i(t)))} \det (C_i'(t),B(C_i(t)),n(C_i(t)))\mathrm{d}t,~i \in \{1,2 \}. \end{aligned}$$

Then, by Theorem 8, \([(-I(C_2,u),I(C_1,u))]\) in \({\mathbb {P}}({\mathbb {R}})\) is the winding number and of course \((I(C_1,u),I(C_2,u))\) is Diophantine if and only if \((-I(C_2,u),I(C_1,u))\) is Diophantine (see also the proof of Proposition 23 in Appendix B.1). This gives the final part of Corollary 3. \(\square \)

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 3-manifold M with boundary. Suppose that B and \(\nabla \times B\) have a mutual first integral \(\rho \) on M with a connected component S of a regular level set diffeomorphic to \({\mathbb {S}}^2\).

Just as observed in [17], there is a neighbourhood U of S foliated by spheres for which \(\rho \) is constant and regular thereon. Then, on such a sphere \(S' \subset U\), by Proposition 6 and Corollary 12, setting \(b = B^{\flat }\) and \(\omega ' = \imath '^*b\), we obtain that \(\omega \) is P-harmonic on \(S'\) with \(P = \Vert \nabla \rho \Vert ^{-1}\vert _S\). However, recall that from Proposition 13, that \({\mathcal {H}}^1_P(S')\) and \(H^1_{\text {dR}}(S')\) are isomorphic. In particular, we have \(H^1_{\text {dR}}(S') = \{0\}\) so that \(\omega ' = 0\). Hence, \(B\vert _S' = 0\). Thus, \(B\vert _U = 0\). \(\square \)

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 \(\lambda \in C^{\infty }(M)\) be a smooth function on an oriented connected Riemannian 3-manifold with boundary which is constant and regular on \(\partial M\), where \(\partial M\) is diffeomorphic to a 2-torus. Then, the space of vector fields B satisfying

$$\begin{aligned} \nabla \cdot B&= 0,&\nabla \times B&= \lambda B, \end{aligned}$$

is at most two dimensional.

Proof

Note that our assumptions on \(\lambda \) imply the vector fields under consideration are tangent to the boundary. Let B be a vector field satisfying \(\nabla \cdot B = 0\) and \(\nabla \times B = \lambda B\). Assume that B is not identically zero. Then, Vainshtein’s Lemma for abstract manifolds given by Gerner [16,  Lemma 2.1] immediately implies that B does not entirely vanish on \(\partial M\). Thus, by Proposition 6 and Theorem 5 we get that B is nowhere zero on \(\partial M\). In particular, denoting by \(\imath : \partial M \subset M\) the inclusion and fixing a point \(p \in \partial M\), the map \(B \mapsto \imath ^*B\vert _p\) is a linear injection from the space of divergence-free Beltrami fields with proportionality factor \(\lambda \) into \(T_p \partial M\). \(\square \)

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 \({\mathbb {R}}^3\) the vector field

$$\begin{aligned} {\hat{B}} = f\frac{\partial }{\partial x} + f\frac{\partial }{\partial y} + h \frac{\partial }{\partial z}, \end{aligned}$$

where

$$\begin{aligned} f&= (z+1)\cos {(x+y)},&h&= (z^2+2z)\sin {(x+y)}. \end{aligned}$$

Eventually, we will lower this vector field into the manifold \(M = ({\mathbb {R}}/2\pi {\mathbb {Z}})^2 \times {\mathbb {R}}\) along with some of its properties. To this end, writing \({\hat{S}}\) for the plane \(\{z=0\}\), one finds

$$\begin{aligned} \nabla \cdot {\hat{B}}&= 0,&h \vert _{{\hat{S}}}&= 0,&\nabla \times {\hat{B}} \cdot \frac{\partial }{\partial z}&= 0. \end{aligned}$$

and denoting by \({\hat{\imath }}: {\hat{S}} \subset {\mathbb {R}}^3\) the inclusion, for any \(u \in C^{\infty }({\hat{S}})\), we get

$$\begin{aligned} \mathrm{d}u(\imath ^*{\hat{B}}) = u_x f\vert _{z = 0} + u_y f\vert _{z = 0} = (u_x+u_y)\cos {(x+y)}. \end{aligned}$$

on the other hand, since \(\frac{\partial }{\partial z}\) is a geodesic vector field of unit length,

$$\begin{aligned} \partial _{{\hat{n}}}{\hat{B}} = \partial _z \cdot [{\hat{B}},\partial _z] = \partial _z({\hat{B}} \cdot \partial _z) = h_z\vert _{z=0} = 2\sin {(x+y)}. \end{aligned}$$

Hence, \(\mathrm{d}u(\imath ^*{\hat{B}}) = \partial _{{\hat{n}}}{\hat{B}}\) has no solutions. Since f and h are \(2\pi \)-periodic in x and y, \({\hat{B}}\) descends to a vector field B on M. Given the inherited oriented Riemannian structure from \({\mathbb {R}}^3\) on M, we have

$$\begin{aligned} \nabla \cdot B&= 0,&\nabla \times B\vert _{S}&= 0,&\nabla \times B\vert _{S} \cdot n&= 0, \end{aligned}$$

where S is the 2-torus embedded in M lowered from \({\hat{S}}\) in \({\mathbb {R}}^3\) with outward unit normal n. The other data shows that there does not exist a solution \(u \in C^{\infty }(S)\) to \(\mathrm{d}u(\imath ^*{\hat{B}}) = \partial _{n}B\). 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 \(\omega \) on a 2-torus S does not explain the topology of the integral curves of \(\omega ^{\sharp }\).

Example 2

Consider the nowhere zero closed 1-forms,

$$\begin{aligned} {\hat{\omega }}&= \mathrm{d}y,&{\hat{\eta }}&= \mathrm{d}\sin {x} + \mathrm{d}y = \cos {x}\mathrm{d}x + \mathrm{d}y. \end{aligned}$$

Then, \({\hat{\omega }}\) and \({\hat{\eta }}\) descend to nowhere zero cohomologous 1-forms \(\omega \) and \(\eta \) on the 2-torus \(S = {\mathbb {R}}^2/2\pi {\mathbb {Z}}^2\). We see that \(\omega ^{\sharp }\) is periodic. However, \(\eta ^{\sharp }\) is not periodic. To see this, consider \(X = {\hat{\eta }}^{\sharp }\) and let C be an integral curve of X and \(T > 0\). Writing \(C = (C^1,C^2)\), we have

$$\begin{aligned} {\dot{C}} = ({\dot{C}}^1,{\dot{C}}^2) = (\cos C^1,1). \end{aligned}$$

In particular,

$$\begin{aligned} {\dot{C}}^1(t)^2 = \cos {(C^1(t))} {\dot{C}}^1(t) = \frac{\mathrm{d}}{\mathrm{d}t} \sin {(C^1(t))}. \end{aligned}$$

Hence,

$$\begin{aligned} \int _0^T \cos ^2{(C^1(t))} \mathrm{d}t = \int _0^T {\dot{C}}^1(t)^2 \mathrm{d}t = \sin {(C^1(T))}-\sin {(C^1(0))}. \end{aligned}$$

Now, assume \(p = C(0)\) is with \(\cos p^1 \ne 0\). Then, by continuity, we must have

$$\begin{aligned} \int _0^T \cos ^2{(C^1(t))} \mathrm{d}t > 0. \end{aligned}$$

So that

$$\begin{aligned} \sin {(C^1(T))} \ne \sin {(C^1(0))}. \end{aligned}$$

Hence

$$\begin{aligned} \not \exists k \in {\mathbb {Z}} \text { such that } C^1(T) = C^1(0) + 2\pi k. \end{aligned}$$

Thus, in S, \(C(T) + 2\pi {\mathbb {Z}}^2 \ne C(0) + 2\pi {\mathbb {Z}}^2\). Hence, \(\eta ^{\sharp }\) is not periodic. On the other hand, the perpendicular vector fields \(W = (\star \omega )^{\sharp }\) and \(H = (\star \eta )^{\sharp }\) have the same winding number with respect to every pair of generators since \([\omega ] = [\eta ]\) and the flows of W and H are explicitly given by

$$\begin{aligned} \psi ^W(t,p + 2\pi {\mathbb {Z}}^2)&= p-t(1,0) + 2\pi {\mathbb {Z}}^2,\\ \psi ^H(t,p + 2\pi {\mathbb {Z}}^2)&= p-t(1,0) + (0,\sin {p^1}+\sin {(t-p^1)}) + 2\pi {\mathbb {Z}}^2. \end{aligned}$$

The curves \(t\mapsto \psi ^W(2\pi t,p+ 2\pi {\mathbb {Z}}^2)\) and \(t \mapsto \psi ^H(2\pi t,p+ 2\pi {\mathbb {Z}}^2)\), \(t \in [0,1]\), are closed and easily seen to be homologous to the curve \(C_1: [0,1] \rightarrow S\) where \(C_1(t) = [-t(1,0)]\). 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.

5 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.

Proposition 18

Let S be an oriented 2-torus. Let \(\omega \) be a closed 1-form. Then, the following are equivalent.

  1. 1.

    \(\delta \omega = 0\) for some metric g on S.

  2. 2.

    \(\omega \) is either identically zero or nowhere zero.

Proof

Suppose that \(\omega \) is intrinsically harmonic. Let g be a metric on S for which \(\omega \) is harmonic on (S, g). Then, as in Proposition 16, there exists a maximal holomorphic atlas \({\mathcal {A}}\) compatible with the smooth structure on S where the component functions x, y of holomorphic charts satisfy \(\star \mathrm{d}x = dy\) where \(\star \) is the Hodge star of S in U. So, S is now a Riemann surface of genus 1. Consider the complex 1-form \(W = \omega +i\star \omega \). This 1-form is holomorphic and thus, from the Riemann-Roch Theorem, W is either identically zero or nowhere zero. The other direction is precisely Theorem 27 in Appendix B.2 for the case of a 2-torus. \(\square \)

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 \(0 < P \in C^{\infty }(S)\). Let g be a metric on S and consider the P-harmonic 1-forms \({\mathcal {H}}_P^1(S,g)\) on (S, g). Let \(\omega ,\eta \in {\mathcal {H}}_P^1(S,g)\) form a basis for \({\mathcal {H}}_P^1(S,g)\). Then, there exists a metric \({\tilde{g}}\) with induced Hodge star \({\tilde{\star }}\) satisfying

$$\begin{aligned} {\tilde{\star }} \omega = \eta \end{aligned}$$

so that \({\mathcal {H}}_P^1(S,g) = {\mathcal {H}}^1(S,{\tilde{g}})\), the harmonic 1-forms on \((S,{\tilde{g}})\).

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 \(\delta _p = P^{-1}\delta P\) on (S, g) has a different kernel to the co-differential \({\tilde{\delta }}\) on \((S,{\tilde{g}})\) when P is non-constant 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 and \({\tilde{g}}\) be Riemannian metrics on S. Let \(0 < P,Q \in C^{\infty }(S)\) and consider the operators

$$\begin{aligned} \delta _p&= P^{-1}\delta P,&{\tilde{\delta }}_q&= Q^{-1}{\tilde{\delta }} Q. \end{aligned}$$

Suppose that

$$\begin{aligned} \ker \delta _p \subset \ker {\tilde{\delta }}_q. \end{aligned}$$

Then g and \({\tilde{g}}\) are conformally equivalent and \(P = c Q\) for some constant \(c > 0\).

Proof

Let \(R = P^{-1}Q\), \(r = \ln \sqrt{R}\) and consider \({\tilde{\delta }}_r = R^{-1}{\tilde{\delta }} R\). Then, we have \(\ker \delta \subset \ker {\tilde{\delta }}_r\). Considering the bundle metrics \(\langle , \rangle \) and \(\langle , \rangle _{\sim }\) on \(\Omega ^1(S)\), we have for any \(f \in C^{\infty }(S)\) and \(\omega \in \Omega ^1(S)\) that

$$\begin{aligned} \delta (f \omega )&= \delta f - \langle \mathrm{d}f,\omega \rangle ,\\ {\tilde{\delta }}_r (f \omega )&= {\tilde{\delta }}_r \omega -\langle \mathrm{d}f,\omega \rangle _{\sim }. \end{aligned}$$

In particular, for any \(f \in C^{\infty }(S)\) and g-harmonic 1-form \(\zeta \), because also \(\delta _w\zeta = 0\), we get

$$\begin{aligned} \delta (f \zeta )&= -\langle \mathrm{d}f,\zeta \rangle ,\\ {\tilde{\delta }}_r (f \zeta )&= -\langle \mathrm{d}f,\zeta \rangle _{\sim }. \end{aligned}$$

Take two g-harmonic 1-forms \(\chi ^1,\chi ^2\) which generate the first cohomololgy on S. Now, set \(\zeta ^i = \star \chi ^i\). Then, for each point \(p \in S\), \((\zeta ^1\vert _p,\zeta ^2\vert _p)\) forms a basis of \(\Lambda ^1(T_pS)\). For \((a,b) \in {\mathbb {R}}^2\), form \(\zeta _{(a,b)} = a\zeta ^1+b\zeta ^2\). We claim that, for all \((a,b) \in {\mathbb {R}}^2\) such that \((a,b) = c q\) for some \(c \in {\mathbb {R}}\) and \(q \in {\mathbb {Q}}^2\), for any point \(p \in S\), there exists \(f \in C^{\infty }(S)\) such that \(df\vert _p \ne 0\) and \(\langle df,\zeta _{(a,b)} \rangle = 0\).

Indeed, let \(0 \ne (a,b) \in {\mathbb {R}}^2\). Consider the dual vector field \(X = \zeta _{(a,b)}^{\sharp }\) in the metric g. Then, we have the nowhere zero form \(\omega = a\chi ^1+b\chi ^2\) with \(\omega (X) = 0\). Hence, X has winding number [(a, b)] and is nowhere zero. By Theorem 9, and Proposition 24 in Appendix B.1, this means there exists a diffeomorphism \(\Phi : S \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\) and a function \(0 < f \in C^{\infty }(S)\) such that

$$\begin{aligned} \Phi _*(X/f) = a\frac{\partial }{\partial x} + b\frac{\partial }{\partial y}. \end{aligned}$$

Now, assume that \((a,b) = c q\) for some \(c \in {\mathbb {R}}\) and \(q \in {\mathbb {Q}}^2\). Then, there exists \(0 \ne (m,n) \in {\mathbb {Z}}^2\) such that \((m,n) \cdot (a,b) = 0\). Consider \(h_1,h_2 \in C^{\infty }({\mathbb {R}}^2)\) given by

$$\begin{aligned} h_1&= \cos (2\pi (mx+ny)),&h_2&= \sin (2\pi (mx+ny)). \end{aligned}$$

We have that, \(h_1,h_2\) are invariant under \({\mathbb {Z}}^2\)-translations. Hence, \(h_1,h_2\) descend to \(h_1',h_2' \in C^{\infty }({\mathbb {R}}^2/{\mathbb {Z}}^2)\). Then, considering \(f_1 = h_1' \circ \Phi , g_2 = h_2' \circ \Phi \in C^{\infty }(S)\), we see that \(df_1(X) = 0 = df_2(X)\) and that for any point \(p \in S\), either \(df_1\vert _p \ne 0\) or \(df_2\vert _p \ne 0\). This proves the claim.

With this, let \(p \in S\). Then, let \((a,b) \in {\mathbb {R}}^2\) such that \((a,b) = c q\) for some \(c \in {\mathbb {R}}\) and \(q \in {\mathbb {Q}}^2\). Then, consider the form \(\eta = a\zeta ^1\vert _p + b\zeta ^2\vert _p \in \Lambda ^1(T_pS)\). Then, take \(f \in C^{\infty }(S)\) such that \(df\vert _p \ne 0\) and \(\langle \mathrm{d}f,\zeta _{(a,b)} \rangle = 0\). Then, we have

$$\begin{aligned} \delta (f \zeta ) = -\langle \mathrm{d}f,\zeta \rangle = 0. \end{aligned}$$

So that

$$\begin{aligned} -\langle \mathrm{d}f,\zeta \rangle _{\sim } = {\tilde{\delta }}_r (f \zeta ) = 0. \end{aligned}$$

Hence, because \(\mathrm{d}f\vert _p \ne 0\),

$$\begin{aligned} \langle {\tilde{\star }}\eta ,\eta \rangle = 0. \end{aligned}$$

Hence, since \({\mathbb {Q}}^2\) is dense in \({\mathbb {R}}^2\), with respect to the topology induced by the inner product \(\langle ,\rangle \) on \(\Lambda ^1(T_pS)\), there exists a dense subset S such that, for all \(\eta \in S\), \(\langle {\tilde{\star }}\eta ,\eta \rangle = 0\). Thus, since \({\tilde{\star }}\) is a linear operator, \(\langle {\tilde{\star }}\eta ,\eta \rangle = 0\) for all \(\eta \in \Lambda ^1(S)\). Now, considering \(\star \) on \(\Lambda ^1(T_pS)\), we see that, for all \(\eta \in \Lambda ^1(T_pS)\), there exists \(c \in {\mathbb {R}}\) such that \({\tilde{\star }} \eta = c \star \eta \). Since \({\tilde{\star }}\) and \(\star \) are linear operators and square to \(-\text {Id}\), it follows that for \(\eta \in \Lambda ^1(S)\), \({\tilde{\star }} \eta = \star \eta \). In particular, \(g\vert _p\) and \({\tilde{g}}\vert _p\) are conformally equivalent. Since \(p \in S\) was arbitrary, g and \({\tilde{g}}\) are conformally equivalent.

On top-forms, \({\tilde{\star }} = \kappa \star \) and as in the above, on 1-forms, \({\tilde{\star }} = \star \). Hence

$$\begin{aligned} {\tilde{\delta }}_r = R^{-1}{\tilde{\delta }} R = \kappa \delta _r. \end{aligned}$$

Hence, \(\ker \delta \subset \ker \delta _r\). Now, we have for \(\omega \in \Omega ^1(S)\) that

$$\begin{aligned} \delta _r \omega = \delta \omega - \langle \mathrm{d}w,\omega \rangle . \end{aligned}$$

Hence, for any g-harmonic 1-form \(\zeta \), we have

$$\begin{aligned} 0 = \delta _r \zeta = \delta \omega - \langle \mathrm{d}r,\omega \rangle = 0 - \langle \mathrm{d}r,\zeta \rangle = \langle \mathrm{d}r,\zeta \rangle . \end{aligned}$$

Thus, r is constant. Hence, R is constant. Thus, \(P = cQ\) for some constant \(c > 0\). \(\square \)

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.