# Existence of global symmetries of divergence-free fields with first
integrals

David Perrella<EMAIL_ADDRESS>The University of Western Australia,
35 Stirling Highway, Crawley WA 6009, Australia  Nathan Duignan  School of
Mathematics and Statistics, University of Sydney, NSW 2050, Australia  David
Pfefferlé  The University of Western Australia, 35 Stirling Highway, Crawley
WA 6009, Australia

(March 28, 2023)

###### Abstract

The relationship between symmetry fields and first integrals of divergence-
free vector fields is explored in three dimensions in light of its relevance
to plasma physics and magnetic confinement fusion. A Noether-type Theorem is
known: for each such symmetry, there corresponds a first integral. The extent
to which the converse is true is investigated. In doing so, a reformulation of
this Noether-type Theorem is found for which the converse holds on what is
called the toroidal region. Some consequences of the methods presented are
quick proofs of the existence of flux coordinates for magnetic fields in high
generality; without needing to assume a symmetry such as in the cases of
magneto-hydrostatics (MHS) or quasi-symmetry.

##  I Introduction

There is a growing interest in divergence-free vector fields $B$ that admit
either a symmetry or a first integral. A symmetry $u$ is a divergence-free
vector field and a first integral $\psi$ is a function which respectively
satisfy

$[u,B]=0,\qquad B\cdot\nabla\psi=0.$

Here $[u,B]$ is the Lie-bracket of vector fields $u$ and $B$. In the context
of plasma physics, one considers magneto-hydrostatics (MHS) under which $B$
satisfies

$J\times B=\nabla p,\qquad J=\nabla\times B$

where $p$ is the pressure and $J$ is the current density. In this case, $J$ is
a symmetry of $B$ and $p$ is a first integral of both $B$ and $J$. Another
class of magnetic fields, called _quasi-symmetric_ , arise when there exists a
divergence-free vector field $u$ which possesses $B$ as a symmetry and
$\|B\|^{2}$ as a first integral [1, 2]. In magnetic confinement fusion
devices, transport of heat and particles along magnetic field-lines is
generally several orders of magnitude faster than across. Promoting first
integrals is a way of guaranteeing temperature and density gradients because
then the field-lines lie on lower-dimensional manifolds. Without substantial
gradients, the core of a fusion device would not be hot enough to sustain
fusion reactions and the outer region cold enough for the device’s walls to
resist damage.

For any $B$ admitting both a symmetry and an integral, the field is integrable
in the broad sense [3] as defined in [4]. On the other hand, divergence-free
fields generally admit no symmetries nor integrals and are chaotic [5, 6].

Between the cases of chaos and integrability lies magnetic fields $B$ which
admit either an integral or a symmetry, but a priori not both. For example, if
one has a vacuum magnetic field, which is a divergence-free vector field $B$
satisfying $\nabla\times B=0$, and it is assumed to have nested flux surfaces,
it is not immediately clear whether or not $B$ also possesses a symmetry.

It has been established [7, 3] that a divergence-free vector field $B$ in 3D
admits a Noether theorem: loosely, this states that for each symmetry field of
$B$, there corresponds a first integral of $B$. However, the converse is
false: there exists $B$ which has a first integral but no corresponding
symmetry (see for instance Example II.12). Fortunately, a partial converse is
available. If $B$ is non-vanishing and has a first integral $\psi$, as
indicated in [8], an extension of an argument due to Kolmogorov [9] shows that
in a neighbourhood $U$ of a compact, regular level-surface of $\psi$, there
exists coordinates
$(\theta_{1},\theta_{2},\psi):U\to\mathbb{T}^{2}\times\mathbb{R}$ and a smooth
positive function $f>0$ such that

$B/f=a(\psi)\partial_{\theta_{1}}+b(\psi)\partial_{\theta_{2}}.$

Thus, the vector field
$u=b(\psi)\partial_{\theta_{1}}-a(\psi)\partial_{\theta_{2}}$ on $U$, for
example, is a symmetry for the rescaled field $B/f$. In this case, $B$ is said
to be semi-rectified in this neighborhood $U$ [10]. However, this converse is
local to a given regular level surface of $\psi$.

The main result of this paper is a _converse conformal Noether theorem for
divergence-free vector fields_ ; while $B$ may not admit a symmetry, a
rescaling of $B$ does. We call such a symmetry a _conformal symmetry_ of $B$.
The proof is independent of Kolmogorov’s original ergodic methods and is more
global than the above construction. The result takes place on a 3-manifold $M$
with (possibly empty) boundary and is centred around flux systems.

######  Definition I.1.

A _flux system_ is a triple $(B,\nu,\mu)$ where $B$ is a vector field on $M$,
$\nu$ is a closed 1-form on $M$, and $\mu$ is a volume form on $M$, such that

$\nu(B)=0,\qquad\mathcal{L}_{B}\mu=0,$

where $\mathcal{L}_{B}$ denotes the Lie derivative with respect to $B$. We
call $(B,\nu,\mu)$ a _tangential flux system_ if $B$ is tangent to $\partial
M$ and the kernel of $\nu$ contains all vectors tangent to $\partial M$.

For most applications $\nu=d\psi$ will be exact so that $\psi$ will be a first
integral of $B$. This occurs automatically if the first cohomology of $M$ is
trivial, $H^{1}(M)=0$. Our main result is split into two parts. The first part
highlights the role played by _adapted 1-forms_.

######  Definition I.2.

A 1-form $\eta\in\Omega^{1}(M)$ is said to be _adapted to a flux system
$(B,\nu,\mu)$_ if

$\eta(B)>0,\qquad d\eta\wedge\nu=0.$

Clearly only non-vanishing $B$ could admit such a 1-form. The first part of
the main result is stated as follows.

######  Theorem I.3.

Let $(B,\nu,\mu)$ be a flux system on $M$. Assume that there exists a 1-form
$\eta$ adapted to the flux system. Then, the vector field $u$ satisfying

$\iota_{u}\mu=\frac{\nu\wedge\eta}{\eta(B)}$

is the unique vector field $u$ such that

$\iota_{u}\iota_{B}\mu=\nu,\qquad\eta(u)=0$

and the vector fields $u$ and rescaling ${\tilde{B}}=B/\eta(B)$

  1. 1.

are linearly independent at $x\in M$ whenever $\nu|_{x}\neq 0$,

  2. 2.

satisfy $\nu({\tilde{B}})=0=\nu(u)$,

  3. 3.

commute $[u,{\tilde{B}}]=0$,

  4. 4.

and preserve the volume form ${\tilde{\mu}}:=\eta(B)\mu$.

Theorem I.3 is constructive and global. It shows that, given a flux system
$(B,\nu,\mu)$, one can use the 1-form $\eta$, if it exists, to uniquely
generate a vector field $u$ which is a symmetry for the rescaled field
${\tilde{B}}$. It will also be shown how adapted 1-forms relate to the theory
in [3], which in turn can be used to prove Theorem I.3 from a more abstract
perspective.

The second part addresses an outstanding question left from the first: for a
given flux system $(B,\nu,\mu)$ with $B$ having no zeroes and $\nu$ having
nowhere dense critical points, does an adapted 1-form always exist? We provide
a partial answer to this question: the semi-global existence of an adapted
1-form. By semi-global, we mean in the sense of [11], that it holds on an open
subset of $M$ determined by the flux system. We call this subset the toroidal
region (defined in Section III). In essence, the toroidal region is the
largest subset of $M$ consisting of invariant tori of $B$ and for which $B$ is
non-vanishing, together with periodic orbits of $B$ neighboured by the tori.
In the context of magnetic fields, these periodic orbits are called _magnetic
axes_. The following example illustrates the toroidal region.

######  Example I.4.

Consider the manifold $M=\mathbb{R}^{2}\times\mathbb{R}/\mathbb{Z}$ with
$x,y,t$ the respective coordinate projections of each factor. Define the flux
system $(B,\nu,\mu)$ where

$B=y\partial_{x}+(2x-4x^{3})\partial_{y}+\partial_{t},\qquad\nu=d\psi,\qquad\mu=dx\wedge
dy\wedge dt,$

and $\psi=\tfrac{1}{2}y^{2}-x^{2}(1-x^{2})$. In this case, the toroidal region
$\mathcal{T}=M\backslash\\{\psi=0\\}$ has three connected components divided
by $\\{\psi=0\\}$ as depicted in Fig. 1.

Figure 1: A constant-$t$ Poincaré section of $B$. The toroidal region is
everything except for $\\{\psi=0\\}$ which intersects the Poincaré section in
a figure eight (dashed).

The second part is stated as follows.

######  Theorem I.5.

Let $(B,\nu,\mu)$ be a tangential flux system on $M$. Then, the toroidal
region $\mathcal{T}$ is an open subset of $M$ for which the flux system
restricted to $\mathcal{T}$ satisfies the assumptions, and therefore the
conclusion, of Theorem I.3.

In fact, motivated by the work of [12, 13] we exhibit an example (Example
IV.3) flux system $(B,d\psi,\mu)$ on a solid torus for which an adapted 1-form
exists on all level sets of $\psi$ except the unique level set contained in
the complement of the toroidal region. Interestingly, this flux system still
possesses a symmetry and $S^{*}$ is in fact a torus.

The techniques used to prove Theorem I.5 also bring about the following
classification of the types of topologies which can occur for the toroidal
region. Some possible toroidal domains are given in Fig. 2.

######  Proposition I.6.

Let $C$ be a connected component of $\mathcal{T}$ considered as an open
submanifold of $M$ with boundary and consider the manifold interior
$\operatorname{int}C=C\backslash\partial C$.

  1. 1.

If $\operatorname{int}C$ is compact, then $C=\operatorname{int}C$ does not
have boundary nor contain axes.

  2. 2.

Otherwise, $C$ is diffeomorphic to either $\mathbb{T}^{2}\times I$ where $I$
is some interval, a solid torus $D\times\mathbb{S}^{1}$ (where $D$ denotes the
closed unit disk), an open solid torus
$\operatorname{int}D\times\mathbb{S}^{1}$, or
$\mathbb{S}^{2}\times\mathbb{S}^{1}$. In all these cases, $\nu$ is exact on
$C$.

Figure 2: Two examples of possible toroidal domains. Left is a knotted
toroidal domain given by the embedding of a solid torus
$D\times\mathbb{S}^{1}$. The case $\operatorname{int}D\times\mathbb{S}^{1}$
and $\mathbb{T}^{2}\times I$ are analagous. On the right is the more exotic
$\mathbb{S}^{2}\times\mathbb{S}^{1}$, demonstrating a toroidal domain with two
axes. The annuli depicted in the right diagram are the invariant tori after
identification of the inner and outer sphere producing the $\mathbb{S}^{1}$
factor. The two axes pass through different poles of the spheres.

The paper proceeds as follows. In Section II, Theorem I.3 is proved and its
relation to dynamical systems theory is discussed with an example. Moreover, a
counterexample to the naive converse of a Noether type theorem for divergence
free fields is also given. In Section III, Theorem I.5 is proved, and the
toroidal region is classified. In Section IV an example exhibiting an
obstruction to the global existence of an adapted 1-form is discussed. In
Section V, we discuss our methods in relation to plasma physics, specifically
to the existence of flux-coordinates. In Section VI, the results are discussed
both mathematically and in relation to plasma physics. The discussion includes
comments on the Noether type theorems and future work in the context of
integrable systems.

##  II The Conformal Noether theorem given an adapted 1-form

The main purpose of this section is to prove Theorem I.3. The remaining
material puts this theorem into its context as a (partial) converse to a
conformal Noether Theorem. In Section II.1, the (forward) conformal Noether
Theorem is presented and an explicit counterexample to the (non-conformal)
converse Noether Theorem is provided. This is then related to Theorem I.3 as a
conformal Noether Theorem. In Section II.2, an elementary proof of Theorem I.3
is given. An alternative proof is then given in Section II.3 by
contextualising the theorem in the theory of presymplectic geometry. From this
more abstract perspective, the theorem can be proved using a coisotropic
embedding of the manifold $M$ into a symplectic manifold $\tilde{M}$. By
revealing the connection to symplectic geometry, this version of the proof of
Theorem I.3 gives insight into why one should expect the conformal Noether
theorem to hold. Lastly, as Theorem I.3 relies on the existence of an adapted
1-form, Section II.5 provides contexts in which adapted 1-forms are guaranteed
to exist globally.

Throughout this section, we will make extensive use of the formulae [14]

$\displaystyle\mathcal{L}_{X}\alpha$
$\displaystyle=d\iota_{X}\alpha+\iota_{X}d\alpha,$
$\displaystyle\iota_{[X,Y]}\alpha$
$\displaystyle=\mathcal{L}_{X}\iota_{Y}\alpha-\iota_{Y}\mathcal{L}_{X}\alpha,$

for any vector fields $X,Y$ and any $k$-form $\alpha$ on $M$. In particular,
if $B$ is a divergence-free vector field with respect to a volume form $\mu$,
$f$ is a positive function and $u$ is a vector field, writing
${\tilde{B}}=B/f$ and ${\tilde{\mu}}=f\mu$, one has
$\mathcal{L}_{{\tilde{B}}}{\tilde{\mu}}=\mathcal{L}_{B}\mu=0$ and therefore

$\iota_{[u,{\tilde{B}}]}{\tilde{\mu}}=\mathcal{L}_{u}\iota_{\tilde{B}}{\tilde{\mu}}-\iota_{\tilde{B}}\mathcal{L}_{u}{\tilde{\mu}}=d\iota_{u}\iota_{B}\mu-\iota_{\tilde{B}}\mathcal{L}_{u}{\tilde{\mu}}=d\iota_{u}\iota_{B}\mu-(\operatorname{div}_{{\tilde{\mu}}}u)\iota_{B}\mu$
(1)

where $\operatorname{div}_{{\tilde{\mu}}}u$ is the divergence of $u$ with
respect to ${\tilde{\mu}}$:
$(\operatorname{div}_{{\tilde{\mu}}}u){\tilde{\mu}}=\mathcal{L}_{u}{\tilde{\mu}}$.

###  II.1 The Conformal Noether theorem

The forward direction of the conformal Noether theorem for divergence-free
vector fields is the same observation as that of [7, Thm. 3.1] but applied to
Equation (1), which allows functions $f\neq 1$.

######  Theorem II.1.

Let $B$ be a vector field on a 3-manifold $M$ with boundary and assume it is
divergence-free with respect to $\mu$. Suppose there exists a vector field $u$
on $M$ and a positive function $f\in C^{\infty}(M)$ such that $[u,B/f]=0$ and
$u$ is divergence-free with respect to ${\tilde{\mu}}=f\mu$. Then the 1-form
$\nu:=\iota_{u}\iota_{B}\mu$ makes $(B,\nu,\mu)$ a flux system.

###### Proof.

Equation (1) reads

$\iota_{[u,{\tilde{B}}]}{\tilde{\mu}}=d\iota_{u}\iota_{B}\mu-(\operatorname{div}_{{\tilde{\mu}}}u)\iota_{B}\mu$

where ${\tilde{B}}=B/f$. In particular, from $[u,{\tilde{B}}]=0$ and
$\operatorname{div}_{{\tilde{\mu}}}u=0$, we get that
$d\iota_{u}\iota_{B}\mu=0$. Hence, $\nu=\iota_{u}\iota_{B}\mu$ is closed.
Moreover, $\nu(B)=0$ (and also $\nu(u)=0$) making $(B,\nu,\mu)$ a flux system.
∎

The $u$ in Theorem II.1 is what we call a conformal symmetry of $B$. Ideally,
a converse to Theorem II.1 would read as follows.

> Converse Conformal Noether “Theorem”: Given a flux system $(B,\nu,\mu)$,
> there exists a function $f$ and a vector field $u$ which is divergence-free
> with respect to $f\mu$ such that $\iota_{u}\iota_{B}\mu=\nu$. Consequently,
> $u$ is a conformal symmetry satisfying $[u,B/f]=0$ and $\nu(u)=0$.

What Theorem I.3 says in relation to this Converse Conformal Noether“Theorem”
is that this “Theorem” is true under the additional assumption that a 1-form
adapted to $(B,\nu,\mu)$ exists. However, the naive converse Noether “Theorem”
is false: that is, the above “Theorem” is false if one requires that $f=1$.
This is seen in Example II.12 in Section II.4.

###  II.2 An elementary proof of Thm. I.3

Here we give a constructive and elementary proof of Theorem I.3. To do so, we
will use the following lemma. Let $M$ be a 3-manifold with (possibly empty)
boundary.

######  Lemma II.2.

Suppose $B$ is divergence-free with respect to the volume form $\mu$, $\eta$ a
1-form such that $\eta(B)>0$, and $\nu$ a closed 1-form. Then $(B,\nu,\mu)$ is
a flux system if and only if there exists a unique vector field $u$ such that
$\iota_{u}\iota_{B}\mu=\nu$ and $\eta(u)=0$, in which case, $u$ satisfies

$\iota_{u}\mu=\frac{\nu\wedge\eta}{\eta(B)}.$

###### Proof.

The proof of this lemma is similar to that of [3, Lemma 3.4]. One direction is
clear; if $\iota_{u}\iota_{B}\mu=\nu$ then
$0=\iota_{B}\iota_{u}\iota_{B}\mu=\iota_{B}\nu$ regardless of whether
$\eta(u)=0$. Hence, $(B,\nu,\mu)$ is a flux system.

To prove the other direction, it suffices to show that the linear map
$\hat{b}:\Gamma(TM)\to\Omega^{1}(M)$ from vector fields to 1-forms defined by

$\hat{b}(X)=\iota_{X}\iota_{B}\mu+\eta(X)\eta$

is a bundle isomorphism with inverse $\hat{b}^{-1}:\Omega^{1}(M)\to\Gamma(TM)$
defined by

$\iota_{\hat{b}^{-1}(\alpha)}\mu=\frac{\alpha\wedge\eta}{\eta(B)}+\frac{\alpha(B)}{\eta(B)^{2}}\iota_{B}\mu$

where products (interior or exterior) and evaluations (of k-forms) are
interpreted are occurring at the according fibres of $TM$ and $T^{*}M$.
Indeed, with this fact, if $u^{\prime}$ is a vector field satisfying
$\iota_{u^{\prime}}\iota_{B}\mu=\nu$ and $\eta(u^{\prime})=0$, then

$\hat{b}(u^{\prime})=\iota_{u^{\prime}}\iota_{B}\mu+\eta(u^{\prime})\eta=\nu.$

On the other hand, the vector field $u=\hat{b}^{-1}(\nu)$ satisfies
$\nu=\iota_{u}\iota_{B}\mu+\eta(u)\eta$ and evaluating this equation on $B$
shows that $\eta(u)=0$ because $\eta(B)>0$.

Now, to establish the isomorphism, let $X\in\Gamma(TM)$ and
$\alpha\in\Omega^{1}(M)$. Then, because $\eta(B)>0$, the following are
equivalent.

$\displaystyle\hat{b}(X)=\alpha$
$\displaystyle\Longleftrightarrow\iota_{X}\iota_{B}\mu+\eta(X)\eta=\alpha$
$\displaystyle\Longleftrightarrow(\iota_{X}\iota_{B}\mu+\eta(X)\eta)(B)=\alpha(B)\text{
and }(\iota_{X}\iota_{B}\mu+\eta(X)\eta)\wedge\eta=\alpha\wedge\eta.$

On the other hand,

$(\iota_{X}\iota_{B}\mu+\eta(X)\eta)(B)=\eta(X)\eta(B)$

and

$\displaystyle(\iota_{X}\iota_{B}\mu+\eta(X)\eta)\wedge\eta$
$\displaystyle=\iota_{X}\iota_{B}\mu\wedge\eta$
$\displaystyle=\iota_{X}(\iota_{B}\mu\wedge\eta)-\eta(X)\iota_{B}\mu$
$\displaystyle=\eta(B)\iota_{X}\mu-\eta(X)\iota_{B}\mu.$

Hence, we have the additional equivalent statements.

$\displaystyle\hat{b}(X)=\alpha$
$\displaystyle\Longleftrightarrow\eta(X)\eta(B)=\alpha(B)\text{ and
}\eta(B)\iota_{X}\mu-\eta(X)\iota_{B}\mu=\alpha\wedge\eta$
$\displaystyle\Longleftrightarrow\eta(X)=\frac{\alpha(B)}{\eta(B)}\text{ and
}\iota_{X}\mu=\frac{\alpha\wedge\eta}{\eta(B)}+\frac{\alpha(B)}{\eta(B)^{2}}\iota_{B}\mu.$

Now, observe that

$\iota_{X}\mu\wedge\nu=\eta(X)\mu,\qquad\left(\frac{\alpha\wedge\eta}{\eta(B)}+\frac{\alpha(B)}{\eta(B)^{2}}\iota_{B}\mu\right)\wedge\eta=\frac{\alpha(B)}{\eta(B)}\mu,$

and therefore

$\iota_{X}\mu=\frac{\alpha\wedge\eta}{\eta(B)}+\frac{\alpha(B)}{\eta(B)^{2}}\iota_{B}\mu\implies\eta(X)=\frac{\alpha(B)}{\eta(B)}.$

Hence, we obtain the equivalence

$\displaystyle\hat{b}(X)=\alpha\Longleftrightarrow\iota_{X}\mu=\frac{\alpha\wedge\eta}{\eta(B)}+\frac{\alpha(B)}{\eta(B)^{2}}\iota_{B}\mu$

and therefore the proof of our claim. ∎

We are now ready to prove Theorem I.3.

###### Proof of Theorem I.3.

Let $(B,\nu,\mu)$ be a flux system on $M$. Assume that $\eta$ is an adapted
1-form for $(B,\nu,\mu)$. Then, by Lemma II.2 the vector field $u$ given by

$\iota_{u}\mu=\frac{\nu\wedge\eta}{\eta(B)}$

is the unique vector field satisfying

$\iota_{u}\iota_{B}\mu=\nu,\qquad\eta(u)=0.$

It remains to show claims 1 to 4 of Theorem I.3 on the relationship between
$u$ and ${\tilde{B}}:=\eta(B)^{-1}B$. Claims 1 and 2 can immediately be seen
from $\iota_{u}\iota_{B}\mu=\nu$. Equation (1) with ${\tilde{B}}$ and
${\tilde{\mu}}=\eta(B)\mu$ shows that claims 3 and 4 are equivalent. Indeed,

$\iota_{[u,{\tilde{B}}]}{\tilde{\mu}}=d\iota_{u}\iota_{B}\mu-(\operatorname{div}_{{\tilde{\mu}}}u)\iota_{B}\mu=d\nu-(\operatorname{div}_{{\tilde{\mu}}}u)\iota_{B}\mu=-(\operatorname{div}_{{\tilde{\mu}}}u)\iota_{B}\mu.$

In addition, claim 4 holds because

$\mathcal{L}_{u}{\tilde{\mu}}=d\iota_{u}{\tilde{\mu}}=d(\nu\wedge\eta)=\nu\wedge
d\eta=0$

where we used that $d\eta\wedge\nu=0$. Thus, all claims hold, completing the
proof. ∎

######  Remark II.3.

From the proof of Lemma II.2, the map $\hat{b}$ can be understood as being a
vector bundle isomorphism $b:TM\to T^{*}M$ whereby

$b(v)=\iota_{v}(\iota_{B}\mu)|_{x}+\eta|_{x}(v)\eta|_{x}$

for $v\in TM$. This clarifies that $\hat{b}$ is a purely linear-algebraic
construction.

###  II.3 A proof of Thm. I.3 through a coisotropic embedding

In this section, theoretical context is given to Theorem I.3. In particular,
we will take the perspective of [3] in viewing a non-vanishing, divergence-
free vector field $B$ on a 3-manifold $M$ as giving $M$ the structure of a
presymplectic manifold. Through this perspective, the connection between
Theorem I.3 and the usual Noether theorem for symplectic manifolds can be
understood.

####  II.3.1 A brief introduction to presymplectic geometry

The vector field $B$ and the volume form $\mu$ which it preserves gives rise
to a presymplectic manifold. This change in perspective is facilitated by the
observation that the 2-form

$\beta=\iota_{B}\mu$

is necessarily closed ($d\beta=0$) due to the fact that $B$ is divergence-
free, and that $\beta$ is of maximal rank by the non-vanishing assumption of
$B$. In the plasma physics literature, the form $\beta$ is often called the
_flux form_.

Any closed, maximal rank 2-form $\beta$ is called _presymplectic_ and makes
$(M,\beta)$ a _presymplectic manifold_. The study of a presymplectic manifold
is called presymplectic geometry. We follow [3] in briefly summarising some
key observations in presymplectic geometry. In particular, we will point out
some differences to symplectic geometry arising from the fact that $\beta$ is
only of maximal rank and not non-degenerate.

######  Definition II.4 (Presymplectic and Hamiltonian Vector Fields).

A vector field $u$ on $M$ is _presymplectic with respect to $\beta$_ or is
_locally Hamiltonian_ if there exists a closed 1-form $\nu$ such that
$\iota_{u}\beta=\nu$. The pair $(u,\nu)$ is called a _presymplectic pair_. If
in addition $\nu$ is exact, that is $\nu=-dH$ for some $H\in C^{\infty}(M)$,
then $u$ is called a _Hamiltonian vector field_ for the Hamiltonian $H$, and
$(u,H)$ is a _Hamiltonian pair_.

######  Remark II.5.

Observe for any vector field $u$ that
$\mathcal{L}_{u}\beta=\iota_{u}d\beta+d\iota_{u}\beta=d\iota_{X}\beta$ and
hence $u$ preserves $\beta$ if and only if $u$ is presymplectic.

Every Hamiltonian pair is presymplectic, and the converse is true if the first
de Rham cohomology of $M$, $H^{1}(M)$, vanishes. It is true that, given a
presymplectic pair, $B$ preserves $\nu$ as shown in the following.

######  Lemma II.6 ([3]).

Suppose that $(u,\nu)$ is a presymplectic pair for $\beta$. Then

  1. 1.

$\nu(B)=0$;

  2. 2.

$\mathcal{L}_{B}\nu=0$;

  3. 3.

$\beta\wedge\nu=0$.

Moreover, if $(u,H)$ is a Hamiltonian pair, $B$ is tangent to surfaces of
constant $H$, that is $\mathcal{L}_{B}H=0$.

Lemma II.6 has the following natural converse which follows immediately from
Lemma II.2.

######  Lemma II.7.

If $\nu$ is a closed 1-form with $\nu(B)=0$, then there exists $u$ such that
$(u,\nu)$ is a presymplectic pair.

It is natural to compare symplectic vector fields to presymplectic vector
fields. A symplectic 2-form $\omega$ is non-degenerate and consequently can
only be defined on some even dimensional manifolds. The non-degeneracy of a
symplectic form implies that, given any function $H\in C^{\infty}(M)$, there
exists a unique Hamiltonian vector field $X_{H}$ defined by
$\iota_{X_{H}}\omega=-dH$. In the presymplectic setting, given a function $H$
on $M$, a Hamiltonian vector field need _not_ exist. Moreover, even if it does
exist, the Hamiltonian vector field is not unique. Indeed, if $u$ is a
Hamiltonian vector field for $\nu$ then any vector field of the form $u+hB$ is
also Hamiltonian for $\nu$, for any function $h$ on $M$.

Perhaps more surprisingly, unlike the symplectic case, not all presymplectic
vector fields are divergence-free.

######  Lemma II.8 ([3]).

A presymplectic vector field $u$ is divergence-free if and only if $[u,B]=0$.

The consequence of Lemmas II.7 and II.8 is that one must be careful when
trying to translate well-known results and theory from symplectic geometry to
presymplectic geometry. In particular, as we will see in the proceeding
sections, one needs additional assumptions in order to provide a Noether
theorem for presymplectic vector fields, and consequently, divergence-free
vector fields.

####  II.3.2 Converse Conformal Noether Theorem

In this section we establish a second proof of Theorem I.3 from the
perspective of presymplectic geometry. Before doing so, we discuss an initial
approach to the Converse Conformal Noether “Theorem” described in Section II.1
from this perspective.

Let $\nu$ be a closed 1-form such that $\nu(B)=0$. Then Lemma II.7 enables us
to fix a vector field $u_{0}$ so that $(u_{0},\nu)$ is a presymplectic pair:
$\iota_{u_{0}}\beta=\nu$. Then, there exists a divergence-free vector field
$u$ satisfying $\iota_{u}\beta=\nu$ if and only if there exists a function
$h\in C^{\infty}(M)$ such that the vector field $u=u_{0}+hB$ satisfies
$\operatorname{div}_{\mu}u=0$ or equivalently that $h$ satisfies the
cohomological equation

$B(h)=-\operatorname{div}_{\mu}u_{0}.$

However, in general, only special vector fields are able to solve given
cohomological equations [15, 16]. In fact, Example II.12 in Section II.4 shows
that such a $h$ need not exist. On the other hand, we have that

$\iota_{[u,B]}\beta=\mathcal{L}_{u}\iota_{B}\beta-\iota_{B}\mathcal{L}_{u}\beta=-\iota_{B}d\nu=0.$

so $[u,B]=\kappa B$ for some function $\kappa$ and $u$ being a conformal
symmetry is equivalent to another cohomological equation. This line of
approach requires a deeper understanding of the vector field $B$ and the
associated cohomological equation to make any conclusion.

An alternative approach comes delivered in a more abstract package. For the
sake of the exposition, assume that $\nu=d\psi$ is exact. The journey begins
with $\eta$-embeddings of the presymplectic manifolds.

######  Theorem II.9 ($\eta$-embedding [3]).

Let $(M,\beta)$ be a presymplectic manifold. Suppose there exists a 1-form
$\eta$ on $M$ such that $\beta\wedge\eta$ is a volume form. Let
$\tilde{M}=M\times\mathbb{R}\ni(z,v)$, define the projection $\pi:\tilde{M}\to
M$ and take $\tilde{\beta}=\pi^{*}\beta,\,\tilde{\eta}=\pi^{*}\eta$. Then,
$(\tilde{M},\omega)$ is a symplectic manifold with

$\omega=\tilde{\beta}+d(v\tilde{\eta}).$

Moreover, $M$ is coisotropically embedded as $v=0$ in $\tilde{M}$.

Note that on a presymplectic manifold $(M,\beta)$, a metric $g$ provides the
required 1-form by setting $\eta:=B^{\flat}$. There is of course some choice
in $\eta$, and we say any specific embedding is an _$\eta$ -embedding of
$(M,\beta)$_.

The $\eta$-embedding provides a mechanism to apply the tools of symplectic
geometry to that of presymplectic geometry. In order to find a conformal
symmetry, we will need to not only embed the presymplectic manifold associated
to $B$ into a symplectic manifold, but do so in such a way as to have the lift
of $B$ to $\tilde{M}$ be part of an integrable system (in the sense of
Liouville). With a little more structure imposed on $\eta$, this is indeed
possible.

######  Theorem II.10.

Assume $\eta$ is a 1-form adapted to the flux system $(B,d\psi,\mu)$. Let
$(M\times\mathbb{R},\omega)$ be the symplectic manifold obtained from the
$\eta$-embedding of $(M,\beta)$, take coordinates $(z,v)\in M\times\mathbb{R}$
and let $\tilde{\psi}=\pi^{*}\psi$. Then $H=v$ and $\tilde{\psi}$ form an
integrable system on $M\times\mathbb{R}$ with associated Hamiltonian vector
fields respectively satisfying

$X_{H}|_{v=0}=\frac{1}{\eta(B)}B,\quad
X_{\tilde{\psi}}|_{v=0}=u-\frac{\eta(u)}{\eta(B)}B,$

for any vector field $u$ such that $\iota_{u}\beta=-d\psi$.

###### Proof.

The proof is identical to that of [3, Thm. 4.11] which has stronger hypotheses
to Theorem II.10. It is enough to observe that the original proof follows with
the weaker hypotheses. The proof is included here for completeness.

By Lemma II.7, there always exists some $u$ such that $\iota_{u}\beta=-d\psi$,
although such a $u$ is not guaranteed to be divergence-free. We begin by
computing the Hamiltonian vector fields associated with $H$ and $\tilde{\psi}$
on $v=0$. They are defined through Hamilton’s equations:

$\iota_{X_{H}}\omega=-dv,\quad\iota_{X_{\tilde{\psi}}}\omega=-d\tilde{\psi}.$

A calculation reveals that $X_{H}|_{v=0}=\frac{1}{\eta(B)}B$ and
$X_{\tilde{\psi}}|_{v=0}=u-\frac{\eta(u)}{\eta(B)}B$, as desired.

To show that $(H,\tilde{\psi})$ form an integrable system on
$M\times\mathbb{R}$ we require the Poisson bracket $\\{\tilde{\psi},H\\}=0$. A
quick way to see this is to pull back $\omega$ to the common level set
$(\tilde{\psi},H)=(\tilde{\psi}_{0},H_{0})$. The condition $H=H_{0}$ implies
$v=H_{0}$. The pullback of $\omega$ to the level set is therefore

$\pi_{\tilde{\psi}_{0},H_{0}}^{*}(\tilde{\beta}+d(v\tilde{\eta}))=0+\pi_{\tilde{\psi}_{0}}^{*}d(H_{0}\tilde{\eta})=H_{0}\pi_{\tilde{\psi}_{0}}^{*}d\tilde{\eta}=0,$

since $\tilde{\beta}=\pi^{*}\beta$ and $d\tilde{\eta}=\pi^{*}d\eta$ both
vanish when pulled back to a $\tilde{\psi}$-surface. It follows that $\omega$
vanishes on the common level sets, and hence, that
$\\{\tilde{\psi},v\\}:=\iota_{X_{H}}\iota_{X_{\tilde{\psi}}}\omega=0$. ∎

By $\eta$-embedding the presymplectic manifold generated from the field $B$ of
a flux system $(B,d\psi,\mu)$, Theorem II.10 provides an integrable system (in
the sense of Liouville) given by functions $u,\tilde{\psi}$ and associated
Hamiltonian vector fields $X_{v},X_{\tilde{\psi}}$. As $u,\tilde{\psi}$
Poisson commute, it follows that $X_{v},X_{\tilde{\psi}}$ commute. In
particular the vector fields

${\tilde{B}}:=X_{v}|_{v=0}=\frac{1}{\eta(B)}B,\qquad\tilde{u}:=X_{\rho}|_{u=0}=u-\frac{\eta(u)}{\eta(B)}B$

will commute. In fact $\iota_{\tilde{u}}\beta=-d\psi$. We thus have what we
want: a conformal symmetry $\tilde{u}$ of $B$. Hence, we have provided an
alternative proof of Theorem I.3 for the case $\nu$ is exact. Theorem II.10
can be adjusted to prove the case where $\nu$ is no longer exact.

###  II.4 A counter-example to the converse naive Noether theorem

To understand the counter-example, one must bear in mind the following
elementary results in the study of Diophantine vector fields (for a definition
of Diophantine vector fields see [15]). The first statement follows directly
from [10, Proposition 23 and 24] while the second statement is in [15,
Proposition 2.6].

######  Proposition II.11.

On the 2-torus $S=(\mathbb{R}/\mathbb{Z})^{2}$, the following holds.

  1. 1.

Let $0<f\in C^{\infty}(S)$ and consider the vector field

$X=f(a\partial_{x}+b\partial_{y})$

where $(a,b)\in\mathbb{R}^{2}$. If $(a,b)$ is not Diophantine, then $X$ is not
smoothly conjugate to a Diophantine vector field. That is, there does not
exist a diffeomorphism $\varphi:S\to S$ such that

$\varphi_{*}X=c\partial_{x}+d\partial_{y}$

for any Diophantine vector $(c,d)\in\mathbb{R}^{2}$.

  2. 2.

A vector field $X$ on $S$ is smoothly conjugate to a Diophantine vector field,
if and only if, for every $h\in C^{\infty}(S)$, there exists a $g\in
C^{\infty}(S)$ and a constant $c\in\mathbb{R}$ solving the cohomological
equation

$X(g)=h-c.$

With this in mind, we now consider the example.

######  Example II.12.

First consider the 2-torus $S=(\mathbb{R}/\mathbb{Z})^{2}$. With the natural
coordinates $x,y$, consider the vector fields

$X_{0}=a\partial_{x}+b\partial_{y},\qquad Y_{0}=c\partial_{x}+d\partial_{y}$

where $(a,b)\in\mathbb{R}^{2}$ be linearly independent from $\mathbb{Z}^{2}$
but not Diophantine and $(c,d)$ is Diophantine and linearly independent from
$(a,b)$. We will show that for some $0<f\in C^{\infty}(S)$, the only vector
fields commuting with $X=fX_{0}$ are constant multiples of $X$.

For the time being, let $f\in C^{\infty}(S)$ be arbitrary and set $X=fX_{0}$.
Any vector field $Z$ may be written, for some $g,h\in C^{\infty}(S)$, as

$Z=gX+hY_{0}.$

In this way, we obtain

$\displaystyle[X,Z]$ $\displaystyle=[X,gX+hY_{0}]$
$\displaystyle=[X,gX]+[fX_{0},hY_{0}]$
$\displaystyle=(fX(g)-hY_{0}(f))X_{0}+fX_{0}(h)Y_{0}.$

Hence, if $[X,Z]=0$, then $X_{0}(h)=0$ and since $X_{0}$ has dense orbits in
$S$, $h$ must be constant. Moreover, we have the cohomological equation

$X(g)=Y_{0}(h\ln f)$ (2)

Applying the first and second statement of Proposition II.11 to $X$, there
exists $H\in C^{\infty}(S)$ such that, for any function $g\in C^{\infty}$ and
$c\in\mathbb{R}$

$X(g)\neq H-c.$

Fix such a $H$. Suppose that $h\neq 0$. Applying the second statement of
Proposition II.11 to $Y_{0}$, there exists function $f$ and a constant
$c^{\prime}$ such that

$Y_{0}(h\ln f)=H-c^{\prime}.$

Fixing this choice of $f$, (2) is false. Therefore, we must have that $h=0$
and so $X(g)=0$. This implies as before that $g$ is constant and $Z=gX$, a
constant multiple of $X$. Hence, our claim is true.

With this in mind, we now consider the manifold with (possibly empty) boundary
$M=S\times I$ where $I$ is some interval. With the natural coordinates
$x,y,z$, consider the flux system $(B,\nu,\mu)$ where

$B=F(a\partial_{x}+b\partial_{y}),\qquad\nu=dz,\qquad\mu=F^{-1}dx\wedge
dy\wedge dz$

and $F(x,y,z)=f(x,y)$ where $f$ is as before. Then, the previous paragraph
shows that $B$ is not rectifiable, not even locally about a single
constant-$z$ torus. In particular, there cannot exist a vector field $u$
linearly independent from $B$ satisfying

$[u,B]=0,\qquad\nu(u)=0$

let alone satisfying $\iota_{u}\iota_{B}\mu=\nu$.

###  II.5 Contexts in which an adapted 1-form is guaranteed globally

With some additional assumptions on a flux system, an adapted 1-form can be
guaranteed to exist globally. Fix a 3-manifold $M$ with (possibly empty)
boundary.

######  Proposition II.13.

Let $B$ be a vector field on $M$. Let $\eta$ be a 1-form such that

$\iota_{B}d\eta=0,\qquad\eta(B)>0.$

If $(B,\nu,\mu)$ is a flux system, then $\eta$ is a 1-form adapted to
$(B,\nu,\mu)$.

###### Proof.

It suffices to note that $d\eta\wedge\nu$ is a top-form on $M$ and hence
$d\eta\wedge\nu=0$ is equivalent to $\iota_{B}(d\eta\wedge\nu)=0$. Moreover,

$\iota_{B}(d\eta\wedge\nu)=(\iota_{B}d\eta)\wedge\nu+d\eta\wedge\iota_{B}\nu=0$

∎

There are some particular cases of Proposition II.13 of interest. If $B$ is
vector field and $\eta$ is a closed 1-form such that $\eta(B)>0$, then the
hypothesis of Proposition II.13 are satisfied. If $M$ is closed, any vector
field $B$ with a global Poincaré section possesses such an $\eta$ by the work
of [17]. In plasma physics, magnetic fields $B$ in a vacuum on an oriented $M$
satisfy $dB^{\flat}=0$ with respect to the metric $g$ placed on $M$. Hence,
all non-vanishing vacuum fields have the closed 1-form $\eta=B^{\flat}$
satisfying the assumptions of Proposition II.13.

Alternatively, if $B$ is a vector field and $\eta$ is a 1-form such that
$d\eta$ is non-vanishing, then $B$ and $\eta$ satisfy the hypothesis of
Proposition II.13 if and only if $\eta$ is a contact form and $B$ is a Reeb-
like vector field of $\eta$. In this context, $B$ is a Reeb-like field if
there exists a non-vanishing function $f\in C^{\infty}(M)$ such that $B/f$ is
the Reeb vector field of $\eta$. For such fields, $B$ is divergence-free with
respect to $\mu=\frac{1}{\eta(B)}\eta\wedge d\eta$. The following is an
example of a contact system giving rise to an adapted 1-form.

######  Example II.14.

Set $M=(\mathbb{R}/2\pi\mathbb{Z})^{3}$ with periodic coordinates
$x,y,z:(\mathbb{R}/2\pi\mathbb{Z})^{3}\to\mathbb{R}/2\pi\mathbb{Z}$. Consider
the contact form

$\eta=\sin zdx+\cos zdy$

whereby $\eta\wedge d\eta=dx\wedge dy\wedge dz$, the standard volume form on
$M$. The associated Reeb vector field is given by

$\displaystyle B$ $\displaystyle=\sin z\partial_{x}+\cos z\partial_{y}.$

With $\nu=dz$, we get a flux system $(B,\nu,\mu)$ and $\eta$ an adapted
1-form.

It is worth highlighting that Example II.14 gives an example of a flux system
such that $B$ has no global Poincaré section, and $\eta$ is not closed. In
fact, the adapted 1-forms for the flux systems which arise from contact forms
can never be closed in the case that $M$ is closed. This is because of the
following elementary fact.

######  Proposition II.15.

Let $B$ be a Reeb-like vector field for a contact form $\eta$ on a manifold
$M$. Then, $B$ has no closed Poincaré section. In particular, if $M$ is
closed, $B$ has no closed 1-form $\kappa$ for which $\kappa(B)>0$.

###### Proof.

Let $S$ be a closed hypersurface of dimension $2n$ of $M$ where $\dim M=2n+1$
($n\geq 1$). Considering the volume-form $\mu=\eta\wedge(d\eta)^{n}$ and the
Reeb vector field $R$, we have $\iota_{R}\mu=(d\eta)^{n}$. Then considering
the inclusion $i:S\subset M$, because $i^{*}(d\eta)^{n}$ is exact on $S$,
Stokes’ Theorem gives

$\int_{S}i^{*}\iota_{R}\mu=\int_{S}i^{*}(d\eta)^{n}=0.$

Hence, $R$ is not transverse to $S$. Thus, $B$ is not transverse to $S$
either. In particular, $B$ has no Poincaré section. Moreover, it follows from
the same techniques as in [18] that a vector field $X$ possesses a closed
transverse section provided there exists a closed 1-form $\kappa$ such that
$\kappa(X)>0$. This concludes the proof. ∎

As an example with relevance to applications, Beltrami fields can be thought
of as generalisations of Reeb-like vector fields in 3 dimensions [19]. If $M$
has a metric and an orientation, a Beltrami field $B$ satisfies

$\nabla\cdot B=0,\qquad\nabla\times B=\lambda B$

for some $\lambda\in C^{\infty}(M)$. In terms of $\beta=\iota_{B}\mu$ where
$\mu$ is the induced volume form, this equation reads

$dB^{\flat}=\lambda\beta$

and therefore $\eta=B^{\flat}$ is a 1-form where the assumptions of
Proposition II.13 are satisfied provided that $B$ is non-vanishing.

##  III An adapted 1-form exists in the toroidal region

The main purpose of this section is to prove Theorem I.5. The proof for a
given tangential flux system $(B,\nu,\mu)$ consists of two steps. In step one,
the local existence of closed 1-forms adapted to the flux system is
established in a neighborhood of each of the torus and axis in the toroidal
region. Step one is achieved using tubular neighborhoods and in the case of
the tori, we use flowouts which are essentially the restriction of a
transverse flow to the torus; these are also used in the second step. In step
two, we patch together these closed 1-forms using a partition of unity. In
step two, care is taken to ensure that the functions in the partition of unity
have differentials that are linearly dependent with $\nu$. The construction of
the partition relies on techniques which also prove Proposition I.6. These
techniques apply more generally to closed 1-forms in a region which we call
the closed region. This generalisation is adopted to clarify the role that
each element of the flux system plays in the proof and because it does not
introduce any additional difficulties to the proof.

This section proceeds as follows. In Section III.1, the toroidal region is
formally defined and some auxiliary definitions used to prove Theorem I.5 are
introduced. Then, in Section III.2, Theorem I.5 is proven under the assumption
that the suitable partition of unity has been constructed. Lastly, in Section
Section III.3, the partition of unity is constructed and Proposition I.6 is
proven.

###  III.1 The toroidal region and the closed region

We begin by introducing some convenient notation used throughout this section.
If $M$ is a manifold with (possibly empty) boundary, $S$ is a submanifold, and
$i:S\subset M$ is the inclusion, then for any $k$-form $\nu$ on $M$, we denote
$S^{*}\nu=i^{*}\nu$.

The toroidal region arises from a more general concept which we call the
closed region. The closed region accounts for the simplest kind of critical
sets which can occur for a closed 1-form $\nu$ on $M$ such that $\partial
M^{*}\nu=0$ and will now be defined.

######  Definition III.1.

Let $\Omega=\\{x\in M:\nu|_{x}\neq 0\\}$ be the set of non-critical points of
$\nu$. The boundary condition $\partial M^{*}\nu=0$ implies that the non-
vanishing closed 1-form $\Omega^{*}\nu$ defines an integrable distribution $D$
on $\Omega$ regarded as an open submanifold with boundary of $M$ (see Appendix
A for a formal review of basic distribution and foliation theory on manifolds
with boundary). Let $\mathcal{F}$ be the associated foliation on $M$ to $D$.

  1. 1.

A closed leaf $S$ of the induced foliation is a _regular closed leaf of
$\nu$_. The _pre-closed region of $\nu$_, denoted $\mathcal{C}^{2}$, is the
union of all regular closed leaves of $\nu$

  2. 2.

A _degenerate closed leaf of $\nu$_ is a connected closed submanifold $K$ of
codimension at least $2$ which is contained in $\operatorname{int}M$ and
satisfying $\nu|_{K}=0$ and possessing a neighbourhood $U$ in $M$ such that
$U\backslash K\subset\mathcal{C}^{2}$.

  3. 3.

Set $\mathcal{C}^{1}$ to be the union of degenerate closed leaves of $\nu$.
Then, the closed region $\mathcal{C}$ is the union
$\mathcal{C}^{1}\cup\mathcal{C}^{2}$.

######  Remark III.2 (on Definition III.1).

The regular closed leaves of $\nu$ coincide with the compact integral manifold
of $D$, since integral manifolds of a distribution which are closed as subsets
of the ambient space are necessarily maximal.

We now define the toroidal region.

######  Definition III.3.

Assume that $\dim M=3$ and let $(B,\nu,\mu)$ be a tangential flux system on
$M$ (see Definition I.1).

  1. 1.

A _regular torus of $(B,\nu,\mu)$_ is a regular closed leaf $S$ of $\nu$ such
that $B|_{S}$ is non-vanishing. The _pre-toroidal region of $(B,\nu,\mu)$_,
denoted $\mathcal{T}^{2}$, is the union of all regular tori.

  2. 2.

An _axis of $(B,\nu,\mu)$_ is a 1-dimensional periodic orbit $\gamma$ of $B$
which is also a degenerate closed leaf of $\nu$.

  3. 3.

Set $\mathcal{T}^{1}$ to be the union of all axes. The _toroidal region of
$(B,\nu,\mu)$_ is then the subset
$\mathcal{T}=\mathcal{T}^{1}\cup\mathcal{T}^{2}$.

######  Remark III.4 (on Definition III.3).

It is well-known from index theory that the only compact orientable surface
admitting a nowhere zero vector field is the 2-torus. In particular, the
regular tori are indeed tori.

###  III.2 Proof of Thm. I.5

Our main tool for proving Theorem I.5 is a flowout. This terminology is
adopted from [14] and for our purposes means the following. Before using the
specific assumptions in Theorem I.5, we will focus more generally on the
closed region because of its relevance to later sections. Let $M$ be a
manifold with boundary and $\nu$ a closed 1-form on $M$ such that $\partial
M^{*}\nu=0$. Let $\Omega=\\{x\in M:\nu|_{x}=0\\}$ and fix a vector field $N$
on $\Omega$ such that $\nu(N)=1$.

######  Definition III.5.

For any $S$ a closed leaf of $\nu$ and $I$ an interval containing $0$, if for
each $x\in S$, the integral curve $\gamma_{x}$ of $N$ starting at $x$ is
defined for all $t\in I$, then the smooth map $\Sigma:S\times I\to M$ such
that $\Sigma(x,t)=\gamma_{x}(t)$ is called the _flowout of $S$ with interval
$I$_.

The following establishes elementary facts about of flowouts we require for
proving Theorem I.5.

######  Lemma III.6.

Let $S$ be a closed leaf of $\nu$. Then either $S\subset\operatorname{int}M$
or $S$ is a connected component of $\partial M$. Moreover, there exists
$\epsilon>0$ and an interval $I$ such that one of the following holds.

  1. 1.

if $S\subset\operatorname{int}M$, then $I=(-\epsilon,\epsilon)$,

  2. 2.

and if $S$ is a component of $\partial M$ and $N$ is inward-pointing on
$S\subset\partial M$, then $I=[0,\epsilon)$ and otherwise $I=(-\epsilon,0]$,

and a flowout $\Sigma$ with interval $I$ which is a diffeomorphism onto its
open image in $M$.

###### Proof.

Note that the boundary $\partial\Omega$ is closed in $\Omega$. Suppose that
$S\cap\partial M\neq\emptyset$. Then, we may fix a point $x\in
S\cap\partial\Omega$. Consider now the connected component $C$ of
$\partial\Omega$ containing $x$. Because $\partial\Omega$ is embedded in
$\Omega$, $C$ is a closed subset of $\partial\Omega$. On the other hand, it is
also an integral manifold containing $x$ of the distribution $D$ since
$\partial M^{*}\nu=0$. In summary, by closedness, $C$ and $S$ are maximal
integral manifolds containing $x$. Thus, we must have that $C=S$ from above.
In particular, $S$ is a closed, open and connected subset of $\partial M$.
Thus, $S$ is a connected component of $\partial M$.

The existence of flowouts now follows directly from the flowout theorems in
[14, Theorem 9.20 and Theorem 9.24]. ∎

To prove openness of the toroidal region (and closed region) along with some
algebraic aspects of the proof of Theorem I.5, we will also make use of the
following properties of flowouts.

######  Lemma III.7.

Let $\Sigma$ be a flowout of $S$ with some interval $I$ containing $0$. Then
the following holds.

  1. 1.

If $t:S\times I\to I\subset\mathbb{R}$ denotes projection followed by
inclusion into $\mathbb{R}$, then $\Sigma^{*}\nu=dt$.

  2. 2.

If $\Sigma^{\prime}$ is another flowout of $S$ with some interval $J$
containing $0$, then $\Sigma$ and $\Sigma^{\prime}$ coincide on $S\times(I\cap
J)$.

  3. 3.

For all $s\in I$, $\Sigma(\cdot,s):S\to M$ is a diffeomorphism onto its image
and its image $S^{\prime}=\Sigma(S\times\\{s\\})$ is a closed leaf of $\nu$.
Moreover, the flowout $\Sigma^{\prime}:S^{\prime}\times I^{\prime}\to M$ of
$S^{\prime}$ with interval $I^{\prime}=I-s$ exists and satisfies, for $x\in S$
and $t^{\prime}\in I^{\prime}$,

$\Sigma^{\prime}(\Sigma(x,s),t^{\prime})=\Sigma(x,s+t^{\prime}).$

###### Proof.

Concerning the first statement, set $\kappa=\Sigma^{*}\nu$. Let $V$ be a
contractible open subset of $S$. Then, we have the contractible open subset of
$V\times I$ so that $(V\times I)^{*}\kappa=df$ for some $f\in
C^{\infty}(V\times I)$. Now, let $p\in V$ and consider the curve
$\gamma_{p}:I\to U$ given by $\gamma_{p}(t)=(p,t)$. Then, because
$\kappa(\partial_{t})=1$, we have

$\displaystyle(f\circ\gamma_{p})^{\prime}(t)=df(\gamma_{p}^{\prime}(t))=1.$

Hence, $f\circ\gamma_{p}=t+f(p,0)$. On the other hand because
$(S\times\\{0\\})^{*}\kappa=0$, we have that $f(\cdot,0)=c$ for some
$c\in\mathbb{R}$. Thus

$f=t+c.$

Hence $\kappa|_{V\times J}=dt|_{V\times J}$. Since $V$ was arbitrary, we
conclude that $\Sigma^{*}\nu=\kappa=dt$. The second statement follows directly
from uniqueness of integral curves. Having established these prior statements,
the third statement now follows directly from the Fundamental Theorem of Flows
in [14, Theorem 9.12]. ∎

We now show the openness of the closed and toroidal regions and verify that
the critical points of $\nu$ are nowhere dense in these regions.

######  Lemma III.8.

The regions $\mathcal{C}^{2}$ and $\mathcal{C}$ of $\nu$ are open in $M$ and
the critical points of $\nu$ in $\mathcal{C}$ are nowhere dense in
$\mathcal{C}$. Moreover, if $(B,\nu,\mu)$ is a tangential flux system on $M$
(so that $\dim M=3$), then the pre-toroidal $\mathcal{T}^{2}$ and toroidal
region $\mathcal{T}$ are open in $M$.

###### Proof.

First, if $S$ is a closed leaf of $\nu$, by Lemma III.6, we can take a flowout
$\Sigma:S\times I\to M$ of $S$ with some interval $I$ which is a
diffeomorphism onto its open image $U=\Sigma(S\times I)$ in $M$. By Lemma
III.7, $U\subset\mathcal{C}^{2}$. Regarding $(B,\nu,\mu)$, if $B$ is non-
vanishing on $S$, then $\\{x\in U:B|_{x}\neq 0\\}=\Sigma(V)$ for some
neighbourhood $V$ of $S\times\\{0\\}$ in $S\times I$. Because $S\times\\{0\\}$
is compact in $S\times I$, it is elementary topology that $V$ contains
$S\times$ for some open subset $J\subset I$. Thus, $\\{x\in U:B|_{x}\neq
0\\}=\Sigma(S\times J)\subset\mathcal{T}^{2}$. In summary, $\mathcal{C}^{2}$
is open and if $(B,\nu,\mu)$ is a tangential flux system, $\mathcal{T}^{2}$ is
open in $M$. It now follows by construction that $\mathcal{C}$ is open in $M$
and that $\mathcal{T}$ are open in $M$ if $(B,\nu,\mu)$ is a tangential flux
system on $M$.

Concerning critical points, if $U$ is an open set in $M$ contained in
$\mathcal{C}$ with $\nu|_{U}=0$, then $U\subset\mathcal{C}^{1}$. If $K$ is a
degenerate closed leaf of $\nu$, then taking an open set $V\supset K$ so that
$V\backslash K\subset\mathcal{C}^{2}$, we have $U\cap V\subset K$. Since each
degenerate closed leaf has nonzero codimension, we must have that $U\cap
K=\emptyset$. Hence, $U=\emptyset$. As required. ∎

######  Remark III.9.

The relationship we are exploiting between the closed region and toroidal
region for a tangential flux system $(B,\nu,\mu)$ can now be clarified.
Clearly we have the relationship $\mathcal{T}\subset\mathcal{C}$. But, since
$\mathcal{T}$ is an open submanifold of $M$, we may consider the closed region
$\mathcal{C}_{\mathcal{T}}$ of the closed 1-form $\mathcal{T}^{*}\nu$ which
satisfies $(\partial\mathcal{T})^{*}(\mathcal{T}^{*}\nu)=0$ because $\partial
M^{*}\nu=0$. It is clear that we have equality
$\mathcal{C}_{\mathcal{T}}=\mathcal{T}$. Thus, statements applying to $\nu$ on
$\mathcal{C}$ also apply to $\nu$ on $\mathcal{T}$ (such as the nowhere
denseness of critial points of $\nu$).

We now establish the local existence an adapted 1-form in a neighbourhood of
each regular torus of a tangential flux system.

######  Lemma III.10.

Let $(B,\nu,\mu)$ be a tangential flux system on $M$. Let $S$ be a regular
torus. Then, there exists a neighbourhood $U$ of $S$ in $\mathcal{T}$ and a
closed 1-form $\eta\in\Omega^{1}(U)$ such that $\eta(B)>0$.

###### Proof.

By Lemma III.6 and Lemma III.7 we may take an interval $I\subset\mathbb{R}$
and an embedding $\Sigma:S\times I\to M$ such that $\Sigma^{*}\nu=dt$ where
$t:S\times I\to\mathbb{R}$ is projection onto $I\subset\mathbb{R}$. Now, the
map $\Sigma$ induces a volume form
${\tilde{\mu}}=\Sigma^{*}\mu\in\Omega^{3}(S\times I)$ and vector field
${\tilde{B}}$ on $S\times I$ which is $\Sigma$-related to $B$. In particular,
we have the relations

$\displaystyle\mathcal{L}_{{\tilde{B}}}{\tilde{\mu}}$ $\displaystyle=0,$
$\displaystyle dt({\tilde{B}})$ $\displaystyle=0.$

On the other hand, with the vector field $\partial_{t}$ on $S\times I$,
observe that

$\iota_{\partial_{t}}(dt\wedge\iota_{\partial_{t}}{\tilde{\mu}})=(\iota_{\partial_{t}}dt)\iota_{\partial_{t}}{\tilde{\mu}}-dt\wedge\iota_{\partial_{t}}\iota_{\partial_{t}}{\tilde{\mu}}=\iota_{\partial_{t}}{\tilde{\mu}}.$

Thus, ${\tilde{\mu}}=dt\wedge\iota_{\partial_{t}}{\tilde{\mu}}$ so that

$0=\mathcal{L}_{{\tilde{B}}}(dt\wedge\iota_{\partial_{t}}{\tilde{\mu}})=(\mathcal{L}_{{\tilde{B}}}dt)\wedge\iota_{\partial_{t}}{\tilde{\mu}}+dt\wedge\mathcal{L}_{{\tilde{B}}}\iota_{\partial_{t}}{\tilde{\mu}}=dt\wedge\mathcal{L}_{{\tilde{B}}}\iota_{\partial_{t}}{\tilde{\mu}}.$

Thus with the induced vector field $B_{0}$ on $S_{0}=S\times\\{0\\}$ from
${\tilde{B}}$ and area form
$\mu_{0}=(S_{0})^{*}\iota_{\partial_{t}}{\tilde{\mu}}$ we have that
$\mathcal{L}_{B_{0}}\mu_{0}=0$. On the other hand, it is well known in general
(see for instance [10]) that any area-preserving vector field $X$ on a 2-torus
possess a closed 1-form $\eta_{0}$ such that $\eta_{0}(X)>0$. In particular,
fixing such an $\eta_{0}$ for $B_{0}$, via the projection $\pi:S\times I\to
S_{0}$, we obtain a closed 1-form $\tilde{\eta}=\pi^{*}\eta_{0}$ on $S\times
I$ such that $\tilde{\eta}({\tilde{B}})|_{S}>0$. Moreover, the set
$\tilde{U}=\\{x\in S\times I:\tilde{\eta}({\tilde{B}})>0\\}$ is an open
neighbourhood of $S_{0}$ in $S\times I$. The result now easily follows from
pushing forward $\tilde{\eta}$ into $M$ with $\Sigma$. ∎

We now establish an adapted 1-form in a neighbourhood of each axis of a
tangential flux system. This follows immediately from the following well-known
result.

######  Proposition III.11.

Let $X$ be a vector field on a manifold $Y$ and $\gamma$ a 1-dimensional
periodic orbit of $X$. Then, there exists a neighborhood $U$ of $\gamma$ and a
closed 1-form $\eta\in\Omega^{1}(U)$ such that $\eta(X)>0$.

###### Proof.

By periodicity, $\gamma$ must be an embedded 1-dimensional submanifold. Thus,
by the Tubular Neighbourhood Theorem there exists a neighbourhood $U^{\prime}$
of $\gamma$ and a retraction $r:U^{\prime}\to\gamma$. We also have the induced
vector field $\tilde{X}$ on $\gamma$ as $X$ is tangent to $\gamma$. Hence, we
have a 1-form $\tilde{\eta}\in\Omega^{1}(\gamma)$ with
$\tilde{\eta}(\tilde{X})>0$. Moreover, $\tilde{\eta}$ is closed as a top-form
on $\gamma$. Now, consider the closed 1-form
$\eta=r^{*}\tilde{\eta}\in\Omega^{1}(U^{\prime})$. Since $r$ is a retraction,
we have $\eta(X)>0$ on $\gamma$. Thus, $U=\\{x\in
U^{\prime}:\eta(X)|_{x}>0\\}$ is the required neighbourhood. ∎

As a corollary, we obtain the following lemma.

######  Lemma III.12.

Let $(B,\nu,\mu)$ be a tangential flux system on $M$. Let $\gamma$ be an axis.
Then, there exists a neighbourhood $U$ of $\gamma$ in $\mathcal{T}$ and a
closed 1-form $\eta\in\Omega^{1}(U)$ such that $\eta(B)>0$.

To patch together the local adapted 1-forms around regular tori and axes, we
require an adapted partition of unity which we now introduce. This may be done
on the closed region. Let $F$ denote the set of leaves and degenerate closed
leaves in $\mathcal{C}$.

######  Definition III.13.

Let $\\{U_{L}\\}_{L\in F}$ be an open cover of $\mathcal{C}$ such that
$U_{L}\supset L$ for all $L\in F$. A _partition of unity adapted to $\nu$ on
$\mathcal{C}$ and subordinated to $\\{U_{L}\\}_{L\in F}$_ is a family
$\\{\psi_{L}:\mathcal{C}\to\mathbb{R}\\}_{L\in A}$ of smooth compactly
supported non-negative functions such that

  1. 1.

the collection of supports $\\{\text{supp}(\psi_{L})\\}_{L\in F}$ is locally
finite and $\text{supp}(\psi_{L})\subset U_{L}$ for each $L\in F$,

  2. 2.

we have $\sum_{L\in F}\psi_{L}=1$, and

  3. 3.

for all $L\in F$, $d\psi_{L}\wedge\nu=0$.

We have the following existence of partitions of unity in the above sense. The
proof is given in Section III.3.

######  Proposition III.14.

Let $\\{U_{L}\\}_{L\in F}$ be an open cover of $\mathcal{C}$ such that
$U_{L}\supset L$ for all $L\in F$. Then, there exists a partition of unity
$\\{\psi_{L}:\mathcal{C}\to\mathbb{R}\\}_{L\in A}$ adapted to $\nu$ on
$\mathcal{C}$ and subordinated to $\\{U_{L}\\}_{L\in F}$.

With this, we may prove Theorem I.5.

###### Proof of Theorem I.5.

Let $F$ denote the set of all axes and regular tori. By Lemmas III.10 and
III.12, for each $L\in F$ we may choose a neighbourhood $U_{L}$ of $L$ in
$\mathcal{T}$ and a closed 1-form $\eta_{L}\in\Omega^{1}(U_{L})$ such that
$\eta_{L}(B)>0$. Then, since $F$ covers $\mathcal{T}$, the indexed set
$(U_{L})_{L\in F}$ is an open cover of $\mathcal{T}$. By Proposition III.14,
there exists a partition of unity
$\\{\psi_{L}:\mathcal{T}\to\mathbb{R}\\}_{L\in F}$ subordinated to
$\\{U_{L}\\}_{L\in F}$ and adapted to $\nu$. Define

$\eta=\sum_{L\in F}\psi_{L}\eta_{L}$

Then, because the $\psi_{L}$ are each non-negative and $\sum_{L\in
F}\psi_{L}=1$, we have

$\eta(B)>0.$

Lastly, we observe that

$d\eta=\sum_{L\in F}d\psi_{L}\wedge\eta_{L}$

so that, because $d\psi_{L}\wedge\nu=0$ for each $L\in F$, we have

$d\eta\wedge\nu=0.$

∎

###  III.3 Construction of the partition of unity and classification of the
toroidal region

Here we will construct the partition of unity and classify the closed region
$\mathcal{C}$ (thereby proving Proposition I.6). In Section III.3.1 we
establish the partition of unity for the case of a toroidal region without
boundary or axes. Then, in Section III.3.2, we will extend the result of
III.3.1 to include the situation where there are axes and boundary points.

####  III.3.1 Without boundary or axes

Let $M$ be a manifold without boundary of dimension $n$ and $\nu$ be a closed
1-form on $M$. Here, we will assume that the region $\mathcal{C}^{2}$
coincides with $M$ (so that $\mathcal{C}=\mathcal{C}^{2}=M$). Denote by
$\mathcal{F}$ the rank $n-1$ foliation by $\nu$ (consisting of all the regular
closed leaves). The following shows that non-injectivity of a flowout implies
compactness.

######  Lemma III.15.

Assume $M$ is connected and there exists a non-injective flowout $\Sigma$ of
some $S\in\mathcal{F}$ for some open interval $I$ containing $0$. Then, $M$ is
compact.

###### Proof.

Because $\Sigma$ is not injective but $\Sigma(\cdot,t):S\to M$ is injective
for each $t\in I$, there exists points $(x,s),(y,t)\in S\times I$ with
$\Sigma(x,s)=\Sigma(y,t)$ and $s<t$. Now by Lemma III.7, we have a flowout
$\Sigma^{\prime}$ of $S^{\prime}=\Sigma(S,s)\in\mathcal{F}$ with interval
$I^{\prime}=I-s$. With the point $x^{\prime}=\Sigma(x,s)\in S^{\prime}$ and
time $0<t^{\prime}=t-s\in I^{\prime}$ we get

$\Sigma^{\prime}(x^{\prime},t^{\prime})=\Sigma^{\prime}(\Sigma(x,s),t^{\prime})=\Sigma(x,t).$

On the other hand, since $x^{\prime}=\Sigma(y,t)$ we have $\Sigma(y,t)\in
S^{\prime}$. So that $\Sigma(x,t)\in S^{\prime}$ and hence
$\Sigma^{\prime}(x^{\prime},t^{\prime})\in S^{\prime}$. Altogether, we
conclude so far that $\Sigma^{\prime}(S^{\prime},t^{\prime})=S^{\prime}$.

Now, consider the compact subset

$K=\Sigma^{\prime}(S^{\prime}\times[0,t^{\prime}])\subset M.$

We show that $K$ is open. Because $\Sigma^{\prime}$ is an open map by Lemma
III.7, it suffices to find $\epsilon>0$ so that
$K=\Sigma^{\prime}(S^{\prime}\times(-\epsilon,t^{\prime}+\epsilon))$. To this
end, we consider the flowout $\Sigma^{\prime\prime}$ of
$S^{\prime}=\Sigma^{\prime}(S^{\prime},t^{\prime})$ with interval
$I^{\prime\prime}=I^{\prime}-t^{\prime}$ and make the following periodicity
argument. On the one hand, for $z\in S^{\prime}$ and $t^{\prime\prime}\in
I^{\prime\prime}$, by Lemma III.7 we have the identity

$\Sigma^{\prime\prime}(\Sigma^{\prime}(z,t^{\prime}),t^{\prime\prime})=\Sigma^{\prime}(z,t^{\prime}+t^{\prime\prime}).$

Since $\Sigma^{\prime}(S^{\prime}\times\\{t^{\prime}\\})=S^{\prime}$, we have
in particular that, for $t^{\prime\prime}\in I^{\prime\prime}$,

$\Sigma^{\prime\prime}(S^{\prime}\times\\{t^{\prime\prime}\\})=\Sigma^{\prime}(S^{\prime}\times\\{t^{\prime}+t^{\prime\prime}\\}).$

On the other hand, $I=I^{\prime}\cap I^{\prime\prime}$ is an open interval
containing $0$ and by Lemma III.7, for each $z\in S^{\prime}$ and $r\in I$, we
have

$\Sigma^{\prime\prime}(z,r)=\Sigma^{\prime}(z,r).$

Thus for each $r\in I$,

$\Sigma^{\prime\prime}(S^{\prime}\times\\{r\\})=\Sigma^{\prime}(S^{\prime}\times\\{r\\}).$

Now, for $\epsilon>0$ sufficiently small we have that
$[0,\epsilon),(t^{\prime}-\epsilon,t^{\prime}]\subset[0,t^{\prime}]$ and
$\pm\epsilon\in I$ so that the above relations imply

$\displaystyle\Sigma^{\prime}(S^{\prime}\times[t^{\prime},t^{\prime}+\epsilon))=\Sigma^{\prime\prime}(S^{\prime}\times[0,\epsilon))=\Sigma^{\prime}(S^{\prime}\times[0,\epsilon))\subset
K,$
$\displaystyle\Sigma^{\prime}(S^{\prime}\times(-\epsilon,0])=\Sigma^{\prime\prime}(S^{\prime}\times(-\epsilon,0])=\Sigma^{\prime}(S^{\prime}\times(t^{\prime}-\epsilon,t^{\prime}])\subset
K.$

This shows that

$K=\Sigma^{\prime}(S^{\prime}\times(-\epsilon,t^{\prime}+\epsilon)).$

Thus, $K$ is open, as claimed. To conclude, $M$ being connected implies $M=K$,
a compact subset of $M$. That is, $M$ is compact. ∎

With Lemma III.15 the following useful lemma can be proved. The lemma will be
used substantially in Section III.3 due to the convenient dichotomy it gives
rise to.

######  Lemma III.16.

Assume that $M$ is connected. Then, one of the following holds.

  1. 1.

For any $S\in\mathcal{F}$, there exists a flowout $\Sigma$ of $S$ along $N$
with an interval $I$ which is open and contains $0$ such that $\Sigma:S\times
I\to M$ is a diffeomorphism.

  2. 2.

$M$ is compact.

###### Proof.

Let $S\in\mathcal{F}$. Then, consider the subset $U$ of points $x\in M$ for
which there exists an open interval $I$ containing $0$ and a flowout $\Sigma$
of $S$ with interval $I$ such that $x\in\Sigma(S\times I)$. This set $U$ is
open because each flowout $\Sigma$ of $S$ with any open interval $I$
containing $0$ is an open map by Lemma III.7 and $U$ a union of images
$\Sigma(S\times I)$ of such maps.

We now show that $U$ is closed. Let $y\in M$ and consider the unique
$S^{\prime}\in\mathcal{F}$ containing $y$. Then, let
$\Sigma^{\prime}:S^{\prime}\times I^{\prime}\to M$ be a flowout of
$S^{\prime}$ with some interval $I^{\prime}$ containing $0$ and consider
$\Sigma^{\prime}(S^{\prime}\times I^{\prime})$. Suppose that
$\Sigma^{\prime}(S^{\prime}\times I^{\prime})\cap U\neq\emptyset$. Then, there
exists a flowout $\Sigma$ of $S$ with interval $I$ such that $\Sigma(S\times
I)\cap\Sigma^{\prime}(S^{\prime}\times I^{\prime})\neq\emptyset$. Then, there
must exist $t\in I$ and $t^{\prime}\in I^{\prime}$ such that
$\Sigma(S\times\\{t\\})=\Sigma^{\prime}(S^{\prime}\times\\{t^{\prime}\\})$.
Consider now the flowout $\Sigma^{\prime\prime}$ of
$S^{\prime\prime}=\Sigma(S\times\\{t\\})$ with interval
$I^{\prime\prime}=I-t^{\prime}$. Recall $\Sigma^{\prime\prime}$ has the
identity, for $x^{\prime}\in S^{\prime}$ and $t^{\prime\prime}\in
I^{\prime\prime}$,

$\Sigma^{\prime\prime}(\Sigma^{\prime}(x^{\prime},t^{\prime}),t^{\prime\prime})=\Sigma^{\prime}(x^{\prime},t^{\prime}+t^{\prime\prime}).$

Now, consider the open interval $\tilde{I}=I\cup(I^{\prime\prime}+t)$ and the
map $\tilde{\Sigma}:S\times\tilde{I}\to M$ given by

$\tilde{\Sigma}(x,s)=\begin{cases}\Sigma(x,s),&\text{if }s\leq t\\\
\Sigma^{\prime\prime}(\Sigma(x,t),s-t)&\text{otherwise}.\end{cases}$

It is clear that $\tilde{\Sigma}$ is a flowout of $S$ with interval
$\tilde{I}$. Lastly,

$\Sigma^{\prime}(S^{\prime}\times
I^{\prime})\subset\tilde{\Sigma}(S\times\tilde{I})\subset U.$

Thus, if $y\in M\backslash U$, then $\Sigma^{\prime}(S^{\prime}\times
I^{\prime})\subset M\backslash U$. Since $\Sigma^{\prime}$ is an open map and
$y\in\Sigma^{\prime}(S^{\prime}\times I^{\prime})$, this is a neighbourhood of
$y$ contained in $M\backslash U$. Thus, $U$ is closed as claimed. Lastly,
since $U\neq\emptyset$ and $M$ is connected, we must have that $U=M$.

Thus, for every $y\in M$, we may choose an open interval $I_{y}$ containing
$0$ such that the flowout $\Sigma_{y}$ of $S$ with interval $I_{y}$ is such
that there exists $x\in S$ and $t\in\iota_{x}$ such that $\Sigma_{y}(x,t)=y$.
Then, consider the open interval $I=\cup_{y\in M}I_{y}$ containing $0$ and the
map $\Sigma:S\times I\to M$ defined by, for $x\in S$,

$\Sigma(x,t)=\Sigma_{y}(x,t)\text{ if there exists }y\in M\text{ such that
}t\in I_{y}.$

It is clear that this is well-defined and is a flowout of $S$ with interval
$I$. Moreover, $\Sigma$ is onto $M$. Supposing that $M$ is not compact, by
Lemma III.15, $\Sigma$ must be injective. Thus, because $\Sigma$ is a
bijection and local diffeomorphism, $\Sigma$ must be a diffeomorphism. ∎

This enables us to establish the partition of unity in the present situation.

######  Lemma III.17.

Let $\\{U_{S}\\}_{S\in\mathcal{F}}$ be an open cover of $M$ such that for each
$S\in\mathcal{F}$, $U_{S}\supset S$. Then, there exists a partition of unity
$\\{\psi_{S}:M\to\mathbb{R}\\}_{S\in\mathcal{F}}$ adapted to $\nu$ on $M$ and
subordinated to $\\{U_{S}\\}_{S\in\mathcal{F}}$.

###### Proof.

First, assume that $M$ is connected and compact. For each $S\in\mathcal{F}$,
an open interval $I_{S}$ containing $0$ with a flowout
$\Sigma_{S}^{\prime}:S\times I_{S}^{\prime}\to M$ which is a diffeomorphism
onto its open image in $M$. Now, for each $S\in\mathcal{F}$,
${\Sigma^{\prime}_{S}}^{-1}(U_{S})$ is a neighbourhood of $S\times\\{0\\}$ in
$S\times I_{S}^{\prime}$. Thus, we may choose an open interval $I_{S}\subset
I_{S}^{\prime}$ containing $0$ such that $\Sigma_{S}^{\prime}(S\times
I_{S})\subset U_{S}$ and hence a flowout $\Sigma_{S}:S\times I_{S}\to M$ which
is a diffeomorphism onto its open image in $M$ contained in $U_{S}$. Now, for
each $S\in\mathcal{F}$, choose a smooth bump function
$\psi_{S}^{\prime}:I_{S}\to\mathbb{R}$ with $0\leq\psi_{S}^{\prime}\leq 1$
which is compactly supported and $\psi_{S}^{\prime}(0)=1$. Then we obtain the
function $\psi_{S}:S\times I_{S}\to\mathbb{R}$ given by
$\psi_{S}(x,t)=\psi_{S}^{\prime}(t)$ which is compactly supported in $S\times
I_{S}$. Moreover, if $t_{S}:S\times I_{S}\to\mathbb{R}$ denotes projection
into $\mathbb{R}\supset I_{S}$, we have the relation

$d\psi_{S}\wedge dt_{S}=0.$

By use of $\Sigma_{S}$, this then gives a function
$\rho_{S}^{\prime}:M\to\mathbb{R}$ which $1$ on $S$ and is compactly supported
in $U_{S}$ and additionally satisfies

$d\rho_{S}^{\prime}\wedge\nu=0.$

Now, the open subsets $V_{S}=\\{\rho_{S}^{\prime}\neq 0\\}\subset U_{S}$ form
a covering of $M$ and so there exists $N\in\mathbb{N}$ and leaves
$S_{1},...,S_{N}\in\mathcal{F}$ such that $(V_{S_{i}})_{i=1}^{N}$ is a
covering of $M$. Then we obtain the smooth positive function

$P=\sum_{i=1}^{N}\rho_{S_{i}}.$

Lastly, we define a family of smooth functions
$\\{\rho_{S}:M\to\mathbb{R}\\}_{S\in\mathcal{F}}$ as follows. If $S=S_{i}$ for
some $i\in\\{1,...,n\\}$, then set $\rho_{S}=\rho_{S_{i}}^{\prime}/P$ and
otherwise, set $\rho_{S}=0$. Then, clearly this is a partition of unity
subordinated to the open cover $\mathcal{O}$ which, by virtue of the identity
$dP\wedge\nu=0$, additionally satisfies

$d\rho_{S}\wedge\nu=0$

for all $S\in\mathcal{F}$.

Now assume that $M$ is connected and non-compact. In this case, we may fix an
open interval $I$ containing $0$, a leaf $S^{*}\in\mathcal{F}$, and a flowout
$\Sigma:S^{*}\times I\to M$ of $S^{*}$ which is a diffeomorphism. Then, we
have the open cover $\\{\Sigma^{-1}(U_{S})\\}_{S\in\mathcal{F}}$ of
$S^{*}\times I$. Now, for $z\in I$, setting $S_{z}=\Sigma(S^{*},z)$,
$\Sigma^{-1}(U_{S_{z}})$ is a neighbourhood of $S^{*}\times\\{z\\}$. Thus, for
each $z\in I$, (using compactness of $S^{*}$), we may choose an open interval
$J_{z}$ of $z$ such that $S^{*}\times J_{z}\subset\Sigma^{-1}(U_{S_{z}})$.
Now, fix a partition of unity $\\{\psi_{z}^{\prime}:I\to\mathbb{R}\\}_{z\in
I}$ subordinated to the open cover $\\{J_{z}\\}_{z\in I}$. For each $z\in I$,
set $\psi_{z}:S^{*}\times I\to\mathbb{R}$ where
$\psi_{z}(x,t)=\psi_{z}^{\prime}(t)$. Then, for each $z\in I$, we have the
identity

$d\psi_{z}\wedge dt=0$

where $t:S^{*}\times I\to\mathbb{R}$ is projection into $\mathbb{R}\supset I$.
Using the diffeomorphism $\Sigma$, and in particular that $I\ni z\mapsto
S_{z}\in\mathbb{F}$ is a bijection, this gives a partition of unity
$\\{\rho_{S}:M\to\mathbb{R}\\}_{S\in\mathcal{F}}$ adapted to $\nu$ on $M$ and
subordinated to $\\{U_{S}\\}_{S\in\mathcal{F}}$.

In summary, this shows that the proposition is true if $M$ is connected. The
general case now follows by considering the connected components of $M$. ∎

We will now extend the foregoing to account for a general closed region
containing both boundary and degenerate closed leaves.

####  III.3.2 With boundary and axes

Let $M$ be a manifold with boundary and $\nu$ be a closed 1-form such that
$\partial M^{*}\nu=0$. If $U\subset\mathcal{C}$ is an open subset partitioned
by closed leaves and degenerate closed leaves of $\nu$, then the 1-form
$U^{*}\nu$ is closed and satisfies $(\partial U)^{*}(U^{*}\nu)=0$ since
$\partial U=\partial M\cap U$. Moreover, the closed region $\mathcal{C}_{U}$
of $U^{*}\nu$ is all of $U$. Let $F_{U}$ denote the set of closed leaves and
degenerate closed leaves in $U$. Let $P(U)$ denote the following statement:

> “For any open cover $\\{U_{L}\\}_{L\in F_{U}}$ of $U$ such that
> $U_{L}\supset L$ for all $L\in A$, there exists a partition of unity adapted
> to $\nu$ on $U$ and subordinated to $\\{U_{L}\\}_{L\in F_{U}}$.”

######  Lemma III.18.

Let $C$ be a connected component of $\mathcal{C}$. Then
$\operatorname{int}C\backslash\mathcal{C}^{1}$ is a connected open
submanifold. Let $K$ be a degenerate closed leaf contained in $C$. Then, for
any neighbourhood $U_{K}$ of $K$, there exists a codimension 0 compact
connected manifold $R$ with boundary such that $\partial R$ is a closed leaf
of $\nu$ and $K\subset\operatorname{int}R\subset U_{K}$. Moreover,
$P(C\backslash K)\Rightarrow P(C)$.

The proof of Lemma III.18 relies upon the following relatively well-known
result.

######  Proposition III.19.

Let $X$ be a connected manifold and $K$ be an embedded submanifold of $X$ of
codimension at least $2$ which is also a closed as a subset of $X$. Then
$X\backslash K$ is connected.

Although Proposition III.19 may hold without assuming that $K$ is a closed
subset, it is sufficient for our purposes to ignore this greater generality.
We now prove Lemma III.18.

###### Proof of Lemma III.18.

First observe that $\operatorname{int}C=C\backslash\partial C$ is connected.
Now, $\mathcal{C}^{1}\cap\operatorname{int}C$ is a union of (disjoint)
degenerate closed leaves where each leaf has a neighbourhood isolating it from
the rest. On the other hand, $\mathcal{C}^{1}\cap C$ is closed in $C$ (being
the critical set of $\nu$ in $C$) and thus
$\mathcal{C}^{1}\cap\operatorname{int}C=\mathcal{C}^{1}\cap C$ is closed in
$\operatorname{int}C$. Hence, any sub-union of these leaves is also a closed
subset of $\operatorname{int}C$. We can write a union

$\mathcal{C}^{1}\cap\operatorname{int}C=K_{0}\cup...\cup K_{n-2}$

where $K_{i}$ is the union of all degenerate closed leaves of dimension $i$
for $i=0,...,n-2$. In particular, each $K_{i}$ is an embedded submanifold of
$C$ of dimension $i$ which is also a closed subset of $C$. Thus, we can use
Proposition III.19 inductively (on $i$) to infer that
$\operatorname{int}C\backslash\mathcal{C}^{1}$ is a connected open
submanifold.

As preliminaries for the remaining, consider
$C_{0}=(\operatorname{int}C\backslash\mathcal{C}^{1})\cup K$ which is also a
connected open submanifold from the above argument. Because $K$ is embedded in
$C_{0}$, by the Tubular Neighbourhood Theorem, there exists a neighbourhood
$U$ of $K$ diffeomorphic to the normal bundle $\mathcal{V}(K)$ of $K$ in
$C_{0}$. Since $U$ deformation retracts to $K$ where $K^{*}\nu=0$ is in
particular exact, $\nu$ is exact in $U$. Moreover, $C_{0}\backslash K$ is
connected by Proposition III.19. Thus, Lemma III.16 gives the existence of a
flowout $\Sigma$ of some closed leaf $S$ of $\nu$ in $C_{0}$ with some open
interval $I$ containing $0$ which is a diffeomorphism $S\times I\to
C_{0}\backslash K$. Recall also that $\Sigma^{*}\nu=dt$ where $t:S\times
I\to\mathbb{R}$ is projection into $\mathbb{R}\supset I$. Altogether, $\nu$ is
also exact on $C_{0}\backslash K$. Lastly, because $U\cap(C_{0}\backslash
K)=U\backslash S$ is connected, $\nu$ is exact on $C_{0}$. Again, because
$K^{*}\nu=0$, there exists a unique $f:C_{0}\to\mathbb{R}$ satisfying $\nu=df$
and $f|_{K}=0$. The formula $\Sigma^{*}\nu=dt$ shows that there exists at most
one closed leaf $S^{\prime}$ in $C_{0}$ for which $(f|_{C_{0}\backslash
K})^{-1}(0)=S^{\prime}$. If there is such an $S^{\prime}$, set $C_{1}$ to be
the connected component $C_{1}$ of $C_{0}\backslash S^{\prime}$ containing
$K$. Otherwise, set $C_{1}=C_{0}$.

We will now simultaneously show the existence of such $R$ in the lemma and
prepare for the implication involving partitions of unity. For this, fix an
open neighbourhood $U_{K}$ of $K$.

The set $K$ is the set of global minima for $\tilde{f}=f|_{C_{1}}$ (or maxima
and in which case, perform the trivial adjustments to the remainder of the
proof). We may apply the Tubular Neighbourhood Theorem in $U_{K}\cap C_{1}$ to
obtain a neighbourhood $\tilde{U}$ of $K$ contained in $U_{K}$ diffeomorphic
to the normal bundle $\mathcal{V}_{1}(K)$ of $K$ in $C_{1}$. We may then fix a
disk bundle $D(\mathcal{V}_{1}(K))$ of $\mathcal{V}_{1}(K)$ [20]; this is a
fibre bundle over $K$ with typical fibre the (closed) unit disk $D$ and is
thus compact. Moreover, $D(\mathcal{V}_{1}(K))$ is a smooth manifold with
boundary. Using the diffeomorphism $\mathcal{V}_{1}(K)\to\tilde{U}$, we thus
obtain a codimension $0$ compact manifold $\tilde{R}$ with boundary such that
$K\subset\operatorname{int}\tilde{R}$. Observe by the choice of $C_{1}$ that
$\tilde{f}$ does not obtain its minimum of $0$ on $\partial\tilde{R}$.
Denoting by $y$ the positive minimum of $\tilde{f}$ on $\partial\tilde{R}$,
the value $y/2$ (for instance) is a regular value of $\tilde{f}$ in
$\tilde{R}$. Thus, we obtain the compact manifold $R=\\{x\in\tilde{R}:f(x)\leq
y/2\\}$ with boundary $\partial R=\\{x\in\tilde{R}:f(x)=y/2\\}$ is a compact
regular level set of $\tilde{f}$ in $\tilde{R}$. It is then easy to see that
$\partial R$ is a closed leaf of $\nu$. Now, consider a smooth function
$h:\mathbb{R}\to\mathbb{R}$ with $0\leq h\leq 1$ supported in $(-y/2,y/2)$
satisfying $h(x)=1$ for $x\in(-y/3,y/3)$ and set $\psi_{k}=h\circ
f:C\to\mathbb{R}$. This is a smooth function with $0\leq\psi_{k}\leq 1$ which
is compactly supported in $U_{K}$ with $\psi_{K}=1$ in a neighbourhood of $K$.

Now, suppose the statement $P(C\backslash K)$ is true. Let $A$ denote the set
of closed leaves and degenerate closed leaves in $C$ and
$\\{U_{\alpha}\\}_{\alpha\in A}$ be an open cover of $C$ whereby
$U_{\alpha}\supset\alpha$ for each $\alpha\in A$. Set
$\tilde{A}=A\backslash\\{K\\}$ and consider the open cover
$\\{U_{\alpha}\cap(C\backslash K)\\}_{\alpha\in\tilde{A}}$ of $C\backslash K$.
Because $P(C\backslash K)$ holds, there exists a partition of unity
$\\{\tilde{\psi}_{\alpha}\\}_{\alpha\in A}$ adapted to $\nu$ on $C\backslash
K$ and subordinated to $\\{U_{\alpha}\\}_{\alpha\in A}$. Then, considering the
neighbourhood $U_{K}$ of $K$ in the open cover $\\{U_{\alpha}\\}_{\alpha\in
A}$, by the above, there exists a smooth function with $0\leq\psi_{k}\leq 1$
which is compactly supported in $U_{K}$ with $\psi_{K}=1$ in a neighbourhood
of $K$. Thus, we get a partition of unity adapted to $\nu$ on $C$ subordinated
to $\\{U_{\alpha}\\}_{\alpha\in A}$ given by
$\\{\psi_{K}\\}\cup\\{\psi_{\alpha}\\}_{\alpha\in\tilde{A}}$ where

$\psi_{\alpha}=(1-\psi_{K})\tilde{\psi}_{\alpha},~{}\alpha\in\tilde{A}$

and multiplication at points of $K$ is interpreted as $0$. ∎

######  Lemma III.20.

Let $C$ be a connected component of $\mathcal{C}$ let $S$ be a connected
component of $\partial C$ (and thus a closed leaf of $\nu$). Then, we have the
implication $P(C\backslash S)\Rightarrow P(C)$.

###### Proof.

This follows directly from considering a flowout from the boundary using a
similar algebraic construction to that of the end of the proof of Lemma
III.18. ∎

####  III.3.3 Classification of the closed region and the toroidal region

Our classification for $\mathcal{C}$ is the following.

######  Lemma III.21.

Let $C$ be a connected component of $\mathcal{C}$. Then the following holds.

  1. 1.

If $C$ is without boundary and contains no degenerate closed leaves, then $C$
is either compact or there exists a regular closed leaf $S$ such that $C$ is
diffeomorphic to $S\times I$ via $\Sigma$, a flowout of $S$ with interval $I$.

  2. 2.

Otherwise, there exists some regular closed leaf $S$ such that
$C\backslash(\partial C\cup\mathcal{C}^{1})$ is diffeomorphic to $S\times I$
via $\Sigma$, a flowout of $S$ with interval $I$. Furthermore, the combined
sum of closed leaves in $\partial C$ and degenerate closed leaves contained in
$C$ is at most $2$. If this sum is realised, then $C$ is compact.

###### Proof of Lemmas III.21 and I.6.

The first case follows immediately from Lemma III.16. For the second case,
first observe that $\partial C\cup(\mathcal{C}^{1}\cap C)$ is a closed subset
of $C$. Hence, $C\backslash(\partial C\cup\mathcal{C}^{1})$ is not compact
because otherwise $\partial C\cup\mathcal{C}^{1}$ would be open, in turn
implying that $\operatorname{int}C\backslash\mathcal{C}^{1}$ has an proper
subset that is both open and closed. This contradicts (at least) the
connectedness of $\operatorname{int}C\backslash\mathcal{C}^{1}$ from Lemma
III.18. Hence, $\operatorname{int}C\backslash(\partial C\cup\mathcal{C}^{1})$
is non-compact and connected. By Lemma III.16, this implies that
$\operatorname{int}C\backslash(\partial C\cup\mathcal{C}^{1})$ is
diffeomorphic to $S\times I$ via $\Sigma$, a flowout of some closed leaf
$S\subset\operatorname{int}C\backslash(\partial C\cup\mathcal{C}^{1})$ with
some open interval $I$ containing $0$.

We now claim the following: if there are at least 2 closed leaves in $\partial
C$ and degenerate closed leaves in $C$, then there are exactly $2$.

###### Proof of the claim.

We will treat the case of there being at least two closed leaves in $\partial
C$. The cases involving axes are similar: the role of a leaf in our case is
played by $\partial R$ by suitably small choice of $R$ as in Proposition
III.18 in the other cases. With this, let $L_{1},L_{2}$ be two distinct such
leaves. By the first paragraph and Lemmas III.6 and III.7, there exist
flowouts connecting each of these closed leaves to a leaf in
$\operatorname{int}C\backslash(\partial C\cup\mathcal{C}^{1})$ which generates
all of $\operatorname{int}C\backslash(\partial C\cup\mathcal{C}^{1})$. From
these connecting flowouts, it is easy to see that $L_{1}\cup
L_{2}\cup\operatorname{int}C\backslash(\partial C\cup\mathcal{C}^{1})$ is
compact, open, and connected in $C$. Hence,

$C=L_{1}\cup L_{2}\cup\operatorname{int}C\backslash(\partial
C\cup\mathcal{C}^{1}).$

So $\partial C$ consists of exactly the two closed leaves $L_{1}$ and $L_{2}$
and there are no degenerate closed leaves. ∎

Proposition I.6 now follows from the above proof after direct adaptations of
Lemmas III.6 and III.7 to flowouts of leaves from $\partial\mathcal{T}$ and
the diffeomorphism flowouts in the connected components of
$\operatorname{int}\mathcal{T}^{2}$ together with the following observations.
Recall, in this situation $M$ is orientable and of dimension $3$ and of
course, the regular tori (or closed leaves) in $\mathcal{T}$ are each 2-tori.
Lastly, concerning axes in $\mathcal{T}$, the $R$ which exist in Lemma III.18
must be solid tori. Indeed, the vector bundles of fixed dimension over the
circle have only two isomorphism classes with only the trivial class being
orientable. Moreover, from the proof of Lemma III.18, $\tilde{R}$ is a disk
subbundle of such a vector bundle making $\operatorname{int}\tilde{R}$
diffeomorphic to a orientable vector bundle over the circle. Thus, because $R$
is embedded in $\operatorname{int}R$ which must be an open solid torus, $R$
may be embedded in $\mathbb{R}^{3}$ and has a torus boundary, making $R$ a
solid torus. This establishes the claim. ∎

##  IV An obstruction to the global existence of adapted 1-forms

Reeb cylinders, and more generally Kuperberg plugs, were identified in [12] as
obstructions to the existence of MHS fields in arbitrary geometries. In this
section, we will use a similar argument to show that the existence of Reeb
cylinders are an obstruction to the global existence of an adapted 1-form.

######  Definition IV.1.

A _Reeb cylinder_ is a pair $(C,X)$ where $C$ is an oriented cylinder and $X$
is a non-vanishing vector field on $C$ tangent to $\partial C$ such that,
considering the connected components $C_{0},C_{1}$ of $\partial C$ whereby, as
a 1-chain, $\partial C=C_{0}-C_{1}$, the following hold:

  1. 1.

$X$ is positively (respectively negatively) oriented along the 1-chain $C_{0}$
and $C_{0}$ is the alpha-limit of each orbit except $C_{1}$.

  2. 2.

$X$ is negatively (positively) oriented along the 1-chain $C_{1}$ and is the
omega-limit for every orbit except $C_{0}$.

The following elementary principle shows why Reeb cylinders are obstructions.

######  Proposition IV.2.

Let $M$ be a 3-manifold with (possibly empty) boundary and $(B,\nu,\mu)$ a
flux system on $M$. Suppose that $(B,\nu,\mu)$ possesses a non-trivial Reeb
cylinder: that there exists a cylinder $C$ immersed in $M$ which $B$ is
tangent to and such that $C$ together with the induced vector field $X$ on $C$
forms a Reeb cylinder, and the critical points of $\nu|_{C}$ are nowhere dense
on $C$. Then, there cannot exist a 1-form adapted to $(B,\nu,\mu)$ on $M$.

###### Proof.

Suppose that $(B,\nu,\mu)$ possesses a Reeb cylinder $(C,X)$. Then, observe
that $C^{*}\nu$ is closed on $C$. Moreover, the periodic orbits of $B$ assumed
on $\partial C$ generate the homology of $C$. Hence, from the fact that
$\nu(B)=0$, we get that $C^{*}\nu=df$ is exact on $C$. However, because of the
orbit structure of $X$ and $df(X)=0$, we have that $f$ is constant. Thus,
$C^{*}\nu=0$.

Now, suppose that $\eta$ is a 1-form on $M$ such that $d\eta\wedge\nu=0$. We
will show that we cannot have $\eta(B)|_{\partial C}>0$ and thus there are no
adapted 1-forms. Indeed, at points $x\in C$ where $\nu|_{x}\neq 0$, because
$C^{*}\nu|_{x}=0$, we obtain that $C^{*}d\eta|_{x}=0$. Since such points are
dense in $C$ by assumption, we conclude that $C^{*}d\eta=0$ so that
$C^{*}\eta$ is a closed 1-form on $C$. Now, regarding $C$ as a 2-chain,
Stokes’ Theorem says

$0=\int_{C}d\eta=\int_{C_{0}}\eta-\int_{C_{1}}\eta.$

However, if $\eta(B)|_{\partial C}>0$, then by the orientation of $X$ along
$\partial C$, accounting for the two cases of orientation of the periodic
orbits, the quantity

$\int_{C_{0}}\eta-\int_{C_{1}}\eta$

is either strictly positive or strictly negative. Thus, $\eta(B)|_{\partial
C}>0$ cannot hold. ∎

We will now exhibit an example of a flux system on the solid torus which
possesses two (non-trivial) Reeb cylinders as depicted in Fig. 3. Despite this
implying that an adapted 1-form $\eta$ cannot globally exist, this example
still has a large amount of additional structure such as a global symmetry.
The example owes its additional structure to the method of its construction,
which was by explicitly carrying out steps suggested in [13, Lemma 5].

Figure 3: The invariant torus $x^{2}+y^{2}=1$ of the flux system $(B,\mu,dH)$
in Example IV.3 embedded into $\mathbb{R}^{3}$. The invariant torus
$x^{2}+y^{2}=1$ contains two Reeb cylinders of the flux system which intersect
in the periodic orbits at the top and the bottom of the torus.

######  Example IV.3.

Set $M=D\times\mathbb{R}/\mathbb{Z}$ where
$D=\\{(x,y):x^{2}+y^{2}\leq\sqrt{2}\\}$ is the closed disk of radius
$\sqrt{2}$ and use the standard coordinates $(x,y,\varphi)$ with
$\varphi:M\to\mathbb{R}/\mathbb{Z}$ denoting projection onto the final factor
of $M$. Consider the vector field $B$ given by

$B=-H_{y}\partial_{x}+H_{x}\partial_{y}+f\partial_{\varphi}$

where the Hamiltonian component of $B$ is generated by the function

$H=(x^{2}+y^{2}-1)(1+y^{2}(x^{2}+y^{2}-2))$

and the toroidal component of $B$ is given by the function

$f=y+(x^{2}+y^{2}-1).$

Observe that $B$ is divergence-free with respect to the volume form
$\mu=dx\wedge dy\wedge d\varphi$. Furthermore, the vector field
$u=\partial_{\varphi}$ is also divergence-free with respect to $\mu$ and
satisfies $\iota_{u}\iota_{B}=dH$. It follows that $u$ is a symmetry of the
flux system $(B,\mu,dH)$ and, since $H$ is constant and regular on $\partial
M$, the flux system is tangential.

We now discuss the topological structure of $H$ and $B$. Concerning $H$, it
suffices to consider the function $h$ defined by the same formula as $H$
except on $D$ and we have the following.

######  Proposition IV.4.

The critical points of $h$ are $(0,0)$ and $(0,\pm 1)$. Moreover, the critical
point $(0,0)$ constitutes the level set $h^{-1}(-1)$ as the minimising set for
$h$. The critical points $(0,\pm 1)$ belong to the level set
$h^{-1}(0)=\\{x^{2}+y^{2}=1\\}$ and the remaining level sets are regular for
$h$ and are circles.

Thus, the level sets are nested in the sense that: for each level set
$h^{-1}(z)$ with $z\neq-1$, $h^{-1}(z)=\partial A_{z}$ where $A_{z}=\\{H\leq
z\\}$ is (diffeomorphically) a disk and for $z^{\prime}<z$, we have the
containment $A_{z^{\prime}}\subset\operatorname{int}A_{z}$ and in particular
$h^{-1}(z^{\prime})\subset\operatorname{int}A_{z}$ so that $h^{-1}(z)$
“contains $h^{-1}(z^{\prime})$ in its interior”.

###### Proof.

First, note that

$h_{x}=2x(2y^{4}+(2x^{2}-3)y^{2}+1),\qquad
h_{y}=2y(x^{4}+(4y^{2}-3)x^{2}+3(y^{2}-1)^{2}).$

In particular, the factor

$x^{4}+(4y^{2}-3)x^{2}+3(y^{2}-1)^{2}=3(y^{2}-1)^{2}+4x^{2}(y^{2}-1)+x^{4}+x^{2}$

is a quadratic in $y^{2}-1$ with discriminant $4x^{2}(x^{2}-3)$. For $(x,y)\in
D$, this discriminant is non-negative if and only if $x=0$. From here, one may
easily deduce that the only critical points of $h$ are $(0,0)$ and $(0,\pm 1)$
whereby $h(0,0)=-1$ and $h(0,\pm 1)=0$. Clearly $h|_{\partial D}=1$ and so
$(0,0)$ is the unique global minimiser of $h$ so that
$h^{-1}(-1)=\\{(0,0)\\}$. Moreover, $h$ may alternatively be written as

$h=(x^{2}+y^{2}-1)(y^{4}+(x^{2}-2)y^{2}+1)$

and a similar quadratic argument as before shows that
$h^{-1}(0)=\\{x^{2}+y^{2}=1\\}$. Now, the level set $h^{-1}(z)$ for $z\neq
0,1$ is regular. Let $L$ be a connected component of $h^{-1}(z)$ so that $L$
is (diffeomorphically) a circle and let $A$ be the unique disk with $\partial
A=L$. Then, $\operatorname{int}A$ must contain a critical point of $h$. If
$\operatorname{int}A$ contains one the $(0,\pm 1)$ critical points, then
because $L\cap\\{x^{2}+y^{2}=1\\}=\emptyset$, we have that
$A\cap\\{x^{2}+y^{2}=1\\}=\operatorname{int}A\cap\\{x^{2}+y^{2}=1\\}$ and thus
$\operatorname{int}A\cap\\{x^{2}+y^{2}=1\\}$ is closed and open in
$\\{x^{2}+y^{2}=1\\}$ and thus $\operatorname{int}A$ contains all of
$\\{x^{2}+y^{2}=1\\}$ and thus all of $\\{x^{2}+y^{2}<1\\}$ and in particular
the critical point $(0,0)$. Thus, in general $\operatorname{int}A$ contains
$(0,0)$. Hence, by compactness, of $\\{(t,0):t\geq 0\\}\cap A$, have a non-
empty intersection $\partial A\cap\\{x\geq 0,y=0\\}\neq\emptyset$. Hence,
every connected component of every level set intersects the curve
$\gamma(t)=(t,0)$ ($0\leq t\leq\sqrt{2}$). On the other hand,
$h(\gamma(t))=t^{2}-1$ which is injective on $\\{0\leq t\leq\sqrt{2}\\}$.
Hence, the level sets of $h$ are connected. ∎

Now addressing the topology of $B$, first observe that $B$ has no zeros: for
the critical set by the above of $H$ is $\\{(0,\pm
1,\varphi),(0,0,\varphi):\varphi\in\mathbb{R}/\mathbb{Z}\\}$ and there we have
the equalities

$f(0,0,\varphi)=-1,\qquad f(0,\pm 1,\varphi)=\pm
1,\qquad\varphi\in\mathbb{R}/\mathbb{Z}.$

Lastly, we claim that the cylinders

$C_{\pm}=\\{(x,y,\varphi)\in M\,|\,x^{2}+y^{2}=1,\,\pm x\geq 0\\}$

are Reeb cylinders. Indeed, from the fact that $H$ is a first integral of $B$
and $C_{\pm}\subset H^{-1}(0)$ with $dH|_{\operatorname{int}C_{\pm}}$ being
non-vanishing, we have that $B$ is tangent to $C_{\pm}$. In fact, concerning
$\partial C_{\pm}$, one may compute that

$B|_{(0,\pm
1,\varphi)}=\pm\partial_{\varphi},\qquad\varphi\in\mathbb{R}/\mathbb{Z}$

so that $\partial C_{\pm}$ consists of two periodic orbits of the required
opposite orientation for $C_{\pm}$ to be a Reeb cylinder. Lastly, on
$\operatorname{int}C_{\pm}$, using that $x^{2}=1-y^{2}$, we have

$dy(B)=H_{x}=2x^{3}$

and $\pm 2x^{3}>0$ on $C_{\pm}$. Hence, we obtain the required alpha and omega
limit property of the orbits in $\operatorname{int}C$. In conclusion, $C$ is a
Reeb cylinder.

To summarise, Example IV.3 contains (non-trivial) Reeb cylinders and this
obstructs the existence of an adapted 1-form. Despite this obstruction,
Example IV.3 has many nice properties, such as being non-vanishing, admitting
a symmetry, and possessing nested invariant tori. The resolution between this
example and our main result Theorem I.5 is that $H$ has critical points at the
periodic orbits of the Reeb cylinder: if $H$ had no critical points, there
would be a contradiction with Lemma III.12.

##  V Flux-coordinates and rectifiability

In the study of magnetically confined plasma, it is often convenient to use
special coordinates adapted to the magnetic field, so called _magnetic_ or
_flux-coordinates_ [21, 22]. The main advantage of flux-coordinates is that
field-lines are straight. From a computational point of view, magnetic
differential equations in flux-coordinates have a simple form and can be
solved by means of Fourier series in the corresponding poloidal and toroidal
angles. Most notably, in the study of magneto-hydrostatics (MHS), particular
choices of flux-coordinates are extensively used, such as Hamada coordinates
[23], PEST coordinates [24], Boozer coordinates [25], etc. The methods
presented in this paper provide the means of proving the existence of flux-
coordinates without assuming that the magnetic field is MHS nor that the
magnetic field is quasi-symmetric, nor that it has any symmetry by assumption.

In this section, quick and rigorous proofs of the existence of flux-
coordinates are given for flux systems in general (Corollaries V.4 and V.9).
It is highlighted that flux-coordinates all come about because of conformal
symmetries; different coordinates correspond to different pairs of commuting
vector fields. It is important to note that the full machinery of Theorem I.5
is not necessary. We only need the technical Lemma III.10, the elementary
Proposition III.11, and the algebraic Theorem I.3. The work in [3] is a
particular case of the results in this section. In Section V.1, we will focus
on flux-coordinates near regular tori. In section Section V.2, we consider
flux-coordinates near axes relying heavily on the work done in [3]. From this
point on in this section, fix a 3-manifold $M$ with boundary.

To establish flux-coordinates, in relation to symmetries, we remark the
following.

######  Remark V.1.

If $(B,\nu,\mu)$ is a flux system on a manifold $M$ with boundary and $\eta$
is an adapted 1-form, then Theorem I.3 implies that that
$({\tilde{B}},\nu,{\tilde{\mu}})$ given by

${\tilde{B}}=B/f,\qquad{\tilde{\mu}}=f\mu,\qquad f=\eta(B)$

is also a flux system on $M$. Moreover, the vector field $u$ satisfying

$\iota_{u}\mu=(\nu\wedge\eta)/f$

is such that

$\iota_{u}\iota_{{\tilde{B}}}{\tilde{\mu}}=\iota_{u}\iota_{B}\mu=\nu,\qquad[u,{\tilde{B}}]=0.$

An adapted 1-form is locally accessible for many flux systems $(B,\nu,\mu)$:
if the flux system is tangential and $L$ is a regular torus, or if $L$ is a
periodic orbit of $B$ contained in $\operatorname{int}M$, by Lemmas III.10 and
III.11 there exists a neighborhood $\mathcal{U}$ of $L$ and a closed 1-form
$\eta\in\Omega^{1}(\mathcal{U})$ such that $\eta(B)>0$. Hence, $\eta$ is
adapted to $(B,\nu,\mu)$ in $\mathcal{U}$ and thus, one may rescale the
tangential flux system on $\mathcal{U}$ and obtain a symmetry as concluded
above.

In plasma physics, there are choices of an adapted 1-form which may
canonically be available. For instance, if $M$ is an oriented Riemannian
3-manifold with boundary, then a non-vanishing divergence-free vector field
$B$ satisfying

$B\cdot\nabla\psi=0,\qquad J\cdot\nabla\psi=0,$

where $\psi$ is a smooth function and $J=\nabla\times B$, has $\eta=B^{\flat}$
as an adapted 1-form. Then, ${\tilde{B}}=B/B^{2}$ and $u=\nabla\psi\times
B/B^{2}$ are commuting vector fields. The resulting coordinates (when $B$ and
$J$ are tangent to $\partial M$) on regular tori generated by $\psi$ are known
as Boozer-coordinates [25]. This was observed in [10] and in [3] for the case
that $B$ is MHS; namely $J\times B=\nabla\psi$.

Another relevant and common adapted 1-form in the context of toroidal
confinement is $\eta=d\varphi=(-ydx+xdy)/(x^{2}+y^{2})$, where, in an abuse of
notation, $\varphi(x,y,z)=\arctan(y/x)$ is the geometric toroidal angle of a
solid torus $M$ embedded in $\mathbb{R}^{3}$ away from the $z$-axis. The
1-form $d\varphi$ is closed (but deceivingly not exact) and the assumption
that $B^{\varphi}=d\varphi(B)>0$ is valid on physical grounds (the toroidal
component is usually dominant in fusion devices such as tokamaks and
stellarators). The related conformal symmetry of $B$ is

$u=\frac{\nabla\psi\times\nabla\varphi}{B^{\varphi}}$

and is purely poloidal: the integral curves lie on constant $\varphi$-lines in
$S$ (and are hence closed). Consequently, the geometric toroidal angle
$\varphi$ can serve as a coordinate, and a poloidal angle (another independent
coordinate), can be found to make the field-lines of $B$ straight (or,
following [10], semi-rectified on $S$). The resulting coordinates are known as
PEST coordinates [24]. They are adopted by many MHS codes including VMEC [26]
extensively used for stellarator optimisation.

We will now discuss how symmetries give rise to flux-coordinates and write
down the consequences for general flux systems.

###  V.1 Near a torus

In this section, we restrict to tangential flux systems. We remark that if one
begins with a flux system on $M$ which is not tangential, then restriction to
$\operatorname{int}M=M\backslash\partial M$ produces a tangential flux system
on $\operatorname{int}M$.

The following theorem can be viewed as the general principle behind
guaranteeing the existence of flux-coordinates near a torus. At the end of
this section, we write down the consequences of this theorem for tangential
flux systems.

######  Theorem V.2.

Let $(B,\nu,\mu)$ be a tangential flux system on $M$ and suppose that $S$ is a
regular torus. Then, the following are equivalent.

  1. 1.

There exists a neighbourhood $\mathcal{U}$ of $S$ and a vector field $u$ on
$\mathcal{U}$ satisfying $\iota_{u}\iota_{B}\mu=\nu$ and $[u,B]=0$.

  2. 2.

There exists a neighbourhood $\mathcal{U}$ of $S$, a vector field $u$ on
$\mathcal{U}$ satisfying $\iota_{u}\iota_{B}\mu=\nu$, and a diffeomorphism
$\Phi=(\theta_{1},\theta_{2},z):\mathcal{U}\to\mathbb{T}^{2}\times I$ where
$I$ is some interval such that

$\begin{split}\nu&=d\psi(z),\\\
B&=a(z)\partial_{\theta_{1}}+b(z)\partial_{\theta_{2}},\\\
u&=c(z)\partial_{\theta_{1}}+d(z)\partial_{\theta_{2}}.\end{split}$ (3)

in $\mathcal{U}$, where $\psi,a,b,c,d:I\to\mathbb{R}$ are smooth.

###### Proof.

Because $S$ is a regular torus, by Lemma III.6 and Lemma III.6, there exists a
neighborhood $U\subset\mathcal{U}$ of $S$ on which $\nu=d\psi$ for some
$\psi\in C^{\infty}(U)$ whereby $S$ is a regular level set of $\psi$. We have
that $[u,B]=0$ and $u,B$ are tangent to the level sets $\psi$ (and $\partial
M$). We conclude that 2 therefore holds by standard Arnol’d-Liouville argument
(the general argument is given in [27] and the current setting is given in
[10, Prop. 26]). The converse is a direct computation. ∎

Following [10], if Equation (3) in Theorem V.2 is satisfied, we say that
$\Phi$ _rectifies $(B,\nu,\mu)$ and $u$_.

It is often important in plasma physics to instead work with the 2-form
$\beta=\iota_{B}\mu$. The following is a well-known reformulation of the
aforementioned rectifiability condition.

######  Proposition V.3.

Let $(B,\nu,\mu)$ be a tangential flux system on $M$, $S$ be a regular torus,
and $\mathcal{U}$ be a neighborhood of $S$ for which $\nu|_{\mathcal{U}}$ is
non-vanishing. Let $u$ be a vector field on $\mathcal{U}$ satisfying
$\iota_{u}\iota_{B}\mu=\nu$, and
$\Phi=(\theta_{1},\theta_{2},z):\mathcal{U}\to\mathbb{T}^{2}\times I$ be a
diffeomorphism where $I$ is some interval. Then, the following are equivalent.

  1. 1.

The diffeomorphism $\Phi$ rectifies $(B,\nu,\mu)$ and $u$.

  2. 2.

The diffeomorphism $\Phi$ is such that the following is satisfied.

$\begin{split}\nu&=d\psi(z)\\\ \beta:=\iota_{B}\mu&=dF(z)\wedge
d\theta_{1}-dG(z)\wedge d\theta_{2},\\\ \vartheta:=\iota_{u}\mu&=dK(z)\wedge
d\theta_{1}-dL(z)\wedge d\theta_{2},\end{split}$ (4)

in $\mathcal{U}$, where $\psi,F,G,K,L:I\to\mathbb{R}$ are smooth.

Following [3], the coordinate functions $(\theta_{1},\theta_{2},z)$ satisfying
Equation 4 in Proposition V.3 are _Hamada coordinates for $(B,\nu,\mu)$ and
$u$_.

###### Proof of Proposition V.3.

Assume that $\nu=d\psi(z)$ for some smooth $\psi:I\to\mathbb{R}$. Write

$\displaystyle B$
$\displaystyle=B^{1}\partial_{\theta_{1}}+B^{2}\partial_{\theta_{2}},$
$\displaystyle u$
$\displaystyle=u^{1}\partial_{\theta_{1}}+u^{2}\partial_{\theta_{2}},$
$\displaystyle\mu$ $\displaystyle=\rho d\theta_{1}\wedge d\theta_{2}\wedge
dz,$

where $B^{1},B^{2},u^{1},u^{2}:\mathcal{U}\to\mathbb{R}$ are smooth and
$\rho:\mathcal{U}\to\mathbb{R}$ is smooth and non-vanishing. Then, we have
that

$\displaystyle\beta$ $\displaystyle=\rho(B^{2}dz\wedge
d\theta_{1}-B^{1}dz\wedge d\theta_{2})$ $\displaystyle\vartheta$
$\displaystyle=\rho(u^{2}dz\wedge d\theta_{1}-u^{1}dz\wedge d\theta_{2})$
$\displaystyle\iota_{u}\iota_{B}\mu$
$\displaystyle=\rho(B^{1}u^{2}-B^{2}u^{1})dz.$

Now, because $\iota_{u}\iota_{B}\mu=\nu=d\psi(z)=\psi^{\prime}(z)dz$, we have
that

$\rho(B^{1}u^{2}-B^{2}u^{1})=\psi^{\prime}(z)$

and since $\nu|_{\mathcal{U}}$ is non-vanishing, both $\psi^{\prime}(z)$ and
$(B^{1}u^{2}-B^{2}u^{1})$ are non-vanishing.

Hence, if $B^{1},B^{2},u^{1},u^{2}$ are functions of $z$, then so is
$\rho=\psi^{\prime}(z)/(B^{2}u^{1}-B^{1}u^{2})$ and thus $\rho B^{1},\rho
B^{2},\rho u^{1},\rho u^{2}$ are functions of $z$. Conversely, if $\rho
B^{1},\rho B^{2},\rho u^{1},\rho u^{2}$ are functions of $z$, then so is the
function

$\frac{(\rho B^{1})(\rho u^{2})-(\rho B^{2})(\rho
u^{1})}{\psi^{\prime}(z)}=\frac{\rho^{2}(B^{1}u^{2}-B^{2}u^{1})}{\rho(B^{1}u^{2}-B^{2}u^{1})}=\rho$

and thus $B^{1},B^{2},u^{1},u^{2}$ are functions of $z$. This establishes the
claim. ∎

By Remark V.1, Theorem V.2, and the proof of Proposition V.3, the following is
immediate.

######  Corollary V.4.

Let $(B,\nu,\mu)$ be a tangential flux system. Let $S$ be a regular torus.
Then, there exists a neighborhood $\mathcal{U}$ of $S$ and a diffeomorphism
$\Phi=(\theta_{1},\theta_{2},z):\mathcal{U}\to\mathbb{T}^{2}\times I$ where
$I$ is some interval such that

$\displaystyle\nu$ $\displaystyle=d\psi(z),$
$\displaystyle\beta:=\iota_{B}\mu$ $\displaystyle=dF(z)\wedge
d\theta_{1}-dG(z)\wedge d\theta_{2},$ $\displaystyle B/f$
$\displaystyle=G^{\prime}(z)/\rho(z)\partial_{\theta_{1}}+F^{\prime}(z)/\rho(z)\partial_{\theta_{2}},$
$\displaystyle f\mu$ $\displaystyle=\rho(z)d\theta_{1}\wedge d\theta_{2}\wedge
dz$

in $\mathcal{U}$, where $f:\mathcal{U}\to\mathbb{R}$,
$\psi,\rho,F,G:I\to\mathbb{R}$ are all smooth and $f,\rho>0$.

###  V.2 Near an axis

In this section, flux-coordinates are established in the neighbourhood of an
axis. The notion of axis readily extends to flux systems which are not
tangential. Indeed, the restriction to $\operatorname{int}M$ gives a
tangential flux system for any flux system on $M$. We will refer in this
section to this generalisation of axis. The results will hold provided the
axis is non-degenerate.

######  Definition V.5.

An axis $\gamma\subset\mathcal{T}^{1}$ of a flux system $(B,\nu,\mu)$ is non-
degenerate if there exists a neighborhood $\mathcal{U}$ of $\gamma$ for which
$\nu=d\psi$ in $\mathcal{U}$ for some $\psi:\mathcal{U}\to\mathbb{R}$ and
$\gamma$ is a Morse-Bott critical manifold of $\psi$.

######  Remark V.6 (on Definition V.5).

The definition is simplified by the following observations about axes of a
flux system $(B,\nu,\mu)$.

  1. 1.

If $\gamma$ is an axis, by Lemma III.18 and the proof of Lemmas III.21 and
I.6, in any neighborhood $U$ of $\gamma$, there exists a solid torus $R\subset
U$ with $R\backslash\gamma$ contained in the pre-toroidal region
$\mathcal{T}^{2}$ with boundary $\partial R$ a regular torus of $(B,\nu,\mu)$.
In particular, $\operatorname{int}R$ is a neighborhood of $\gamma$ for which
$\nu$ is exact and $\gamma$ is the critical set of
$\nu|_{\operatorname{int}R}$.

  2. 2.

By the Morse-Bott Lemma, an axis $\gamma$ of $(B,\nu,\mu)$ is non-degenerate
if and only if there exists a neighborhood $\mathcal{U}$ of $\gamma$ and a
diffeomorphism $\Phi=(x,y,\varphi):\mathcal{U}\to D^{2}\times\mathbb{T}$ such
that

$\nu=ds$

where $s=\tfrac{1}{2}(x^{2}+y^{2})$.

Analogously to the existence of flux-coordinates near a regular torus, we have
the following theorem on the existence flux-coordinates near a non-degenerate
axis for any flux system $(B,\nu,\mu)$. Having made Remark V.6, the forward
direction follows immediately from the proofs of Theorem V.6 on near-axis
Hamada coordinates and Theorem IV.3 in [3] in the so-called elliptic case. The
converse is a direct application of Equation (1).

######  Theorem V.7.

Suppose that $(B,\nu,\mu)$ is a flux system and that that $\gamma$ is an axis.
Then, the following are equivalent.

  1. 1.

There exists a tubular neighbourhood $\mathcal{U}$ of $\gamma$ and a vector
field $u$ on $\mathcal{U}$ such that $\iota_{u}\beta=\nu$ and $[u,B]=0$ in
$\mathcal{U}$ and $\gamma$ is non-degenerate.

  2. 2.

There exists a tubular neighbourhood $\mathcal{U}$ of $\gamma$, a vector field
$u$ on $\mathcal{U}$ such that $\iota_{u}\beta=\nu$, and a diffeomorphism
$\Phi=(x,y,\varphi):\mathcal{U}\rightarrow D^{2}\times\mathbb{T}$ such that

$\begin{split}\nu&=d\psi(s),\\\ \beta&=F(s)dx\wedge dy+dG(s)\wedge d\varphi\\\
\vartheta&=\iota_{u}\mu=K(s)dx\wedge dy+dL(s)\wedge d\varphi,\end{split}$ (5)

in $\mathcal{U}$, where $F,G,K,L,\psi:[0,1/2)\to\mathbb{R}$ are smooth and
$s=\tfrac{1}{2}(x^{2}+y^{2})$.

Following [3] the coordinate functions $(x,y,\varphi)$ satisfying Equation 5
in Theorem V.7 are known as _near-axis Hamada coordinates for $(B,\nu,\mu)$
and $u$_. We can also reformulate this condition to one explicitly on the
vector fields $B$ and $u$ as follows.

######  Proposition V.8.

Let $(B,\nu,\mu)$ be a flux system on $M$, $\gamma$ be a non-degenerate axis,
and $\mathcal{U}$ be a neighborhood of $\gamma$ for which $\nu|_{\mathcal{U}}$
has $\gamma$ as its critical set. Let $u$ be a vector field on $\mathcal{U}$
satisfying $\iota_{u}\iota_{B}\mu=\nu$, and $\Phi=(x,y,\varphi):\mathcal{U}\to
D^{2}\times\mathbb{T}$ be a diffeomorphism where $D^{2}$ is the open unit
disk. Then, the following are equivalent.

  1. 1.

The diffeomorphism $\Phi$ gives near axis Hamada coordinates for $(B,\nu,\mu)$
and $u$

  2. 2.

The diffeomorphism $\Phi$ is such that the following is satisfied.

$\displaystyle\nu$ $\displaystyle=d\psi(s),$ $\displaystyle B$
$\displaystyle=a(s)(y\partial_{x}-x\partial_{y})+b(s)\partial_{\varphi},$
$\displaystyle u$
$\displaystyle=c(s)(y\partial_{x}-x\partial_{y})+d(s)\partial_{\varphi}.$

in $\mathcal{U}$, where $a,b,c,d,\psi:[0,1/2)\to\mathbb{R}$ are smooth and
$s=\tfrac{1}{2}(x^{2}+y^{2})$.

The proof is essentially the same as that of Proposition V.3. In particular,
as in the case of a regular torus, by Remark V.1, Theorem V.7, the following
is immediate.

######  Corollary V.9.

If $\gamma$ is a non-degenerate axis of a flux system $(B,\nu,\mu)$ then there
exists a tubular neighborhood $\mathcal{U}$ of $\gamma$, a diffeomorphism
$\Phi=(x,y,\varphi):\mathcal{U}\to D^{2}\times\mathbb{T}$ where $D^{2}$ is the
open unit disk such that

$\displaystyle\beta$ $\displaystyle=F(s)dx\wedge dy+dG(s)\wedge d\varphi,$
$\displaystyle B/f$
$\displaystyle=G^{\prime}(s)/\rho(s)(y\partial_{x}-x\partial_{y})+F(s)/\rho(s)\partial_{\varphi},$
$\displaystyle f\mu$ $\displaystyle=\rho(s)dx\wedge dy\wedge d\varphi$

in $\mathcal{U}$, where $s=\tfrac{1}{2}(x^{2}+y^{2})$ and
$a,b,\rho:[0,1/2)\to\mathbb{R}$ are smooth and $\rho>0$.

##  VI Discussion

This work considered flux systems on a manifolds with (possibly empty)
boundary. Theorem I.3 identified an algebraic condition, namely the existence
of an adapted 1-form, in order to conclude that the flux system had a
conformal symmetry. This can be viewed as a partial converse to a conformal
Noether theorem (Theorem II.1). Assuming that the flux system was tangential,
in other words, certain natural boundary conditions were obeyed, Theorem I.5
showed that the toroidal region is an apriori open set on which an adapted
1-form exists and in turn, a conformal symmetry. This provides a converse to a
conformal Noether theorem on the topologically simplest region generated by
the flux system. However, there remains some open questions and interesting
further directions from the perspective of both plasma physics, and of a
purely mathematical view.

In the context of plasma physics, this work demonstrates that structural
features of magnetic fields, such as the existence of certain flux-
coordinates, or symmetries, can be deduced from a flux function (first
integral) alone, and the stronger assumption that the field satisfies MHS is
not required. This has a direct implication for Grad’s Conjecture [28] which
loosely states, under some non-degeneracy assumptions appealing to plasma
confinement, that an MHS field with nested flux surfaces must possess a
continuous geometric symmetry. This paper reveals that these structural
features are purely topological in nature and that, because they are present
for a large family of divergence-free vector fields, they alone do not give
any insight into Grad’s conjecture.

It is not known whether symmetry and first integrals are essential for good
confinement; chaotic fields are in some scenarios desirable for the exhaust of
heat and deleterious impurities [29]. In the face of Grad’s conjecture, if
symmetry and first integrals are mathematically impossible beyond Euclidean
symmetry, the flavour of the question changes to how ”good” or ”close” fields
are to ”useful” or ”approximate” symmetry. These questions are primordial for
the viability of stellarators and other non-axisymmetric magnetic confinement
devices.

Turning to a more mathematical view of this paper, a first theoretical
question is that of the global existence of an adapted 1-form $\eta$, for any
flux system $(B,\nu,\mu)$. In this work, Theorem I.5 was proved, guaranteeing
for any flux-system the existence of an adapted 1-form in the toroidal region.
This region is the most relevant for flux systems appearing in plasma physics.
However, mathematically it is crucial to ask whether the toroidal region is
the largest possible region on which $\eta$ exists. The answer is no. Indeed,
the flux system $(B,\nu,\mu)$ on $M=\mathbb{R}^{2}\times\mathbb{R}/\mathbb{Z}$
given in Example I.4 with

$B=y\partial_{x}+(2x-4x^{3})\partial_{y}+\partial_{t},\qquad\nu=d\psi,\qquad\mu=dx\wedge
dy\wedge dt$

and $\psi=\tfrac{1}{2}y^{2}-x^{2}(1-x^{2})$, is such a counter-example. The
toroidal region is $\mathcal{T}=M\backslash\\{\psi=0\\}$ and yet $\eta=dt$ is
a 1-form adapted $(B,\nu,\mu)$ which is defined globally. More generally, any
flux-system satisfying the hypothesis of Proposition II.13 for which $M$ is
not the toroidal region serves as a counter-example. Besides including Example
I.4, this also includes contact systems and cosymplectic systems.

Having established the fact that $\eta$ can exist on a larger region than the
toroidal region, the natural question is then what is the largest region
$\eta$ can be guaranteed. Unfortunately, finding an adapted 1-form for a flux
system in high generality appears difficult. One approach to start finding an
adapted 1-form of a flux system $(B,\nu,\mu)$ in a general setting can be
roughly summarised as follows. On the non-critical set $U=\\{\nu\neq 0\\}$,
assuming that $(B,\nu,\mu)$ is tangential, the leaves $L$ of the foliation
generated by $\nu$ in $U$ have the property that $B$ is divergence-free with
respect to the area-form $\mu_{L}=L^{*}i_{N}\mu$ in $L$ where $N:U\to TM$ is a
vector field satisfying $\nu(N)=1$. To obtain an adapted 1-form in $U$, one
needs a closed 1-form $\eta_{L}$ on $L$ such that $\eta_{L}(B)>0$. This
problem is solved in this paper when $L$ is a 2-torus by Kolmogorov’s Theorem
[10]. In general, finding $\eta_{L}$ is purely about structural properties of
non-vanishing, area preserving vector fields on $L$ and thus may depend on the
differential topology of $L$. Due to the non-vanishing and area preserving
properties, the possible dynamics are reduced and hence the desired $\eta_{L}$
for each $L$ might be obtained. However, even if each $\eta_{L}$ is obtained,
one still needs to glue each of the $\eta_{L}$’s in $U$ together. This
involves the structure of the foliation and hence the structure of $\nu$.
Restricting to the toroidal region made the gluing relatively easy because the
$L$ were compact and flowouts preserving the foliation could be used. Such
tools are perhaps not usable for general $L$.

If $\eta$ can be found on the region where $\nu$ is non-critical, it still
remains to show $\eta$ exists on regions including the critical points. The
most natural hypothesis for the critical region $K=\\{\nu=0\\}$ (or a subsets
thereof) is that it consists of integral curves of $B$ which are open in the
relative topology of $K$. If these integral curves are 1-dimensional and
periodic and contained in $\operatorname{int}M$, then as observed in Section V
there exists an adapted 1-form. This local construction has nothing to do with
$\nu$ or the toroidal region and can be done anywhere there is such an
integral curve. An interesting consequence of these local adapted 1-forms,
both a mathematically and physically, is that the work in [3] should directly
imply that flux-coordinates exist for these more general periodic orbits under
some Morse-Bott non-degeneracy assumptions. This remains to be carefully
checked in the so-called hyperbolic case of their work.

However, this problem of gluing in general cannot be resolved. This was
exhibited by the flux system in Example IV.3 in Section IV. An interesting
feature of this example is it in fact possesses a global symmetry despite
there not existing any global adapted 1-form. In particular, Theorem I.3 is
not the mechanism through which every symmetry of a flux system is generated.
This shows that the problem of constructing a global adapted 1-form faces
problems of orientation which are not faced by symmetries. Indeed, considering
for instance a Reeb cylinder $(C,X)$ where

$C=\mathbb{R}/2\pi\times[-\pi/2,\pi/2],\qquad X=\sin y\partial_{x}+\cos
y\partial_{y}$

with standard coordinates $x,y$, we have that $\partial_{x}$ is a symmetry of
$X$ and its integral curves are positively oriented with those of $X$ along
the boundary component $C_{1}=\mathbb{R}/2\pi\times\\{\pi/2\\}$ but negatively
oriented with those along the boundary component
$C_{0}=\mathbb{R}/2\pi\times\\{-\pi/2\\}$. This is impossible for a closed
1-form to achieve: more precisely, as seen in Proposition IV.2, there cannot
be a closed 1-form $\eta_{C}$ on $C$ such that $\eta_{C}(X)>0$.

While adapted 1-forms are not the only mechanism producing symmetries, Theorem
I.3 generates all symmetries of a certain type. More precisely, if
$(B,\mu,\nu)$ is a flux system with an adapted 1-form $\eta$, then Theorem I.3
there exists a vector field $u$, a positive function $f$ and a 1-form
$\eta^{\prime}$ such that

$i_{u}i_{B}\mu=\nu,\qquad\mathcal{L}_{u}f\mu=0,\qquad\eta^{\prime}(u)=0,\qquad\eta^{\prime}(B)>0$
(6)

where of course, as discussed, the first two equations imply that $u$ is a
conformal symmetry of $B$. Under our assumptions, $\eta^{\prime}$ may be taken
as $\eta$ and $f$ as $\eta(B)$. Conversely, if one starts with a symmetry $u$
for which there exists an $\eta^{\prime}$ and an $f$ as in Equation (6), then
the 1-form $\eta=f\eta^{\prime}/\eta^{\prime}(B)$ is an adapted 1-form.
Indeed, we have $[u,B/f]=0$, $\eta(B)=f$, and $\eta(u)=0$ so that

$d\eta(u,B/f)=u(\eta(B)/f)-B(\eta(u))-\eta([u,B/f])=0-0-0=0$

and consequently $d\eta\wedge\nu=0$, as required.

One necessary condition for a conformal symmetry $u$ to posses such a 1-form
$\eta^{\prime}$ as in Eq. 6 is that $u$ vanishes at the critical points of
$\nu$. However, this is not sufficient, as the following example illustrates.

######  Example VI.1.

In $\mathbb{R}^{3}$, set $B=\partial_{x}$ and
$u=(y^{2}+z^{2})(\partial_{x}+z\partial_{y}-y\partial_{z})$. Then $u$ and $B$
are divergence-free with respect to the standard volume-form $\mu=dx\wedge
dy\wedge dz$ and additionally satisfy

$\iota_{u}\iota_{B}\mu=d\psi$

where $\psi=-\frac{1}{4}(y^{2}+z^{2})^{4}$. The critical set of $\psi$ is
$\\{y=0=z\\}$ which is nowhere dense in $\mathbb{R}^{3}$ and $u$ only vanishes
on this critical set. However, there does not exist a 1-form $\eta^{\prime}$
such that

$\eta^{\prime}(u)=0,\qquad\eta^{\prime}(B)>0.$

Indeed, letting $\eta^{\prime}=\eta^{1}dx+\eta^{2}dy+\eta^{3}dz$ be a 1-form
such that $\eta^{\prime}(u)=0$, then

$0=\eta^{\prime}(u)=(y^{2}+z^{2})(\eta^{\prime}(B)+z\eta^{2}-y\eta^{3})$

and consequently, because $\\{y=0=z\\}$ is nowhere dense in $\mathbb{R}^{3}$,

$\eta^{\prime}(B)+z\eta^{2}-y\eta^{3}=0$

and hence $\eta^{\prime}(B)$ must vanish on the critical set of $\psi$.

On the other hand, $(B,d\psi,\mu)$ possesses the adapted 1-form $dx$ and by
Theorem I.3, we must have a symmetry field $\tilde{u}$ generated from $dx$.
One finds that
$\tilde{u}=(y^{2}+z^{2})(z\partial_{y}-y\partial_{z})=u-(y^{2}+z^{2})B$. Thus,
in this instance, a modification of the symmetry $u$ into another symmetry
$\tilde{u}$ was necessary for there to exist an $\eta^{\prime}$ such that
$\eta^{\prime}(\tilde{u})=0$ and $\eta^{\prime}(B)>0$.

Although in Example VI.1 there was a way of modifying the given symmetry $u$
into a symmetry $\tilde{u}$ which posesses an $\eta^{\prime}$ as in Eq. 6,
this modification is not always possible: as Example IV.3 has no adapted
1-form, yet still possess a symmetry.

In conclusion, there is still much to understand about the global existence of
$\eta$ and more generally, on the global existence of a conformal symmetry $u$
for a given flux-system. Future work would do well to investigate for which
domains $\eta$ or these conformal symmetries may exist. The global existence
of a conformal symmetry, whether that is through $\eta$ or not, is required to
ensure the conformal Noether theorem demonstrated in this paper is as strong
as the well-known Noether theorem in symplectic geometry.

##  VII Acknowledgements

The Authors would like to thank the Max Planck Institute in Greifswald for
their hospitality during which the core ideas of the paper were formulated.
Nathan would like to thank the Simons Collaboration on Hidden Symmetries and
Fusion Energy for their financial support during this stay in Greifswald.
David Perrella would like to thank Daniel Peralta-Salas for pointing out the
idea of Example IV.3 and providing financial support for his visit to the
Instituto de Ciencias Matemáticas in Madrid. This paper was written while
David Perrella received an Australian Government Research Training Program
Scholarship at The University of Western Australia.

##  VIII Author Declarations

###  VIII.1 Conflict of Interest

The authors have no conflicts to disclose.

###  VIII.2 Author Contributions

David Perrella: Conceptualization (lead), Writing – original draft,
Visualization. Nathan Duignan: Conceptualization, Writing – original draft,
Visualization. David Pfefferlé: Supervision, Writing – original draft
(supporting).

##  Appendix A Tangential distributions and foliations on manifolds with
boundary

This appendix serves as a reference for a formal treatment of distributions
and foliations which are tangent to the boundary of the underlying manifold.
The purpose of this is that the authors have found no formal treatment in the
literature. In Appendix A.1, tangential distributions and integrability are
discussed along with a proof of what we will call The Tangential Stefan-
Sussman Theorem. The Stefan-Sussman Theorem reduces to the Frobenius Theorem
in the context of the latter. In Appendix A.2 integrable tangential
distributions are related with foliations. The treatment will be rapid due to
the close similarity to the case without boundary presented in the books of
Lee [14] and Rudolph and Schmidt [30]. We will largely follow the latter.

###  A.1 Tangential Stefan-Sussman Theorem

Let $M$ be a manifold with boundary. We first introduce tangential
distributions.

######  Definition A.1.

Fix a subset $D\subset TM$. A $D$-section is a (smooth) vector field on $M$
taking values in $D$ and a local $D$-section is a local vector on $M$ taking
values in $D$. The subset $D$ is said to be a distribution on $M$ if for all
$x\in M$

  1. 1.

$D_{x}\coloneqq D\cap T_{x}M$ is a vector subspace of $T_{x}M$,

  2. 2.

and for all $v\in D_{x}$, there exists a $D$ section $X$ such that $X|_{x}=v$.

In this case, for $x\in M$, the number $\dim D_{x}$ is called the rank of $D$
at $x$. Lastly, $D$ is said to be tangential if for all $x\in\partial M$, the
vectors $v\in D_{x}$ are tangent to the boundary.

We now introduce integrability. For clarity, in what follows, we will abuse
notation by omitting pullbacks and pushfowards of tangent spaces via
inclusions.

######  Definition A.2.

Fix a distribution $D$ on $M$. A connected submanifold $L\subset M$ is said to
be an integral manifold of $D$ if $T_{x}L=D_{x}$ for all $x\in L$. The
distribution $D$ is said to be integrable if through all points $x\in M$,
there passes an integral manifold $L$ of $D$.

######  Remark A.3.

An integrable distribution is necessarily tangential.

We will now introduce the other notions for the Tangential Stefan-Sussman
Theorem (Theorem A.6).

######  Definition A.4.

Let $D$ be a tangential distribution $D$. Then $D$ is called

  1. 1.

involutive if whenever $X,Y$ are $D$-sections, so is $[X,Y]$.

  2. 2.

homogenous if for any local $D$-section $X$ and any point $(x,t)$ in the
domain of the flow $\psi^{X}$ of $X$ (which is an open subset of
$M\times\mathbb{R}$ because $X$ is tangent to the boundary), we have

$T\psi^{X}_{t}|_{x}D_{x}=D_{\psi^{X}_{t}(x)}.$

The remaining notion is that of an adapted chart, which is important for the
connection with foliations. Set $n=\dim M$. We will denote by
$\mathbb{H}^{n}=\\{x^{n}\geq 0\\}$ the upper half plane.

######  Definition A.5.

Let $D$ be a tangential distribution on $M$. Set $n=\dim M$. Let $x\in M$ and
$r$ be the rank of $D$ at $x$. A local chart $(U,\varphi)$ on $M$ is said to
be adapted to $D$ at $x$ if

  1. 1.

$\varphi(x)=0$ and there exists $\epsilon>0$ such that
$\varphi(U)=(-\epsilon,\epsilon)^{n}\cap\mathbb{H}^{n}$ if $x\in\partial M$
and $\varphi(U)=(-\epsilon,\epsilon)^{n}$ otherwise,

  2. 2.

the coordinate vector fields
$\partial_{\varphi^{1}},...,\partial_{\varphi^{r}}$ are local $D$-sections

  3. 3.

for all $c\in\mathbb{R}^{n-r}$, the rank of $D$ along
$U_{c}\coloneqq\varphi^{-1}((-\epsilon,\epsilon)^{r}\times\\{c\\})$ is
constant.

With respect to these definitions, The Tangential Stefan-Sussman Theorem reads
as follows.

######  Theorem A.6 (Tangential Stefan-Sussman).

Let $D$ be a tangential distribution on $M$. Then the following are
equivalent.

  1. S1.

$D$ is integrable,

  2. S2.

$D$ is involutive and has constant rank along integral curves of local $D$
sections,

  3. S3.

$D$ is homogenous,

  4. S4.

for every $x\in M$, there exists a local chart adapted to $D$ at $x$.

We will now set out to prove Theorem A.6; that the statements S1, S2, S3 and
S4 are all equivalent. Using the construction of the double $D(M)$ (see [14,
Example 9.32]), we may fix a manifold (without boundary) $\tilde{M}$ for which
$M$ is a regular domain in $\tilde{M}$.

######  Remark A.7.

Recall, $M\subset\tilde{M}$ being a regular domain means that the inclusion
$i:M\subset\tilde{M}$ is a proper embedding and that $\dim\tilde{M}=n$. The
embeddingness of the inclusion is equivalent to $M\subset\tilde{M}$ having the
subspace topology. The properness of the inclusion is equivalent to $M$ being
closed in $\tilde{M}$ [14, Proposition 5.5]. Because of this, and extension
lemmas [14, Lemma 2.26, Lemma 8.6], if $U$ is open in $\tilde{M}$ and $F$ is a
function or local vector field defined on $U\cap M$, then there exists an
extension $\tilde{F}$ of $F$ defined on $U$.

Let $D$ be a tangential distribution on $M$. Define $\tilde{D}\subset
T\tilde{M}$ by

$\displaystyle\tilde{D}_{x}$ $\displaystyle=D_{x}\text{ for }x\in M,$
$\displaystyle\tilde{D}_{x}$ $\displaystyle=T_{x}\tilde{M}\text{ for
}x\in\tilde{M}\backslash M.$

Using this as auxiliary, we prove Theorem A.6.

###### Proof of Theorem A.6.

Because Theorem A.6 is true for $\tilde{M}$, it suffices to establish that the
subset $\tilde{D}$ is a distribution on $\tilde{M}$ and that for $i=1,2,3,4$,
$D$ satisfies statement Si on $M$ if and only if $\tilde{D}$ satisfies
statement Si on $\tilde{M}$.

To this end, let $X$ is a local $D$-section defined on $U\cap M$ where $U$ is
some open subset of $\tilde{M}$. Because $M\subset\tilde{M}$ is a regular
domain, there exists an extension of $X$ to $\tilde{X}$, a local vector field
on $\tilde{M}$ defined on $U$. Moreover, any such extension is a local
$\tilde{D}$-section. In particular, this shows that $\tilde{D}$ is a
distribution on $\tilde{M}$. The equivalence between $D$ and $\tilde{D}$
satisfying S2 or S3 also follows from this observation and the naturality of
Lie brackets and flows.

For S1, and another argument (Proposition A.12) we have the following lemma.

######  Lemma A.8.

The following are equivalent.

  1. 1.

$L$ is an integral manifold of $D$

  2. 2.

$L$ is an integral manifold of $\tilde{D}$ with $L\cap M\neq\emptyset$.

###### Proof of Lemma A.8.

For the less obvious direction, let $L$ is an integral manifold of $\tilde{D}$
with $L\cap(\tilde{M}\backslash M)\neq\emptyset$. Then $\dim L=n$ and so
because the rank of $\tilde{D}$ is at most $n-1$ on $\partial M$, we have that
$L\cap\partial M=\emptyset$. On the other hand $\operatorname{int}M$ and
$\tilde{M}\backslash M$ are open disjoint sets covering
$\tilde{M}\backslash\partial M$. Thus, because $L$ is a connected subset of
$M$, we must have that $L\subset\tilde{M}\backslash M$. Contrapositively, if
$L$ is an integral manifold of $\tilde{D}$ with $L\cap M\neq\emptyset$, then
$L\subset M$. Because $M$ is a regular domain in $\tilde{M}$, $L\subset M$ is
a submanifold. Thus, $L$ is an integral manifold of $D$. ∎

Continuing with S1, observe that the connected components of
$\tilde{M}\backslash M$ are integral manifolds of $\tilde{D}$ covering
$\tilde{M}\backslash M$. It is then clear by Lemma A.8 that $D$ satisfies S1
on $M$ if and only if $\tilde{D}$ satisfies S1 on $\tilde{M}$.

For S4, first observe that for $x\in\tilde{M}\backslash M$, any chart
$(U,\varphi)$ about $x$ with $U\subset\tilde{M}\backslash M$ satisfying
$\varphi(x)=0$ and $\varphi(U)=(-\epsilon,\epsilon)^{n}$ for some $\epsilon>0$
is adapted to $\tilde{D}$ at $x$. Let $x\in\operatorname{int}M$ then if
$(U,\varphi)$ is a chart adapted to $D$ at $x$, $(U,\varphi)$ is a chart
adapted to $\tilde{D}$ at $x$. Conversely, if $(U,\varphi)$ is a chart adapted
to $\tilde{D}$ at $x$, we may choose $\epsilon^{\prime}>0$ sufficiently small
so that
$U^{\prime}\coloneqq\varphi^{-1}((-\epsilon^{\prime},\epsilon^{\prime})^{n})\subset\operatorname{int}M$.
Then, the restricted chart
$(U^{\prime},\varphi^{\prime}:U\to(-\epsilon^{\prime},\epsilon^{\prime})^{n})$
is adapted to $D$ at $x$.

Continuing with S4, suppose that $x\in\partial M$. Let $(U,\varphi)$ be a
chart adapted to $D$ at $x$. We have $U=U^{\prime}\cap M$ for some
$U^{\prime}$ open in $\tilde{M}$. The coordinate functions
${\varphi}^{i}:U\to\mathbb{R}$ may be extended to functions
${\varphi^{\prime}}^{i}:U^{\prime}\to\mathbb{R}$. Then, consider the map
$\varphi^{\prime}=({\varphi^{\prime}}^{1},...,{\varphi^{\prime}}^{n}):U^{\prime}\to\mathbb{R}^{n}$.
Then by The Inverse Function Theorem, because $(U,\varphi)$ is a chart, there
exists $\epsilon>0$ so that $\varphi^{\prime}$ restricts to a chart
$(U^{\prime\prime},\varphi^{\prime\prime})$ on
$U^{\prime\prime}=\varphi^{\prime-1}((-\epsilon,\epsilon)^{n})$. Because
$(U,\varphi)$ is adapted to $D$ at $x$, and $\tilde{D}_{y}=T_{y}\tilde{M}$ for
all $y\in\tilde{M}\backslash M$, $(U^{\prime\prime},\varphi^{\prime\prime})$
is a chart adapted to $\tilde{D}$ at $x$.

Finally, let $(U,\varphi)$ be a chart adapted to $\tilde{D}$ at $x$. Here we
adapt ideas in the proof of the Stefan-Sussman Theorem found in [30]. Consider
a chart $(V,\kappa)$ of $\tilde{M}$ at $x$ such that $V\cap
M=\\{\kappa^{n}\geq 0\\}$, and $V\cap\partial M=\\{\kappa^{n}=0\\}$. Setting
$r$ to be the mutual rank of $D$ and $\tilde{D}$ at $x$, extend
$\partial_{\varphi^{1}}|_{x},...,\partial_{\varphi^{r}}|_{x}$ into a basis

$\partial_{\varphi^{1}}|_{x},~{}...,~{}\partial_{\varphi^{r}}|_{x},~{}v_{n-r},~{}...,~{}v_{n-1},~{}\partial_{\kappa^{n}}|_{x}$

of $T_{x}\tilde{M}$ where the $v_{i}$ are tangent to $\partial M$. Then, we
may extend the $v_{i}$ to vector fields $X_{i}$ on $\tilde{M}$. Then, consider
the local vector fields $X_{i}^{\prime}$ defined on $U\cap V$ where

$\displaystyle X_{i}^{\prime}$ $\displaystyle=\partial_{\varphi^{i}},$
$\displaystyle i$ $\displaystyle=1,...,r,$ $\displaystyle X_{i}^{\prime}$
$\displaystyle=X_{i}-d\varphi^{n}(X_{i})X_{n}^{\prime},$ $\displaystyle i$
$\displaystyle=n-r,...,n-1,$ $\displaystyle X_{n}^{\prime}$
$\displaystyle=\partial_{\kappa^{n}}.$

Considering the local flows $\psi^{i}\coloneqq\psi^{X_{i}^{\prime}}$, there
exists $\epsilon>0$ sufficiently small so that the map
$F:(-\epsilon,\epsilon)^{n}\to\tilde{M}$ given by

$F(t_{1},...,t_{n})=\psi^{1}_{t_{1}}\circ...\circ\psi^{n}_{t_{n}}(x)$

is well-defined. Because $X^{\prime}_{1}|_{x},...,X^{\prime}_{n}|_{x}$ forms a
basis, The Inverse Function Theorem ensures a sufficiently small
$\epsilon^{\prime}>0$ so that
$F|_{(-\epsilon^{\prime},\epsilon^{\prime})^{n}}$ is a diffeomorphism onto its
open image. Moreover, since the $X_{1},...,X_{n-1}$ is tangent to $\partial
M$, an integral curve $\gamma_{i}$ of $X_{i}$ intersects $\partial M$ if and
only if $\gamma_{i}\subset\partial M$. This invariance implies that

$\displaystyle
F((-\epsilon^{\prime},\epsilon^{\prime})^{n}\cap\mathbb{H}^{n})$
$\displaystyle\subset\\{\kappa^{n}\geq 0\\}$ $\displaystyle
F((-\epsilon^{\prime},\epsilon^{\prime})^{n}\backslash\mathbb{H}^{n})$
$\displaystyle\subset\\{\kappa^{n}<0\\}.$

thus, restriction of $F$ to
$(-\epsilon^{\prime},\epsilon^{\prime})^{n}\cap\mathbb{H}^{n}$ gives a chart
$(U^{\prime},\varphi^{\prime})$ of $M$ at $x$. Now, because the
$X_{1},...,X_{r}$ commute and the construction of $F$, it is easy to see that

$\displaystyle\partial_{{\varphi^{\prime}}^{i}}$
$\displaystyle=\partial_{\varphi^{i}},$ $\displaystyle i$
$\displaystyle=1,...,r.$

so that for each constant $c^{\prime}\in\mathbb{R}^{n-r}$, there exists a
constant $c\in\mathbb{R}^{n-r}$ such that

$\varphi^{\prime-1}((-\epsilon^{\prime},\epsilon^{\prime})\times\\{c^{\prime}\\})\subset\varphi^{\prime-1}((-\epsilon^{\prime},\epsilon^{\prime})\times\\{c\\}).$

From these observations it immediately follows that
$(U^{\prime},\varphi^{\prime})$ is an adapted chart to $D$ at $x$. This
completes the equivalence between $D$ and $\tilde{D}$ satisfying S4 and thus
the proof. ∎

We will now discuss integrable distributions in relation to foliations.

###  A.2 Integral manifolds and foliations

Let $M$ be a manifold with boundary. Just as in the case of manifolds (see
[14] and [30]), the following definition serves as a specification of
regularity of immersed submanifolds.

######  Definition A.9.

An (immersed) submanifold $L\subset M$ is said to be weakly embedded in $M$ if
for any manifold $N$ and smooth map $F:N\to M$ such that $F(N)\subset L$, we
have that the restricted map $f:N\to L$ is smooth.

######  Remark A.10.

Just as in the case of manifolds without boundary [14, Theorem 5.33], if
$L\subset M$ is a weakly embedded submanifold, then $L$ has the unique
topology and smooth structure making it into a submanifold of $M$. That is, if
$L^{\prime}\subset M$ is a submanifold having the same underlying set of
points as $L$, then, as manifolds, $L^{\prime}=L$.

Let $D$ be an tangential distribution on a manifold $M$ with boundary. We will
now introduce maximality of integral manifolds of $D$.

######  Definition A.11.

An integral manifold $L$ of $D$ is said to be maximal if for any integral
manifold $K$ of $D$ with $L\cap K\neq\emptyset$, we have that $K\subset L$ is
an open submanifold.

To prove the following proposition which involves the definitions just
introduced, fix a manifold $\tilde{M}$ for which $M\subset\tilde{M}$ is a
regular domain. Moreover, we consider the associated distribution $\tilde{D}$
just as in Section A.1.

######  Proposition A.12.

The following holds.

  1. 1.

At every point $x\in M$, there passes a maximal integral manifold of $D$.

  2. 2.