Abstract
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 \([B,\nabla \times B] = 0\). Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution \(u: S \rightarrow {\mathbb {R}}\) to the cohomological equation \(B\vert _S(u) = \partial _n B\) on a toroidal surface S mutually invariant to B and \(\nabla \times B\). The right hand side \(\partial _n B\) 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 \(B\vert _S/\Vert B\Vert ^2 \vert _S\) being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality \(\Vert B\Vert ^2 \vert _S\)). 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 \(B\vert _S\) itself is rectifiable. The rectifiability and semi-rectifiability of \(B\vert _S\) is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.
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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,
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 \(\partial M\).
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
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
-
(i)
B is a Beltrami field, namely \(\nabla \times B = \lambda B\) for some function \(\lambda \),
-
(ii)
\(\lambda \) is a first integral and the closed regular level sets of \(\lambda \) are unions of tori, and
-
(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 \(\rho \),
Let S be a closed connected component of a regular level set of \(\rho \). Then, the following are equivalent
-
1.
B is not identically zero on S and S is a 2-torus
-
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
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 \(\lambda \in {\mathbb {R}}\),
Consider \(B\vert _{S}\), the vector field B along S. Then, the following are equivalent.
-
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.
The vector field \(B\vert _{S}\) preserves a top-form \(\mu \) on S.
-
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
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.
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.
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.
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
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
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
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 \(u = -\ln \Vert \nabla \rho \Vert \vert _S\),
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
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.
\(\dim {\mathcal {H}}_P^1(S) = 2\).
-
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.
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
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.
\(\dim {\mathcal {H}}_P V(S) = 2\).
-
2.
Let \(X \in {\mathcal {H}}_P V(S)\). Then either X is identically zero or X is nowhere zero.
-
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.
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.
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.
X preserves a top-form \(\mu \) on S.
-
2.
X is winding.
-
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
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,
\(\square \)
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
Proof
We may write
We also have the \(\imath \)-relatedness
Hence,
With this,
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,
In particular, \(\imath ^*i_{[B,X]}\mu = g([B,X],X)\vert _S\mu _S\). Thus,
Hence, we arrive at
\(\square \)
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,
Moreover, if \(\nabla \cdot B\vert _S = 0\), then if \(P,u \in C^{\infty }(S)\) are with \(P = e^{u}\), then,
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
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,
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
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
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
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
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
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
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
With this, given a continuous differential W on S and a closed continuously differentiable curve \(C: [0,1] \rightarrow S\), the number
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
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
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
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,
Then, as previously discussed, considering the 1-form \({\tilde{\omega }} = P^{-1/2}\omega \), with \(p = \ln \sqrt{P}\), 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 section \({\hat{\omega }}\) of \(T_{{\mathbb {C}}}^*S\) given by
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
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
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
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
Moreover, for a curve \(C: [0,1] \rightarrow S\) we have
So, if \(\gamma _1,\gamma _2\) are represented by curves \(C_1\) and \(C_2\),
\(\square \)
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
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
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
Now, let \(b = B^{\flat }\) and \(\omega = \imath ^*b\). Then, from Proposition 10, denonting by \(\mu _S\) the area element on S,
From Corollary 12, taking \(P = e^u\), the codifferential \(\delta _S\) on S gives
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,
Next, \(B(\rho ) = 0\) and \(N(\rho ) = g(N,\nabla \rho ) = {\tilde{u}}\) so that,
Hence, since B is tangential, setting \(u = -\ln \Vert \nabla \rho \Vert \vert _S = -\ln {\tilde{u}}\vert _S\), we have
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 \(\rho \),
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,
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
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,
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 \(\lambda \in {\mathbb {R}}\),
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
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
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
where
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
and denoting by \({\hat{\imath }}: {\hat{S}} \subset {\mathbb {R}}^3\) the inclusion, for any \(u \in C^{\infty }({\hat{S}})\), we get
on the other hand, since \(\frac{\partial }{\partial z}\) is a geodesic vector field of unit length,
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
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,
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
In particular,
Hence,
Now, assume \(p = C(0)\) is with \(\cos p^1 \ne 0\). Then, by continuity, we must have
So that
Hence
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
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.
\(\delta \omega = 0\) for some metric g on S.
-
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
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
Suppose that
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
In particular, for any \(f \in C^{\infty }(S)\) and g-harmonic 1-form \(\zeta \), because also \(\delta _w\zeta = 0\), we get
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
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
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
So that
Hence, because \(\mathrm{d}f\vert _p \ne 0\),
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
Hence, \(\ker \delta \subset \ker \delta _r\). Now, we have for \(\omega \in \Omega ^1(S)\) that
Hence, for any g-harmonic 1-form \(\zeta \), we have
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.
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This paper was written while the first author received an Australian Government Research Training Program Scholarship at The University of Western Australia.
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Appendices
Appendix A: The normal surface derivative and a Hodge-star formula
The following addresses the correctness of Definition 1.
Proposition 21
(Correctness of Definition 1) Let S be an oriented codimension 1 Riemannian embedded submanifold with boundary of a manifold with boundary M. Then the following holds.
-
1.
Let \({\mathcal {V}}: S \rightarrow TM\) be a nowhere zero normal-pointing vector field on S. Then for any point \(p \in S\), there exists a local vector field \(V: U \rightarrow TM\) extending \({\mathcal {V}}\vert _{U \cap S}\).
-
2.
Let B be a vector field which is tangent to S. Let \(p \in S\) and \(V_{i}: U_i \rightarrow TM\) for \(i \in \{1,2\}\) be local vector fields on neighbourhoods \(U_i\) of p such that \(V_i\vert _{U_i \cap S} = {\mathcal {V}}_{U_j \cap S}\). Then \(g\vert _p([V_1,B]\vert _p,V_1\vert _p) = g\vert _p([V_2,B]\vert _p,V_2\vert _p)\).
Proof
Set \(n = \dim M\). Use the boundaryless double as in [23, Example 9.32] to give a manifold \({\tilde{M}}\) such that M is a regular domain of \({\tilde{M}}\). That is, a properly embedded codimension 0 submanifold with boundary of \({\tilde{M}}\). Then, S is an codimension 1 embedded submanifold of \({\tilde{M}}\) and thus obeys a local \((n-1)\)-slice condition for submanifolds with boundary [23, Theorem 5.51]. That is, for each point \(p \in S\), there exists a chart \(({\tilde{U}},\varphi )\) in \({\tilde{M}}\) such that either \({\tilde{U}} \cap S = \{\varphi ^n = 0,~\varphi ^{n-1} \ge 0\}\) or \({\tilde{U}} \cap S = \{\varphi ^n = 0\}\). From this, we easily obtain the following. For any \(p \in S\), there exists a neighbourhood U in M and a regular function \(f \in C^{\infty }(U)\) which is constant on \(U \cap S\) and for any \(h \in C^{\infty }(S)\), there exists a \(H \in C^{\infty }(U)\) with \(H\vert _{U \cap S} = h\vert _{U \cap S}\).
For the first part, since \({\mathcal {V}}\) is nowhere zero, \(h = \Vert {\mathcal {V}}\Vert \in C^{\infty }(S)\). Moreover, for any \(p \in S\), there exists a neighbourhood U in M and a regular function \(f \in C^{\infty }(U)\) which is constant on \(U \cap S\) and a \(H \in C^{\infty }(U)\) with \(H\vert _{U \cap S} = h\vert _{U \cap S}\). Then, since both \({\mathcal {V}}\) and \(\nabla f\vert _S\) are normal-pointing, the local vector field \(V = \frac{H}{\Vert \nabla f\Vert } \nabla f\) extends \({\mathcal {V}}\vert _{U \cap S}\).
For the second part, take a neighbourhood \(U \subset U_1 \cap U_2\) of p in M a regular function \(f \in C^{\infty }(U)\) which is constant on \(U \cap S\). Consider the local vector field \(\nabla f/\Vert \nabla f\Vert \) on U. We get
Moreover,
Now, \(\mathrm{d}(B(f))(V_i)\vert _p = \mathrm{d}(B(f))\vert _p({\mathcal {V}}\vert _p)\) and denoting by \(\imath : S \subset M\) we have
From this, we see that \(g\vert _p([V_1,B]\vert _p,V_1\vert _p) = g\vert _p([V_2,B]\vert _p,V_2\vert _p)\). \(\square \)
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.
Proposition 22
Let \((V,g,{\mathcal {O}})\) be an oriented inner product space of dimension \(n \ge 2\). Let S be codimension 1 vector subspace with a unit normal n. Consider the induced oriented inner product space \((S,i^*g,{\mathcal {O}}_S)\) where \(i: S \subset V\) is the inclusion. Consider the Hodge star \(\star \) on V and \(\star _S\) on S. Then, for any k-form \(\omega \in \Lambda ^k(V)\),
Proof
Let \(\eta ,\omega \in \Lambda ^k(S)\). Consider the projection \(T: V \rightarrow S\) such that, \(T(v) = v-g(n,v)n\). Consider then \({\tilde{\eta }} = T^*\eta \in \Lambda ^1(V)\). Since \(T \circ i\) is the identity on S, \(\eta = i^*{\tilde{\eta }}\). Now,
Now, since \(T(n) = 0\), we have \(i_n{\tilde{\eta }} = 0\) so that
Hence,
Let \((e_2,...,e_n)\) be an orthonormal basis of S with the correct orientation. Then, \((e_1,e_2,...,e_n)\) is an orthonormal basis of V where \(e_1 = n\). Then,
Moreover, if \(j_i = 1\) for some \(i \in \{1,...,k\}\), we have \(T(e_{j_i}) = 0\) and hence, \({\tilde{\eta }}(e_{j_1},...,e_{j_k})\omega (e_{j_1},...,e_{j_k}) = 0\). Setting \((f_1,...,f_{n-1}) = (e_2,...,e_n)\) we have
Then, for \(i \in \{1,...,n-1\}\) since \(f_i \in S\), \(T(f_i) = f_i\). Hence, for \(1 \le j_1,...,j_k \le n-1\), we have
Hence,
One may easily check with the orthonormal bases introduced that \(\mu _S = i^*(i_n\mu )\) so that,
Thus,
Hence,
\(\square \)
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.
1.1 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.
Proposition 23
(Correctness of Definition 2) The following holds.
-
1.
If \(\omega _0\) is a nowhere zero closed 1-form, then \([\omega _0] \ne 0\).
-
2.
Let X be a vector densely-nowhere zero vector field. Let \(\gamma _1,\gamma _2 \in H_1(S)\) generate \(H_1(S)\). Assume that \(\omega ,\eta \) are closed 1-forms with non-trivial cohomology classes \([\omega ] \ne 0 \ne [\eta ]\) such that \(\omega (X) = 0 = \eta (X)\). Then, setting
$$\begin{aligned} \omega _i&= \int _{\gamma _i}\omega ,&\eta _i&= \int _{\gamma _i}\eta ,~i \in \{1,2\}, \end{aligned}$$we have that \((-\omega _2,\omega _1) \ne 0 \ne (-\eta _2,\eta _1)\) and, in \({\mathbb {P}}({\mathbb {R}})\), \([(-\omega _2,\omega _1)] = [(-\eta _2,\eta _1)]\).
-
3.
Let \({\tilde{\gamma }}_1,{\tilde{\gamma }}_2 \in H_1(S)\) generate \(H_1(S)\) and set
$$\begin{aligned} {\tilde{\eta }}_i = \int _{{\tilde{\gamma }}_i}\eta ,~i \in \{1,2\}. \end{aligned}$$Then \((-\omega _2,\omega _1)\) is a Diophantine if and only if \((-{\tilde{\eta }}_2,{\tilde{\eta }}_1)\) is a Diophantine.
Proof of statement 1
If \([\omega _0] = 0\), then \(\omega _0 = \mathrm{d}f\) for some \(f \in C^{\infty }(S)\). At an extremising point \(p \in S\) of f, \(\omega _0\vert _p = \mathrm{d}f\vert _p = 0\). Such a point exists by compactness of S. Hence, we must have \([\omega _0] \ne 0\). \(\square \)
Proof of statement 2
First let \(\alpha ,\beta \) be closed 1-forms such that
Denoting by \({\mathcal {Z}}^1(S)\) the set of closed 1-forms on S, from the Poincaré Duality Theorem, we have the isomorphism \(\text {PD}: H_{\text {dR}}^1(S) \rightarrow H_{\text {dR}}^1(S)^*\), given by
Then, since \(\text {PD}([\alpha ])(\lambda [\alpha ]) = 0\) for all \(\lambda \in {\mathbb {R}}\), we get by The Rank-Nullity Theorem and the fact that \(\dim H^1_{\text {dR}}(S) = 2\), that, \([\alpha ]\) and \([\beta ]\) must be linearly dependent. With this, letting \(\mu \in \Omega ^2(S)\) be a volume element and \(f \in C^{\infty }(S)\) such that \(\omega \wedge \eta = f\mu \), we have
Now, for all \(p \in S\) with \(X\vert _p \ne 0\), we have \(i_X\mu \vert _p \ne 0\) so that \(f\vert _p = 0\). Thus, f is densely-vanishing so that, since \(f \in C^{\infty }(S)\), \(f = 0\). Thus,
Hence, by the above, \([\omega ]\) and \([\eta ]\) are linearly dependent. Hence, the vectors \((-\omega _2,\omega _1), (-\eta _2,\eta _1)\) are linearly dependent in \({\mathbb {R}}^2\). Moreover, since \([\omega ] \ne 0 \ne [\eta ]\) we have by de Rham’s Theorem that \((-\omega _2,\omega _1) \ne 0 \ne (-\eta _2,\eta _1)\). Thus, in \({\mathbb {P}}({\mathbb {R}})\), \([(-\omega _2,\omega _1)] = [(-\eta _2,\eta _1)]\). \(\square \)
Proof of statement 3
Consider first the vectors
Now, there exist integer matrices \(A,{\tilde{A}} \in {\mathbb {Z}}^{2\times 2}\) such that, \({\tilde{\gamma }}_i = A_{ij}\gamma _j\) and \(\gamma _i = {\tilde{A}}_{ij}{\tilde{\gamma }}_j\). From linear independence, we see that \(A{\tilde{A}} = {\tilde{A}}A = I\), the identity \(2\times 2\) matrix. Hence, \(A \in GL(2,{\mathbb {Z}})\). Hence,
The \(2\times 2\) matrix R with \(R(x,y) = (-y,x)\) for \((x,y) \in {\mathbb {R}}^2\) is in \(GL(2,{\mathbb {Z}})\). Moreover,
and \(RAR \in GL(2,{\mathbb {Z}})\). Now, in general, let \(L \in GL(2,{\mathbb {Z}})\) and \(u \in {\mathbb {R}}^2\) be a Diophantine vector. So that, there exists \(\gamma > 0\) and \(\tau > 1\) such that, for all \(k \in {\mathbb {Z}}^2\backslash \{0\}\),
Consider \({\tilde{u}} = L u\). Then, fix a constant \(C > 0\) such that, \(\Vert L^Tu'\Vert \le C\Vert u'\Vert \) for all \(u' \in {\mathbb {R}}^2\) and set \({\tilde{\gamma }} = \gamma C^{-\tau }\). Then, for \(k \in {\mathbb {Z}}^2 \backslash \{0\}\) non-zero, we have \(L^Tk \in {\mathbb {Z}}^2 \backslash \{0\}\) so that,
Hence, \(L u = {\tilde{u}}\) is Diophantine. With this, we see that \((-\omega _2,\omega _1)\) is Diophantine if and only if \((-{\tilde{\omega }}_2,{\tilde{\omega }}_1)\) is Diophantine. Moreover, by statement 2, we have that \([(-{\tilde{\omega }}_2,{\tilde{\omega }}_1)] = [(-{\tilde{\eta }}_2,{\tilde{\eta }}_1)]\) so that clearly \((-{\tilde{\omega }}_2,{\tilde{\omega }}_1)\) is Diophantine if and only if \((-{\tilde{\eta }}_2,{\tilde{\eta }}_1)\). In total, \((-\omega _2,\omega _1)\) is Diophantine if and only if \((-{\tilde{\eta }}_2,{\tilde{\eta }}_1)\) is Diophantine. \(\square \)
We will now show that the winding number appearing in this paper is compatible with vector fields lying on straight lines.
Proposition 24
Let X be a densely nowhere zero vector field such that, for some \(f \in C^{\infty }({\mathbb {R}}^2/{\mathbb {Z}}^2)\), numbers \(a,b \in {\mathbb {R}}\) and diffeomorphism \(\Phi : S \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\), we have
Then, \((a,b) \ne (0,0)\) and X is winding. Considering the standard homology generators \(\gamma _1,\gamma _2\) of \(H_1({\mathbb {R}}^2/{\mathbb {Z}}^2)\), X has winding number [(a, b)] with respect to the pulled back generators \(\Phi ^*\gamma _1 = (\Phi ^{-1})_*\gamma _1\) and \(\Phi ^*\gamma _2 = (\Phi ^{-1})_*\gamma _2\).
Proof
There exists \(p \in S\) with \(X\vert _p \ne 0\). Hence, \((a,b)\ne 0\). Now, in \({\mathbb {R}}^2/{\mathbb {Z}}^2\), consider the vector fields \(\frac{\partial }{\partial x}\),\(\frac{\partial }{\partial y}\) and the closed 1-forms \(\mathrm{d}x\), \(\mathrm{d}y\). Then, the form \({\tilde{\omega }}_0 = b\mathrm{d}x - a\mathrm{d}y\) is closed, nowhere zero, and
Then, the pull-back \(\omega _0 = \Phi ^*{\tilde{\omega }}_0\) is a nowhere zero closed 1-form and satisfies \(\omega _0(X) = 0\). Since \(\omega _0\) is nowhere zero, X is winding and \([\omega _0] \ne 0\) from Proposition 23. Moreover,
Hence, the winding number of X with respect to \(\Phi ^*\gamma _1,\Phi ^*\gamma _2\) is [(a, b)]. \(\square \)
1.2 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 \(\Phi : S \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\) and numbers \(a,b \in {\mathbb {R}}\) such that
Proof
In the following, we will use the standard terminology in [23, Page 162 and Pages 550 to 552]. Let \(\psi ^X,\psi ^Y\) denote the complete flows of X and Y respectively. Define the map \(\Psi : {\mathbb {R}}^2 \times S \rightarrow S\) by
Since \([X,Y] = 0\), the flows commute \(\psi ^X_s \circ \psi ^Y_t = \psi ^Y_t \circ \psi ^X_s\) for \(s,t \in {\mathbb {R}}\). Thus, \(\Psi \) defines a smooth group action on S. Now, for \(p \in S\), set \(\Psi _p = \Psi (\cdot ,p): {\mathbb {R}}^2 \rightarrow S\). Then, at \(0 \in {\mathbb {R}}^2\), with the curves \(C_1,C_2: {\mathbb {R}} \rightarrow {\mathbb {R}}^2\) given by \(C_1(t) = (t,0)\) and \(C_2(t) = (0,t)\) and linear independence of \(X\vert _p\) and \(Y\vert _p\), we see that \(T(\Psi _p)\vert _0\) is invertible. Hence, since \(\Psi \) is a group action, we get for all \(p \in S\) and \(u \in {\mathbb {R}}^2\) that \(T(\Psi _p)\vert _u\) is invertible. Hence, by the Inverse Function Theorem, \(\Psi _p\) is a local diffeomorphism.
In particular, the orbits \({\mathbb {R}}^2 \cdot p = \Psi _p({\mathbb {R}}^2)\) for \(p \in S\) form an open disjoint covering of S. Thus, by connectedness of S, the action \(\Psi \) is transitive. That is, S is a homogeneous \({\mathbb {R}}^2\)-space with the action \(\Psi \). From now on, fix a point \(p \in S\). Considering the isotropy subgroup \({\mathbb {R}}^2_p = \{u \in {\mathbb {R}}^2: \Psi (u, p) = p\}\) of \({\mathbb {R}}^2\), we have that the map \(F: {\mathbb {R}}^2/{\mathbb {R}}^2_p \rightarrow S\) defined by \(F(u+{\mathbb {R}}^2_p) = \Psi (u,p)\) is an equivariant diffeomorphism [23, Theorem 21.18]. Then, since the vector fields \(\frac{\partial }{\partial x},\frac{\partial }{\partial y}\) in \({\mathbb {R}}^2\) are translation invariant, they descend to vector fields \({\tilde{X}}\) and \({\tilde{Y}}\) in the quotient \({\mathbb {R}}^2/{\mathbb {R}}^2_p\).
Now, since \(\Psi _p\) is a local diffeomorphism, we also get that \({\mathbb {R}}^2_p\) is a discrete subgroup of \({\mathbb {R}}^2\). Thus (see for instance [28, Lemma 5.14]), we must have either \({\mathbb {R}}^2_p = {\mathbb {Z}}u\) for some \(0 \ne u \in {\mathbb {R}}^2\) or \({\mathbb {R}}^2_p = {\mathbb {Z}}u \oplus {\mathbb {Z}} v\) for some linearly independent \(u,v \in {\mathbb {R}}^2\). Since the quotient \({\mathbb {R}}^2/{\mathbb {R}}^2_p\) is compact, we must have the latter. In particular, there exists an invertible matrix \(A \in \text {GL}(2,{\mathbb {R}})\) such that \(A{\mathbb {Z}}^2 = {\mathbb {R}}^2_p\) which induces a diffeomorphism \({\hat{A}}: {\mathbb {R}}^2/{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}^2/{\mathbb {R}}^2_p\). By definition of F, one sees that, \(F_* {\tilde{X}} = X\) and \({\hat{A}}_* {\tilde{Y}} = Y\). The result then follows with the diffeomorphism \(\Phi = ({\hat{A}} \circ F)^{-1}\). \(\square \)
Directly related to Arnold’s structure Theorems is the following proposition which is a variant of Proposition 25 for multiple tori in three dimensions.
Proposition 26
Consider the product manifold \(M = S \times I\) with boundary where S is a compact connected 2-manifold and I is an interval with, for some \(\epsilon > 0\), either \(I = [0,\epsilon )\) or \(I = (-\epsilon ,\epsilon )\). Let X, Y be commuting vector fields with \(\mathrm{d}z(X) = 0 = \mathrm{d}z(Y)\) where \(z: M \rightarrow I\) is projection onto the second factor. Then, there exist smooth functions \(a,b: I \rightarrow {\mathbb {R}}\) and a diffeomorphism \(\Phi : M \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2 \times I\) such that
Proof
Consider the map \(F: {\mathbb {R}}^2 \times M \rightarrow M\) given by
Fix a point \(p_0 \in S\) and consider the map \(G: {\mathbb {R}}^2 \times I \rightarrow M\) given by
As seen in the proof of Proposition 25, we have for all \(z \in I\) that there exists a rank 2 lattice \(\Lambda _z \subset {\mathbb {R}}^2\) such that the map \(G_z: {\mathbb {R}}^2/\Lambda _z \rightarrow M\) given by
is an embedding on to (S, z) in M satisfying
where \(\frac{\partial }{\partial x}\) and \(\frac{\partial }{\partial y}\) are the constant vector fields lowered to the quotient \({\mathbb {R}}^2/\Lambda _z\).
To turn the \(G_z\)s into a diffeomorphism with domain \({\mathbb {R}}^2/{\mathbb {Z}}^2 \times I\), we must first check that the lattice \(\Lambda _z\) smoothly varies with \(z \in I\). To this end, consider the vector field frame (X, Y, Z) where \(Z = \frac{\partial }{\partial z}\) is the vector field on M induced by the factor I. Consider now the induced co-frame \((\alpha ,\beta ,\gamma )\) of 1-forms to (X, Y, Z) so that
Note that \(\gamma = \mathrm{d}z\). Now, fix smooth curves \(C_1,C_2: [0,1] \rightarrow S\) which generate the first homology of S. For \(z \in I\) and \(i \in \{1,2\}\) set \(C_i^{z} = (C_i,z): [0,1] \rightarrow M\) and
We claim that \((v_{1,z},v_{2,z})\) forms a lattice basis for \(\Lambda _z\) for \(z \in J\). Indeed, let \(z \in I\). Then, we get curves \(c_i: [0,1] \rightarrow {\mathbb {R}}^2/\Lambda _z\) induced by the embedding \(G_z\) and curves \(C_i\) (\(i \in \{1,2\}\)). We also have that
where \(\mathrm{d}x,\mathrm{d}y\) are the constant 1-forms lowered to the quotient \({\mathbb {R}}^2/\Lambda _z\). Now, write \(\Lambda _z = {\mathbb {Z}}u_1 \oplus {\mathbb {Z}}u_2\) for some \(u_1,u_2\) linearly independent in \({\mathbb {R}}^2\). Then, consider the curves \(D_1,D_2: [0,1] \rightarrow {\mathbb {R}}^2/\Lambda _z\) given by
Then, \(D_1\) and \(D_2\) generate the first homology of \({\mathbb {R}}^2/\Lambda _z\). Hence, since both \(([c_1],[c_2])\) and \(([D_1],[D_2])\) are generators for the first homology, there exists a matrix \(A \in GL(2,{\mathbb {Z}})\) such that
With this, we get that
and similarly \(v_{z,i}^2 = A_{ij}u_j^2\). In total, we have \(v_{z,i} = A_{ij}u_j\). Hence, since \((u_1,u_2)\) is a generator for \(\Lambda _z\), so is \((v_{z,1},v_{z,2})\). Hence, our claim holds.
We will now use our \(v_{z,i}\) (\(i \in \{1,2\}\)) to make a diffeomorphism. To this end, for each \(z \in J\), form the matrix
Then, we have the smooth maps \({\hat{A}},{\hat{B}}: {\mathbb {R}}^2 \times I \rightarrow {\mathbb {R}}^2 \times I\) given by
whereby \({\hat{A}} \circ {\hat{B}} = \text {Id} = {\hat{B}} \circ {\hat{A}}\) so that in particular, \({\hat{A}}\) is a diffeomorphism. Moreover, we see that TG is everywhere invertible and that \(G(\partial ({\mathbb {R}}^2 \times I)) = \partial M\). Similarly to the proof of Proposition 25, the tangent map TG is invertible everywhere. Hence, in the case of \(I = (-\epsilon ,\epsilon )\), the Inverse Function Theorem gives that G is a local diffeomorphism. In the case of \(I = [0,\epsilon )\), one may, for instance, globally extend the vector fields X and Y to vector fields \({\tilde{X}}\) and \({\tilde{Y}}\) defined on a neighborhood of \(S \times (-\epsilon ,\epsilon )\) tangent to the compact \(S\times \{z\}\) for all \(z \in I\), construct the suitable \({\tilde{G}}\), and apply the Inverse Function Theorem to \({\tilde{G}}\) using the fact that \(S \times \{0\}\) is left invariant by X and Y. Hence, in any case, we have the local diffeomorphism
Now, we have the product map
where \(\pi : {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\) is the quotient map so that \(\Pi \) is an onto local diffeomorphism. With this, since \(A_z = (v_{z,1},v_{z,2})\) is a matrix of generators of \(\Lambda _z\) for each \(z \in I\), then there exists a unique map \(\Psi : {\mathbb {R}}^2/{\mathbb {Z}}^2 \times I \rightarrow M\) such that
Hence, \(\Psi \) is a local diffeomorphism. It is also clear that \(\Psi \) is bijective. Hence, \(\Psi \) is a diffeomorphism and the desired map is \(\Phi = \Psi ^{-1}\). \(\square \)
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 \(\omega \in \Omega ^1(M)\) be closed and nowhere zero. Then \(\omega \) is intrinsically harmonic; that is, there exists a metric g on M such that \(\delta \omega = 0\).
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, \(\Phi : S \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\) be a diffeomorphism and set X such that
where (a, b) is a Diophantine vector. Then, for any \(v \in C^{\infty }(S)\), there exists \(c \in {\mathbb {R}}\) and a solution \(u \in C^{\infty }(S)\) to the cohomological equation
That is, setting \(\mu = \Phi ^*\mu _0\), where \(\mu _0\) is the standard volume form on \({\mathbb {R}}^2/{\mathbb {Z}}^2\), for any \(v \in C^{\infty }(S)\), there exists a solution \(u \in C^{\infty }(S)\) to the cohomological equation
if and only if
We will now prove Theorem 9.
Proof of Theorem 9
Let X be a nowhere zero vector field on a 2-torus S.
Statement 1 implies statement 2
Suppose that X preserves a top-form \(\mu \in \Omega ^2(S)\). Setting \(\omega = i_X\mu \), we have \(\omega (X) = 0\) and
\(\square \)
Statement 2 implies statement 3
Suppose that X is winding. Let \(\omega \) be a closed nowhere zero 1-form such that \(\omega (X) = 0\). Since \(\omega \) is nowhere zero, by Theorem 18, there exists an Riemannian metric g on S for which \(\omega \) is harmonic on (S, g). Then, consider \(\eta = \star \omega \). Then, \(\eta \) is closed and the top form \(\omega \wedge \eta \) is nowhere zero. Thus, since X is nowhere zero, \(i_X(\omega \wedge \eta )\) is nowhere zero. On the other hand,
Hence, \(\eta (X)\) is nowhere zero. In particular, we have
From this, consider the unique vector field Y on S such that
Then, X and Y are point-wise independent and \([X/\eta (X),Y] = 0\). Hence, from Proposition 25, \(X/\eta (X)\) is rectifiable and so X is semi-rectifiable. \(\square \)
Statement 3 implies statement 1
Suppose that X is semi-rectifiable. Then, for some \(0 < f \in C^{\infty }(S)\), numbers \(a,b \in {\mathbb {R}}\) and diffeomorphism \(\Phi : S \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\) such that
Then, considering the standard top-form \(\mu _0 \in \Omega ^2({\mathbb {R}}^2/{\mathbb {Z}}^2)\), and setting \(\mu = \frac{1}{f}\Phi ^{*}\mu _0\), we see that \({\mathcal {L}}_X \mu = 0\). \(\square \)
Lastly, suppose that X has Diophantine winding number. Then, from the above, together with Propositions 23 and 24, we get for some \(0 < f \in C^{\infty }(S)\), Diophantine vector \((a,b) \in {\mathbb {R}}^2\), and diffeomorphism \(\Phi : S \rightarrow {\mathbb {R}}^2/{\mathbb {Z}}^2\), that
With this, consider the vector fields \({\tilde{X}}\) and Y such that
Since (a, b) is Diophantine, so is \((-b,a)\). With this, let \(u \in C^{\infty }(S)\) and consider the vector field
First, notice that X and \(Y_u\) are point-wise linearly independent. Moreover,
Now, set \(\mu = \Phi ^*\mu _0\) where \(\mu _0\) is the standard top form on \({\mathbb {R}}^2/{\mathbb {Z}}^2\). Setting \(v = Y(-1/f)\), using Proposition 28 on Y, we have that
Hence, using Proposition 28 on X, there exists a solution \(u \in C^{\infty }(S)\) to the cohomological equation
Thus, with this choice of u,
Thus, from Proposition 25, X is rectifiable. \(\square \)
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Perrella, D., Pfefferlé, D. & Stoyanov, L. Rectifiability of divergence-free fields along invariant 2-tori. Partial Differ. Equ. Appl. 3, 50 (2022). https://doi.org/10.1007/s42985-022-00182-3
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DOI: https://doi.org/10.1007/s42985-022-00182-3