[intoc]

# On Darboux theorems for geometric structures
induced by closed forms

aXavier Gràcia , bJavier de Lucas , cXavier Rivas , aNarciso Román-Roy
aDepartment of Mathematics,
Universitat Politècnica de Catalunya, Barcelona, Spain.
bUW Institute for Advanced Studies,
Department of Mathematical Methods in Physics, University of Warsaw, Warszawa,
Poland.
cEscuela Superior de Ingeniería y Tecnología,
Universidad Internacional de La Rioja, Logroño, Spain.
e-mail<EMAIL_ADDRESS>ORCID: 0000-0003-1006-4086e-mail:
<EMAIL_ADDRESS>ORCID: 0000-0001-8643-144Xe-mail:
<EMAIL_ADDRESS>ORCID: 0000-0002-4175-5157e-mail<EMAIL_ADDRESS>ORCID: 0000-0003-3663-9861

###### Abstract

This work reviews the classical Darboux theorem for symplectic, presymplectic,
and cosymplectic manifolds (which are used to describe regular and singular
mechanical systems), and certain cases of multisymplectic manifolds, and
extends it in new ways to $k$-symplectic and $k$-cosymplectic manifolds (all
these structures appear in the geometric formulation of first-order classical
field theories). Moreover, we discuss the existence of Darboux theorems for
classes of precosymplectic, $k$-presymplectic, $k$-precosymplectic, and
premultisymplectic manifolds, which are the geometrical structures underlying
some kinds of singular field theories. Approaches to Darboux theorems based on
flat connections associated with geometric structures are given, while new
results on polarisations for ($k$-)(pre)(co)symplectic structures arise.

Keywords: Darboux theorem, flat connection, $k$-cosymplectic manifold,
$k$-precosymplectic manifold, $k$-presymplectic manifold, $k$-symplectic
manifold, multisymplectic manifold, premultisymplectic manifold. MSC 2020:
Primary: 53C15, 53C12.  Secondary: 53D05, 53C10.

###### Contents

  1. 1 Introduction
  2. 2 Darboux theorems, flat connections, and symplectic-like structures
    1. 2.1 Symplectic and presymplectic manifolds
    2. 2.2 Cosymplectic and precosymplectic manifolds
  3. 3 $k$-symplectic and $k$-presymplectic manifolds
  4. 4 $k$-cosymplectic and $k$-precosymplectic manifolds
  5. 5 Multisymplectic and premultisymplectic structures
  6. 6 Conclusions and outlook

##  1 Introduction

Since its very origins, differential geometry has been applied to many
branches of mathematical physics to study different kinds of physical systems,
and it has led to many developments. Symplectic geometry, namely the study of
closed non-degenerate two-forms, the so-called symplectic forms, was one of
the first areas of differential geometry to be introduced. Symplectic geometry
has its origins in the study of celestial mechanics [45], it has a relevant
role in classical mechanics [2, 10], and it has inspired the development of
many other useful geometric structures with relevant applications [1, 6, 27,
41, 42, 44].

One of the fundamental results in symplectic geometry is the Darboux theorem,
which describes the local structure of finite-dimensional symplectic manifolds
[18]. Roughly speaking, the Darboux theorem states that a symplectic form can
be locally written as a differential form with constant coefficients of a
particular type, namely as the canonical symplectic form of a cotangent bundle
in adapted coordinates, $\omega=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$ [1, 43].
There exist several types of infinite-dimensional symplectic manifolds, and
some of them do not admit a Darboux theorem [46]. Hereafter, we focus on
finite-dimensional manifolds, unless otherwise stated.

The Darboux theorem can be proved in different ways [1, 9], and its proof can
be extended to presymplectic forms, namely closed two-forms of constant rank
[17]. It is well-known that symplectic and presymplectic forms describe the
phase spaces for autonomous regular and singular dynamical systems in
mechanics. For non-autonomous mechanical systems, the suitable structures are
the so-called cosymplectic and precosymplectic manifolds [16, 20].

As a preliminary goal, this paper reviews the theory of Darboux theorems for
symplectic and presymplectic manifolds, and it analyses their relation to the
so-called flat compatible symplectic and presymplectic connections [7, 34,
61]. Connections are hereafter assumed to be linear and torsionless, being the
second condition usual in the literature and a key to the description of
certain features of the differential forms and integrable distributions to be
studied in this work. We also provide proofs of the Darboux theorems for
cosymplectic and precosymplectic manifolds. The Darboux theorem for
precosymplectic structures is assumed in many references but its proof seems
to be absent in the literature [16].

To achieve a geometrical covariant description of (first-order) classical
field theories, the above-mentioned structures have been generalised in
several ways. The simplest ones are the so-called $k$-symplectic manifolds,
introduced by A. Awane [4, 6] and used later by M. de León et al. [22, 24, 25]
and L.K. Norris [48, 51] for describing first-order classical field theories.
They coincide with the polysymplectic manifolds described by G.C. Günther [39]
(although these last ones are different from those introduced by G.
Sardanashvily et al. [35, 56] and I.V. Kanatchikov [40], that are also called
polysymplectic). This structure is used to give a geometric description of
regular field theories whose Lagrangian and/or Hamiltonian functions, in a
local description, do not depend on the space-time coordinates (or the
analogous). In the degenerate case we use $k$-presymplectic structures, which
allow us to describe the corresponding field theories given by singular
Lagrangian functions [37]. It is worth stressing that there exist several ways
of defining $k$-presymplectic manifolds, some of which have apparently been
proposed and studied in the present work for the first time.

A natural extension of $k$-symplectic manifolds are $k$-cosymplectic
manifolds, which enable one to generalise the cosymplectic description of non-
autonomous mechanical systems to regular field theories whose Lagrangian
and/or Hamiltonian functions, in a local description, depend on the space-time
coordinates (or the analogous) [21, 23]. As previously, the singular case of
these theories leads to the introduction of $k$-precosymplectic manifolds,
which can be defined in different manners, as shown in this paper and studied
in previous works [38].

The Darboux theorem was generalised and proved for $k$-symplectic manifolds in
[4, 25, 29] and for $k$-cosymplectic manifolds in [21]. The Darboux theorem
plays a relevant role in these theories since, for instance, it significantly
simplifies the proofs of many results [52]. In this work, we provide a (as far
as we know) new Darboux theorem for $k$-symplectic and $k$-cosymplectic linear
spaces. We also provide new proofs for the Darboux theorems for $k$-symplectic
and $k$-cosymplectic manifolds. Our proofs reveal new properties of such types
of manifolds concerning the properties of their Lagrangian submanifolds. In
particular, new details about the existence of the hereafter called
polarisations for $k$-symplectic and $k$-cosymplectic manifolds are obtained.
Moreover, classical proofs of the $k$-symplectic manifold rely on coordinates
and special, rather lengthy calculations [6]. Others are focused on
connections and give indirect proofs [29]. Meanwhile, our proof of the
$k$-symplectic Darboux theorem is intrinsic and short. Moreover, our proof
could have been made shorten by relying on known results, but we decided to
give a full explanation of all the structures and results involved, which made
it longer than strictly needed to prove the canonical form of $k$-symplectic
manifolds.

Darboux theorems for $k$-symplectic manifolds are closely related to the
notion of polarisation [4]. This means that we search for coordinates where
the $k$-(co)symplectic structures take a form with constant coefficients of a
particular type. Nevertheless, one could find new coordinates where the
$k$-(co)symplectic forms would take constant coefficients of another different
type. This would potentially lead to Darboux coordinates of other types.

Moreover, one may try to find coordinates to put the differential forms of a
$k$-symplectic structure on a canonical manner. This leads to the existence of
a certain type of associated distribution. Notwithstanding, Darboux
coordinates can be defined to additionally put a basis of the distribution in
a particular manner. It is worth noting that, in the case of $k$-symplectic
manifolds, the conditions to obtain Darboux coordinates putting the associated
differential forms into canonical form ensure that there exists a canonical
basis of the distribution too. Meanwhile, our $k$-cosymplectic Darboux theorem
shows that the conditions to put the differential forms associated with a
$k$-cosymplectic manifold are different from those needed to ensure also a
canonical form for a basis of the associated distribution. Moreover, our
analysis also sheds some light in the existence of Darboux coordinates for
$k$-cosymplectic manifolds, and it complements the results given in previous
works [32, 49]. In particular, it is worth noting that Theorem II.4 and
Theorem 5.2.1 in [21, 49] can be slightly misleading, as part of the
assumptions needed to prove such theorems are only described in Remark 2.5 and
Note 5.2.1, after them, respectively.

Then, we study $k$-presymplectic manifolds. These structures appear as a side
problem in $k$-symplectic or multisymplectic theories [29]. We here prove that
the very definition of a $k$-presymplectic manifold can be set in different
ways, depending on the features that we want them to have, e.g. to fit the
analysis of systems we are dealing with. Some of these notions of
$k$-presymplectic manifold do not admit a Darboux theorem of the initially
expected form, even for the linear $k$-presymplectic cases. Then, we study
some different possible definitions of $k$-presymplectic manifolds, and we
provide some counterexamples showing that a Darboux theorem does not need to
exist for them. This is quite unexpected, as it was previously assumed that
Darboux theorems must be satisfied for them. It is worth noting that the
authors in [37] remark that the existence of a Darboux theorem for
$k$-presymplectic manifolds is an open problem, although they skip this by
giving intrinsic proofs of their results. The same happens when we consider
$k$-precosymplectic manifolds [38] in order to deal with non-autonomous field
theories described by singular Lagrangian functions.

As in the $k$-presymplectic manifolds case, one has the same type of problems
and similar solutions are given. A Darboux theorem for precosymplectic
manifolds has been provided. Although this result has been used in the
literature [16, 26], it seems that a proof is missing. More generally, we have
provided definitions of $k$-pre(co)symplectic manifolds admitting Darboux
theorems. This gives an alternative approach to previous point-wise and local
Darboux theorems in [29] for the $k$-presymplectic case. Moreover, our point-
wise and local $k$-precosymplectic Darboux theorems seem to be new. Note also
that $k$-precosymplectic manifold do not have canonically defined Reeb vector
fields. This causes that the Darboux theorems may not involve the existence of
a basis of them given in a canonical form. Moreover, the distribution defined
to put the differential forms of $k$-precosymplectic manifolds in a canonical
manner does not admit a canonical basis unless additional conditions are
given.

Finally, we have the multisymplectic manifolds first introduced by J.
Kijowski, W.M. Tulczyjew, and other authors [31, 36, 41, 42], which constitute
one of the most generic structures for studying the behaviour of Lagrangian
and Hamiltonian field theories (see [54] and references therein).
Nevertheless, although there are some partial results [47], a Darboux-type
theorem for multisymplectic manifolds in general is not known. In particular,
a class of multisymplectic manifolds with a local structure defined by Darboux
type coordinates was characterised in [13], and certain kinds of
multisymplectic manifolds admitting Darboux coordinates have been described in
[19], giving a sufficient condition that guarantees the existence of Darboux
charts.

While studying the different geometric structures, we analyse the existence of
linear connections compatible with them. Some of our results are known, see
for instance symplectic connections [7, 34], $k$-symplectic connections [8],
$k$-cosymplectic [49], multisymplectic and polysymplectic connections [30]. On
the other hand, some of the connections compatible with these and other
structures are here proposed. Moreover, we here review the subject and serves
as a reference point for further works.

The structure of the paper goes as follows. Section 2 reviews the Darboux
theorems for symplectic, presymplectic, cosymplectic, and precosymplectic
structures and its relation to flat compatible connections. In Section 3, we
provide a new proof of the Darboux theorem for $k$-symplectic manifolds, which
is simpler than previous proofs [4, 6, 53]. We also discuss the existence of
Darboux coordinates in $k$-presymplectic manifolds and show that in order to
ensure its existence, some very restrictive hypothesis are required. Section 4
is devoted to study the existence of Darboux coordinates in
$k$-(pre)cosymplectic manifolds. We give a new proof of the Darboux theorem
for $k$-cosymplectic manifolds [21]. We also see that it is not possible to
ensure the existence of Darboux coordinates, unless some additional conditions
are imposed. In Section 5 we review the existing results on Darboux
coordinates for (pre)multisymplectic structures. Some new results on this
topic are presented. Finally, Section 6 summarises our results and gives some
hints on future work. It is worth noting that we explain how the use of flat
connections with torsion may be used to study geometric structures related to
differential forms that are not closed, such as contact ones. This will be the
topic of another paper.

##  2 Darboux theorems, flat connections, and symplectic-like structures

Let us set some general assumptions to be used throughout this work. It is
hereafter assumed that all structures are smooth. Manifolds are real,
Hausdorff, connected, second countable, and finite-dimensional. Differential
forms are assumed to have constant rank, unless otherwise stated. Sum over
crossed repeated indices is understood. Sometimes, the summation sign,
$\Sigma$, will be used to make clear the range of the indexes we are summing
over. All our considerations are local, to avoid technical problems concerning
the global existence of quotient manifolds and similar issues. Hereafter, $M$
and $Q$ are assumed to be manifolds, $\mathfrak{X}(M)$ and $\Omega^{k}(M)$
stand for the $\mathscr{C}^{\infty}(M)$-modules of vector fields and
differential $k$-forms on $M$. Moreover, connections are assumed to be linear
and torsion-free.

More particularly, this section reviews (co)symplectic and (co)presymplectic
manifolds and give the corresponding Darboux theorems. It also analyses the
relation of Darboux theorems with compatible flat connections. We will also
introduce the concept of characteristic distribution, as it will play an
important role when generalising, in Sections 3 and 4, the results of this
section.

###  2.1 Symplectic and presymplectic manifolds

This section reviews the definition of symplectic and presymplectic manifolds,
and it also analyses their corresponding Darboux theorems. In the context of
presymplectic manifolds, we recall the definition of their characteristic
distributions. For symplectic and presymplectic manifolds, the relation
between compatible connections and Darboux coordinates is studied.

######  Definition 2.1.

A symplectic manifold is a pair $(M,\omega)$, where $M$ is a manifold and
$\omega$ is a closed differential two-form on $M$ that is non-degenerate, i.e.
the contraction $\operatorname{\iota}_{X}\omega=0$, for a vector field $X$ on
$M$, if and only if $X=0$.

The canonical model for symplectic manifolds is the cotangent bundle of a
manifold $Q$, namely $(\mathrm{T}^{\ast}Q,\omega_{Q})$, where
$\omega_{Q}\in\Omega^{2}(\mathrm{T}^{\ast}Q)$ is the canonical symplectic two-
form in $\mathrm{T}^{\ast}Q$, whose local expression in adapted coordinates
$\\{q^{i},p_{i}\\}$ of $\mathrm{T}^{\ast}Q$ on their associated coordinated
open subset of $\mathrm{T}^{\ast}Q$ is
$\omega_{Q}=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$.

A symplectic manifold $(M,\omega)$ gives rise to the musical (vector bundle)
isomorphism $\flat:\mathrm{T}Q\rightarrow\mathrm{T}^{\ast}Q$ and its inverse
$\sharp:\mathrm{T}^{\ast}Q\to\mathrm{T}Q$ naturally induced by the
$\mathscr{C}^{\infty}(M)$-module isomorphisms

$\displaystyle\flat\colon\mathfrak{X}(M)$
$\displaystyle\longrightarrow\Omega^{1}(M)$ $\displaystyle X$
$\displaystyle\longmapsto\operatorname{\iota}_{X}\omega$

and $\sharp=\flat^{-1}$. Note that a vector bundle morphism $\flat$ can be
defined for every two-form $\omega$, but $\sharp$ only exists when $\flat$ is
invertible or, equivalently, $\omega$ is non-degenerate.

######  Definition 2.2.

Let $(M,\omega)$ be a symplectic manifold. Given a distribution
$D\subset\mathrm{T}M$, the symplectic orthogonal of $D$ is defined by
$D^{\perp}=\coprod_{x\in M}D^{\perp}_{x}$, where

$D_{x}^{\perp}=\\{v\in\mathrm{T}_{x}M\mid\omega_{x}(v,u)=0,\,\forall u\in
D_{x}\\}\,,$

and $\displaystyle\coprod_{x\in M}D_{x}^{\perp}$ stands for the disjoint sum
of all $D_{x}^{\perp}$ over $x\in M$.

Symplectic orthogonals allow us to introduce several types of submanifolds of
symplectic manifolds.

######  Definition 2.3.

Let $(M,\omega)$ be a symplectic manifold and consider a submanifold $N\subset
M$. Then,

  * •

the submanifold $N$ is said to be isotropic, if
$\mathrm{T}N\subset\mathrm{T}N^{\perp}$.

  * •

the submanifold $N$ is coisotropic, if
$\mathrm{T}N^{\perp}\subset\mathrm{T}N$.

  * •

the submanifold $N$ is Lagrangian if it is isotropic and coisotropic, namely
if $\mathrm{T}N^{\perp}=\mathrm{T}N$. Lagrangian submanifolds are also called
maximally isotropic and then $2\dim N=\dim M$.

######  Definition 2.4.

Two symplectic manifolds $(M_{1},\omega_{1})$ and $(M_{2},\omega_{2})$ are
symplectomorphic if there exists a diffeomorphism $\phi:M_{1}\to M_{2}$ such
that $\phi^{*}\omega_{2}=\omega_{1}$.

The classical Darboux theorem states that every symplectic manifold is locally
symplectomorphic to a cotangent bundle endowed with its canonical symplectic
structure [1, 9]. The Darboux theorem was initially proved by Darboux [18],
but its modern standard proof relies on the so-called Moser’s trick [43,
Theorem 22.13]. The statement of the Darboux theorem for symplectic manifolds
goes as follows.

######  Theorem 2.5.

Let $(M,\omega)$ be a symplectic manifold. Then, for every $x\in M$, there
exist local coordinates $\\{q^{i},p_{i}\\}$ around $x$ where
$\omega=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$.

Note that the Darboux theorem amounts to saying that there exist, on a
neighbourhood of any $x\in M$, two foliations by transversal Lagrangian
submanifolds.

In infinite-dimensional manifolds, one can still define a symplectic form, but
the induced musical morphism $\flat:\mathfrak{X}(M)\rightarrow\Omega^{1}(M)$
is, in general, only injective. This gives rise to the so-called weak
symplectic manifolds. Meanwhile, if
$\flat:\mathfrak{X}(M)\rightarrow\Omega^{1}(M)$ is an isomorphism, then the
symplectic manifold is said to be a strong symplectic manifold. There exists
no Darboux theorem for general weak symplectic manifolds [46]. Nevertheless,
by requiring appropriate additional conditions, an analogue can be derived
[59].

Let us introduce the notion of symplectic connection [7, 57]. Note that the
torsion-free assumption is common in the literature, and it is a key for
certain results to be developed. Indeed, we will show in Section 6 that
skipping it leads to a more general theory, but more involved, inappropriate,
and unnecessary for our present work. Apart from that last comment to be given
in the conclusions of this work, all connections are assumed to be torsion-
free, unless otherwise stated.

######  Definition 2.6.

A symplectic connection on a symplectic manifold $(M,\omega)$ is a connection
$\nabla$ on $M$ such that $\nabla\omega=0$. The symplectic form $\omega$ is
said to be parallel relative to $\nabla$.

Every symplectic manifold admits local symplectic connections. Indeed, as a
consequence of the Darboux theorem, one can construct a local symplectic flat
connection for $(M,\omega)$ around every $x\in M$ by assuming that its
Christoffel symbols vanish in some Darboux coordinates defined around $x$.
Such a connection is flat and torsion-free. In general, flat symplectic
connections cannot be globally defined, as it is known that the curvature of a
connection is linked to the topology of the manifold where it is defined on.
Milnor proved and surveyed in [50] several results on the existence of
connections on the tangent bundle to a manifold. For instance, the tangent
bundle to a closed and oriented surface of genus $g$ has no flat connection if
$|2-2g|\geq g$. The sphere $\mathbb{S}^{2}$ has zero genus. Hence, there is no
flat connection on $\mathbb{S}^{2}$, which admits a natural symplectic
structure $\omega_{\mathbb{S}^{2}}$ that can be defined by considering
$\mathbb{S}^{2}$ embedded in $\mathbb{R}^{3}$ and setting

$(\omega_{\mathbb{S}^{2}})_{x}(v_{x},v^{\prime}_{x})=\langle x,v_{x}\times
v^{\prime}_{x}\rangle\,,\qquad\forall x\in\mathbb{R}^{3}\,,\quad\forall
v_{x},v^{\prime}_{x}\in\mathrm{T}_{x}\mathbb{S}^{2}\subset\mathrm{T}_{x}\mathbb{R}^{3}\simeq\mathbb{R}^{3}\,,$

where tangent vectors at $x\in\mathbb{S}^{2}$ are naturally understood as
vectors in $\mathbb{R}^{3}$ and, hence, their vector products are defined.
Meanwhile, $\langle\cdot,\cdot\rangle$ stands for the natural scalar product
in $\mathbb{R}^{3}$. Let us recall that, if a connection is flat, the parallel
transport of a tangent vector along a path contained in a small open set $U$
does not depend on the path. Thus, a basis $\\{e_{1},\ldots,e_{n}\\}$ of
$\mathrm{T}_{x}M$ gives rise, by parallel transport, to a family of vector
fields $X_{1},\ldots,X_{n}$ on $U\subset M$, such that $X_{i}(x)=e_{i}$ for
$i=1,\ldots,n$. Then, $\nabla_{X_{i}}X_{j}=0$ and, since $\nabla$ is torsion-
free, one has that

$0=T(X_{i},X_{j})=\nabla_{X_{i}}X_{j}-\nabla_{X_{j}}X_{i}-[X_{i},X_{j}]=-[X_{i},X_{j}]\,,\qquad\forall
i,j=1,\ldots,n\,.$

Hence, there exist coordinates $\\{x^{1},\ldots,x^{n}\\}$ on a neighbourhood
of $x$ such that $X_{i}=\partial/\partial x^{i}$ for $i=1,\ldots,n$. Moreover,
the Christoffel symbols of the connection vanish on $U$.

Using the above result and assuming the local existence of a flat torsion-free
connection compatible with a symplectic form, one may prove the Darboux
theorem in a very easy manner. In fact, every symplectic form $\omega$ on $M$
can be put into canonical form at any point $x\in M$ for a certain basis
$\\{e_{1},\dotsc,e_{n}\\}$ of $\mathrm{T}^{\ast}_{x}M$, i.e.

$\omega_{x}=\sum_{i=1}^{n}e^{2i-1}\wedge e^{2i}\,.$

Recall that on a neighbourhood of every point $x\in M$, one can define a
coordinate system $\\{x^{1},\ldots,x^{n}\\}$ around $x$ so that there exists
vector fields $X_{i}=\partial/\partial x^{i}$, with $i=1,\ldots,n$, such that

$\nabla_{X^{i}}X^{j}=0\,,\qquad\forall i,j=1,\ldots,n\,,$

and $X_{i}(x)=e_{i}$ for $i=1,\ldots,n$. Since $\nabla$ is a compatible
symplectic connection for $\omega$, one has that

$\nabla_{X^{i}}\omega(X^{j},X^{k})=0\,,\quad\forall i,j,k=1,\dotsc,n\,.$

Hence, one has

$\omega=\sum_{i=1}^{2n}\mathrm{d}x^{2i-1}\wedge\mathrm{d}x^{2i}\,.$

In a similar way, but weakening the conditions in Definition 2.1, we can
introduce the concept of presymplectic manifold. Recall that we assume
differential forms to have constant rank.

######  Definition 2.7.

A presymplectic form on $M$ is a closed two-form $\omega\in\Omega^{2}(M)$ of
constant rank. The pair $(M,\omega)$ is called a presymplectic manifold.

Let us construct a prototypical example of presymplectic manifold. Let
$(M,\omega)$ be a symplectic manifold, and let $N$ be a submanifold of $M$.
Consider the canonical embedding denoted by $\jmath_{N}\colon N\hookrightarrow
M$, and endow $N$ with the induced two-form
$\omega_{N}=\jmath_{N}^{\ast}\omega$, which is closed. Then, $(N,\omega_{N})$
is a presymplectic manifold provided the rank of $\omega_{N}$ is constant. To
see that the condition on the rank of $\jmath_{N}^{*}\omega$ is necessary, let
us consider the counter example given by the canonical two form,
$\omega=\mathrm{d}x\wedge\mathrm{d}p_{x}+\mathrm{d}y\wedge\mathrm{d}p_{y}$, on
$\mathrm{T}^{\ast}\mathbb{R}^{2}$ and the immersed submanifold given by
$\jmath_{\mathrm{T}^{\ast}\mathbb{R}}:(x,p_{x})\in\mathrm{T}^{\ast}\mathbb{R}\mapsto(x,p_{x}^{2}/2,0,p_{x})\in\mathrm{T}^{\ast}\mathbb{R}\times\mathrm{T}^{\ast}\mathbb{R}\simeq\mathrm{T}^{\ast}\mathbb{R}^{2}$.
Then,
$\jmath_{\mathrm{T}^{\ast}\mathbb{R}}^{*}\omega=p_{x}\mathrm{d}x\wedge\mathrm{d}p_{x}$,
which is not symplectic at the zero section of $\mathrm{T}^{\ast}\mathbb{R}$.

Before introducing the characteristic distribution associated with a
presymplectic manifold, let us fix some terminology about distributions. A
(generalised) distribution on $M$ is a subset $D\subset\mathrm{T}M$ such that
$D\cap\mathrm{T}_{x}M$ is a vector subspace of $\mathrm{T}_{x}M$ for every
$x\in M$. A distribution $D$ on $M$ is said to be smooth if, for every $x\in
M$, there exists a neighbourhood $U_{x}$ of $x$ and (smooth) vector fields
$X_{1},\ldots,X_{k}$ on $U_{x}$ so that $D_{y}=\langle
X_{1}(y),\ldots,X_{k}(y)\rangle$ for every $y\in U_{x}$. A generalised
distribution $D$ is regular if it is smooth and has constant rank. A
(generalised) codistribution on $M$ is a subset $C\subset\mathrm{T}^{\ast}M$
such that $C_{x}=C\cap\mathrm{T}^{\ast}_{x}M$ is a vector subspace of
$\mathrm{T}^{\ast}_{x}M$ for every $x\in M$. The smooth and/or regular notions
introduced for distributions also apply to codistributions.

######  Definition 2.8.

Given a presymplectic manifold $(M,\omega)$, its characteristic distribution
is the distribution

$\mathcal{C}_{\omega}=\ker\omega=\\{v\in\mathrm{T}M\mid\omega(v,\cdot)=0\\}\,.$

A vector field $X\in\mathfrak{X}(M)$ belonging to $\mathcal{C}_{\omega}$, i.e.
such that $\operatorname{\iota}_{X}\omega=0$, is called a characteristic
vector field of $(M,\omega)$.

Note that $\mathcal{C}_{\omega}=\ker\flat$. In the case of symplectic
manifolds, $\flat$ is a vector bundle isomorphism, and thus
$\mathcal{C}_{\omega}=\\{0\\}$. Moreover, $\mathcal{C}_{\omega}$ is a
distribution because $\omega$ has constant rank. But the kernel of a general
closed two-form does not need to be a smooth generalised distribution. For
example, $\omega_{P}=(x^{2}+y^{2})\mathrm{d}x\wedge\mathrm{d}y$ is a closed
two-form on $\mathbb{R}^{2}$, but it is not presymplectic, as its rank is not
constant. Moreover, $\ker\omega_{P}$ is a generalised distribution
$(\ker\omega_{P})_{(0,0)}=\mathrm{T}_{(0,0)}\mathbb{R}^{2}$ while
$(\ker\omega_{P})_{(x,y)}=0$ for every $(x,y)\in\mathbb{R}^{2}$ different from
$(0,0)$. Indeed, $\ker\omega_{P}$ is not even a smooth generalised
distribution.

######  Proposition 2.9.

The characteristic distribution $\mathcal{C}_{\omega}$ of a presymplectic
manifold $(M,\omega)$ is integrable.

###### Proof.

The integrability of $\mathcal{C}_{\omega}$ follows from the closedness of the
symplectic form $\omega$, the constancy of his rank, and the Frobenius
theorem. ∎

If $\omega$ is a presymplectic form on $M$, its characteristic distribution is
integrable. Moreover, around every $x\in M$, there exists an open
neighbourhood $U$ of $x$ such that the space of integral leaves of
$\mathcal{C}_{\omega}$, let us say $U/\mathcal{C}_{\omega}$, admits a natural
manifold structure and the projection $\pi:U\rightarrow
U/\mathcal{C}_{\omega}$ is a submersion. Let us prove this fact. Since
$\mathcal{C}_{\omega}$ is integrable, the Frobenius theorem ensures that, for
every $x\in M$, there exists a local basis of vector fields,
$\\{\partial/\partial x^{1},\ldots,\partial/\partial x^{k}\\}$, spanning
$\mathcal{C}_{\omega}$ on a coordinated neighbourhood $U_{x}$ of $x$ with
coordinates $\\{x^{1},\ldots,x^{n}\\}$. In a small enough open subset $U$ of
$U_{x}$ containing $x$, one can assume that $x^{1},\ldots,x^{n}$ take values
in an open ball of $\mathbb{R}^{n}$. Then, the space of leaves of
$U/\mathcal{C}_{\omega}$ is a manifold of dimension $n-k$ and the mapping
$\pi:U\rightarrow\mathbb{R}^{n-k}$ is an open submersion. We will then say
that $\mathcal{C}_{\omega}$ is simple on $U$. Since $\omega$ is invariant
relative to the elements of its characteristic distribution, and it vanishes
on them, there exists a unique two-form $\widetilde{\omega}$ on
$U/\mathcal{C}_{\omega}$ such that $\pi^{*}\widetilde{\omega}=\omega$. In this
way, $\widetilde{\omega}$ is closed and nondegenerate because if
$\operatorname{\iota}_{X_{\mathcal{C}}}\widetilde{\omega}=0$, then there
exists a vector field $X$ on $U$ such that $\pi_{*}X=X_{\mathcal{C}}$, and
then $\operatorname{\iota}_{X}\omega=0$. Since
$\mathcal{C}_{\omega}=\ker\omega=\ker\mathrm{T}\pi$, then $X_{\mathcal{C}}=0$.

With this in mind, we are ready to state the Darboux theorem for presymplectic
forms. Note that this theorem can be stated since presymplectic forms are
assumed to have constant rank. Otherwise, it would be difficult to establish a
series of canonical forms for closed two-forms even in the most simple cases,
e.g. on $\mathbb{R}^{2}$.

######  Theorem 2.10 (Darboux theorem for presymplectic manifolds).

Consider a presymplectic manifold $(M,\omega)$. Around every point $x\in M$,
there exist local coordinates $\\{q^{i},p_{i},z^{j}\\}$, where $i=1,\dotsc,r$
and $j=1,\dotsc,d$, such that

$\omega=\sum_{i=1}^{r}\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}\,,$

where $2r$ is the rank of $\omega$. In particular, if $r=0$, then $\omega=0$
and $d=\dim M$. If $2r=\dim M$, then $d=0$ and $\\{q^{i},p_{i}\\}$ give a
local coordinate system of $M$.

###### Proof.

Consider an open neighbourhood $V$ of $x\in M$ where the integral foliation
$\mathcal{F}$ defined by the distribution $\mathcal{C}_{\omega}=\ker\omega$ is
simple. Let $P$ be the manifold of leaves of
${\left.\kern-1.2pt\mathcal{F}\right|_{V}}$ and let $\pi\colon V\to P$ be the
canonical projection. There exists a symplectic form $\bar{\omega}$ on $P$
given by $\omega=\pi^{\ast}\bar{\omega}$. The Darboux theorem for symplectic
manifolds ensures that there exists an open coordinate neighbourhood
$\bar{U}\subset P$ of $\pi(x)$ with local coordinates
$\\{\bar{q}^{1},\dotsc,\bar{q}^{r},\bar{p}_{1},\dotsc,\bar{p}_{r}\\}$ such
that
$\displaystyle\bar{\omega}=\sum_{i=1}^{r}\mathrm{d}\bar{q}^{i}\wedge\mathrm{d}\bar{p}_{i}$
on $\bar{U}$. Define $q^{i}=\bar{q}^{i}\circ\pi$ and
$p_{i}=\bar{p}_{i}\circ\pi$ for $i=1,\ldots,r$ and we choose $d=\dim M-2r$
other functions $z^{1},\ldots,z^{d}$, functionally independent relative to the
previous ones. This gives rise to a local coordinate system
$\\{q^{1},\dotsc,q^{r},p_{1},\dotsc,p_{r},z^{1},\dotsc,z^{d}\\}$ around $x$.
This chart satisfies the conditions of the theorem. ∎

The definition of a presymplectic connection is a straightforward
generalisation of Definition 2.6 to the presymplectic realm (see [61]).

######  Definition 2.11.

A presymplectic connection relative to a presymplectic manifold $(M,\omega)$
is a connection $\nabla$ on $M$ such that $\nabla\omega=0$.

The Darboux theorem for presymplectic forms implies that there exists,
locally, a flat presymplectic connection. The other way around, the existence
of a flat presymplectic connection for a presymplectic manifold $(M,\omega)$
allows us to prove the Darboux theorem as in the case of symplectic forms. In
particular, we have proved the following.

######  Lemma 2.12.

Every presymplectic manifold $(M,\omega)$ admits locally defined flat
presymplectic connections $\nabla$, i.e. $\nabla\omega=0$.

At this point, it becomes clear that if a differential form admits a
compatible flat torsion-less connection, it must be closed. Hence, no flat
torsion-less compatible connection exist for contact forms, locally
conformally symplectic forms, and other differential forms that are not closed
[33, 60]. We have stressed the word “torsion-less” unless every connection in
this work is assumed to be so, because in the conclusions of this work will
show that removing this condition may lead to deal with no-closed differential
forms.

###  2.2 Cosymplectic and precosymplectic manifolds

Let us review the definition of cosymplectic [3, 11, 26] and precosymplectic
[16] manifolds, their corresponding Darboux theorems, and their relations to
flat cosymplectic and precosymplectic connections.

######  Definition 2.13.

A cosymplectic structure in $M$ is a pair $(\omega,\eta)$, where
$\omega\in\Omega^{2}(M)$ and $\eta\in\Omega^{1}(M)$ are closed differential
forms such that $\eta$ does not vanish and
$\ker\eta\oplus\ker\omega=\mathrm{T}M$. The triple $(M,\omega,\eta)$ is said
to be a cosymplectic manifold.

Note that a cosymplectic structure on $M$ implies that $M$ is odd-dimensional.
The fact that $\eta$ is non-vanishing implies that
$\langle\eta\rangle\oplus\operatorname{Im}\omega=\mathrm{T}^{\ast}M$ and $\dim
M=2n+1$ for $n\geq 0$. Then, $(M,\omega,\eta)$ is a cosymplectic manifold if
and only if $\eta\wedge\omega^{n}$ is a volume form on $M$, where we assume
that $\omega^{0}=1$. In particular, a cosymplectic manifold $(M,\omega,\eta)$
yields a presymplectic manifold $(M,\omega)$. Note that the case $\dim M=1$
may give rise to a cosymplectic manifold according to our definition [58].

The characteristic distribution of a cosymplectic manifold $(M,\omega,\eta)$
is the rank one distribution given by $\mathcal{C}_{\omega}=\ker\omega$, and
it is often called the Reeb distribution. The following proposition states
that $(M,\omega,\eta)$ induces a unique distinguished vector field, called
Reeb vector field, taking values in $\ker\omega$.

######  Proposition 2.14.

Given a cosymplectic manifold $(M,\omega,\eta)$, there exists a unique vector
field $R\in\mathfrak{X}(M)$ that satisfies

$\operatorname{\iota}_{R}\eta=1\,,\quad\operatorname{\iota}_{R}\omega=0\,.$

A cosymplectic manifold $(M,\omega,\eta)$ induces a
$\mathscr{C}^{\infty}(M)$-module isomorphism
$\flat\colon\mathfrak{X}(M)\to\Omega^{1}(M)$ given by
$\flat(X)=\operatorname{\iota}_{X}\omega+(\operatorname{\iota}_{X}\eta)\eta$,
whose inverse map is denoted by $\sharp=\flat^{-1}$. Then, the Reeb vector
field $R$ reads

$R=\sharp\eta\,.$

Consider the product manifold $\mathbb{R}\times\mathrm{T}^{\ast}Q$ and the
projections $\pi_{1}\colon\mathbb{R}\times\mathrm{T}^{\ast}Q\to\mathbb{R}$ and
$\pi_{2}\colon\mathbb{R}\times\mathrm{T}^{\ast}Q\to\mathrm{T}^{\ast}Q$ onto
the first and second manifolds in $\mathbb{R}\times\mathrm{T}^{\ast}Q$. If $t$
is the natural coordinate in $\mathbb{R}$ and $\omega_{Q}$ is the canonical
symplectic form on $\mathrm{T}^{*}Q$, then the triple

$(\mathbb{R}\times\mathrm{T}^{\ast}Q,\pi_{2}^{\ast}\omega_{Q},\pi_{1}^{\ast}\mathrm{d}t)$
(1)

is a cosymplectic manifold. Let us consider the pull-back of $t$ to
$\mathbb{R}\times\mathrm{T}^{\ast}Q$ via $\pi_{1}$, and the pull-back of some
Darboux coordinates $\\{q^{i},p_{i}\\}$ for $\omega_{Q}$ to
$\mathbb{R}\times\mathrm{T}^{\ast}Q$ via $\pi_{2}$. Let us denote such pull-
backs in the same way as the original coordinates to simplify the notation.
Then, in the coordinates $\\{t,q^{i},p_{i}\\}$, the Reeb vector field of
$(\mathbb{R}\times\mathrm{T}^{\ast}Q,\pi_{2}^{\ast}\omega_{Q},\pi_{1}^{\ast}\mathrm{d}t)$
read $\partial/\partial t$. Locally,
$\pi_{2}^{*}\omega_{Q}=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$ and
$\pi_{1}^{*}\mathrm{d}t=\mathrm{d}t$.

######  Theorem 2.15 (Cosymplectic Darboux theorem [26]).

Given a cosymplectic manifold $(M,\omega,\eta)$, there exists, around each
point $x\in M$, local coordinates $\\{t,q^{i},p_{i}\\}$, where $1\leq i\leq
n$, such that

$\eta=\mathrm{d}t\,,\quad\omega=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}\,.$

###### Proof.

Since $(M,\omega)$ is a presymplectic manifold and $\omega$ has corank one,
there exist for any point $x\in M$ a neighbourhood $U$ of $x$ with coordinates
$\\{s,q^{i},p_{i}\\}$, with $i=1,\ldots,n$, so that
$\omega=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$. Consider now a potential
function of $\eta$, which exists because $\eta$ is closed, and denote it by
$t$. Since $\eta\wedge\omega^{n}$ is a volume form, $\\{t,q^{i},p_{i}\\}$ is a
coordinate system around $x$ and $\eta=\mathrm{d}t$ and
$\omega=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$. ∎

The Darboux theorem for cosymplectic manifolds states that every cosymplectic
manifold is locally diffeomorphic to the canonical model (1) (see [11, 26]).
In Darboux coordinates, the Reeb vector field $R$ for a cosymplectic manifold
$(M,\omega,\eta)$ is written as $R=\frac{\partial}{\partial t}$.

The Darboux theorem for cosymplectic structures implies that there exists,
around each point, a flat connection $\nabla$ such that $\nabla\eta=0$ and
$\nabla\omega=0$. Indeed, $\nabla$ can be chosen to be the connection with
zero Christoffel symbols relative to some Darboux coordinates. This justifies
the following definition.

######  Definition 2.16.

A cosymplectic connection relative to $(M,\omega,\eta)$ is a connection on $M$
such that $\nabla\eta=0$ and $\nabla\omega=0$.

Let us show that the existence of flat cosymplectic connections allows us to
prove the Darboux theorem for $(M,\omega,\eta)$. At a point $x\in M$, the fact
that $\ker\eta_{x}\oplus\ker\omega_{x}=\mathrm{T}_{x}M$ implies that there
exists a basis of $\mathrm{T}_{x}M$ of the form $\\{e_{1},\ldots,e_{2n+1}\\}$
so that $\eta_{x}=e^{2n+1}$ and $\omega_{x}=\sum_{i=1}^{n}e^{2i-1}\wedge
e^{2i}$ relative to the dual basis $\\{e^{1},\ldots,e^{2n+1}\\}$ in
$\mathrm{T}^{\ast}_{x}M$. Due to the fact that $\nabla$ is flat, there exists
a family of commuting parallel vector fields $X_{1},\ldots,X_{2n+1}$ such that
$X_{i}(x)=e_{i}$ for $i=1,\ldots,2n+1$. Since

$\nabla_{X_{i}}[\eta(X_{j})]=0\,,\qquad\nabla_{X_{i}}[\omega(X_{j},X_{k})]=0\,,\qquad
i,j,k=1,\ldots,2n+1\,,$

the dual basis of differential one-forms $\tau^{1},\ldots,\tau^{2n+1}$ to
$X_{1},\ldots,X_{2n+1}$ is such that

$\eta=\tau^{2n+1},\qquad\omega=\sum_{i=1}^{n}\tau^{2i-1}\wedge\tau^{2i}\,.$

Since $X_{1},\ldots,X_{2n+1}$ admit a coordinate system so that
$X_{i}=\partial/\partial x_{i}$, with $i=1,\ldots,2n+1$, then
$\tau^{i}=\mathrm{d}x^{i}$ for $i=1,\ldots,2n+1$, and the Darboux theorem for
cosymplectic manifolds follows. Note that this is due to the fact that the
connection is assumed to be torsion-free.

Cosymplectic manifolds can be generalised by assuming that
$\eta\in\Omega^{1}(M)$ and $\omega\in\Omega^{2}(M)$ are closed forms on $M$,
but $\ker\eta\cap\ker\omega$ is a distribution of fixed rank that is not
necessarily zero. This implies that $\omega$ is a presymplectic form on $M$.
This gives rise to the definition of a precosymplectic manifold. When
$\ker\eta\cap\ker\omega=\\{0\\}$, one retrieves the definition of a
cosymplectic manifold.

######  Definition 2.17.

A precosymplectic structure in $M$ is a pair $(\omega,\eta)$, where
$\omega\in\Omega^{2}(M)$ and $\eta\in\Omega^{1}(M)$ are closed differential
forms such that $\ker\eta\cap\ker\omega$ is a regular distribution strictly
included in $\ker\omega$ at every $x\in M$. If
$\operatorname{rank}\omega=2r<\dim M$, the triple $(M,\omega,\eta)$ is said to
be a precosymplectic manifold of rank $2r$.

It is worth stressing that the fact that $\ker\eta\cap\ker\omega$ is a regular
distribution strictly contained in $\ker\omega$ implies that
$\eta\wedge\omega^{r}$ is a non-vanishing form and $\omega^{r+1}=0$ for a
certain fixed $r$, and conversely. Therefore, $\omega$ has constant rank $2r$,
with $2r<\dim M$.

######  Remark 2.18.

Let $(P,\omega)$ be a presymplectic manifold with Darboux coordinates
$\\{q^{i},p_{i},z^{j}\\}$. Consider the manifold $\mathbb{R}\times P$ with the
induced coordinates $\\{t,q^{i},p_{i},z^{j}\\}$ obtained as usual, namely,
$q^{i},p_{i},z^{j}$ are the pull-back to $\mathbb{R}\times P$ of the chosen
variables in $P$. Then, $(\mathbb{R}\times
P,\pi_{2}^{*}\omega,\pi^{*}_{1}\mathrm{d}t)$ is a precosymplectic manifold. In
the obtained local coordinates,
$\pi^{*}_{2}\omega=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$ while
$\pi^{*}_{1}\mathrm{d}t$ is denoted by $\mathrm{d}t$ to simplify the notation.

######  Remark 2.19.

Consider the regular distribution $D=\ker\omega\cap\ker\eta$ of a
precosymplectic manifold $(M,\omega,\eta)$. Then, $D$ is involutive because
$\ker\omega$ and $\ker\eta$ are so. The foliation associated with $D$ defines
a local projection

$\pi\colon M\to\widetilde{M}=M/(\ker\omega\cap\ker\eta)\,,$ (2)

where $\widetilde{M}$ is the quotient manifold of the leaves of $D$. Recall
that we are assuming that $\widetilde{M}$ is a manifold for simplicity.
Indeed, one of the general assumptions of our paper is that manifold
structures and other existing mathematical local structures are defined
globally. In reality, one can only ensure that for every $x\in M$ and a local
neighbourhood $U_{x}$ of $x$, the space $M/(\ker\omega\cap\ker\eta)$ is a
manifold. Hence, by our general assumptions, there exists a unique
cosymplectic structure $(\widetilde{\omega},\widetilde{\eta})$ on
$\widetilde{M}$ such that $\pi^{\ast}\widetilde{\omega}=\omega$ and
$\pi^{\ast}\widetilde{\eta}=\eta$.

As in the case of cosymplectic manifolds, we can define special types of
vector fields for precosymplectic manifolds.

######  Definition 2.20.

Given a precosymplectic manifold $(M,\omega,\eta)$, a vector field
$X\in\mathfrak{X}(M)$ satisfying

$\operatorname{\iota}_{X}\omega=0\,,\quad\operatorname{\iota}_{X}\eta=1\,,$

is called a Reeb vector field. The space generated by Reeb vector fields,
namely $\ker\omega$, is called the Reeb distribution of $(M,\omega,\eta)$.

Note that, if $R\in\mathfrak{X}(M)$ is a Reeb vector field, then $R+Y$ is also
a Reeb vector field for every $Y\in\ker\omega\cap\ker\eta$. In other words,
Reeb vector fields for precosymplectic manifolds need not be univocally
defined.

Finally, let us state the Darboux theorem for precosymplectic manifolds, whose
proof seems, as far as we know, to be absent in the literature. Nevertheless,
it is always implicitly assumed that it holds [16, 26] and it is quite
straightforward.

######  Theorem 2.21 (Darboux Theorem for precosymplectic manifolds).

Let $(M,\omega,\eta)$ be a precosymplectic manifold with
$\operatorname{rank}\omega=2r\leq\dim M-1$. For every $x\in M$, there exist
local coordinates $\\{t,q^{i},p_{i},z^{j}\\}$ around $x$, where $1\leq i\leq
r$ and $1\leq j\leq\dim M-1-2r$, such that

$\eta=\mathrm{d}t\,,\quad\omega=\sum_{i=1}^{r}\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}\,.$
(3)

###### Proof.

Since $\omega$ is a presymplectic form, there exist coordinates
$\\{q^{i},p_{i},z^{\prime}_{i}\\}$ on a neighbourhood $U$ of $x$ such that
$\omega=\sum_{i=1}^{r}\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$. Since
$(\ker\eta\cap\ker\omega)^{\circ}=\langle\eta\rangle\oplus{\rm Im}\,\omega$
and $\eta$ does not vanish, one has that $\rank\omega\leq\dim M-1$. One the
other hand, $\eta$ is closed and, therefore, there exists a function $t$ on
$U$, where $U$ can be chosen smaller if necessary, such that
$\eta=\mathrm{d}t$ and $\omega=\mathrm{d}q^{i}\wedge\mathrm{d}p_{i}$. Since
$\eta\wedge\omega^{r}$ does not vanish, $\\{t,q^{i},p_{i}\\}$ are functionally
independent functions. Finally, one can choose additional functionally
independent coordinates, $z^{1},\dotsc,z^{n}$ with respect to
$\\{t,q^{i},p_{i}\\}$ and (3) will hold. ∎

As in the previous cases, there exists a locally defined flat connection
$\nabla$ whose Christoffel symbols vanish on the chosen Darboux coordinates.
Then, $\eta$ and $\omega$ become parallel differential forms relative to
$\nabla$. This motivates the following natural definition.

######  Definition 2.22.

A precosymplectic connection relative to a precosymplectic manifold
$(M,\eta,\omega)$ is a connection on $M$ such that $\nabla\eta=0$ and
$\nabla\omega=0$.

Note that, as previously, the existence of a flat precosymplectic connection
allows one to provide a brief proof of the Darboux theorem for precosymplectic
manifolds.

##  3 $k$-symplectic and $k$-presymplectic manifolds

Let us introduce and provide Darboux theorems for $k$-symplectic manifolds.
This will give a new, complementary approach, to the classical results [4, 14]
and some new more modern approaches [29]. Moreover, we will discuss the
existence of Darboux theorems for $k$-presymplectic manifolds. Furthermore,
this will be done by providing new simpler, shorter and more geometrical
proofs of Darboux theorems for $k$-symplectic manifolds while giving more
details and, as far as we know, a new Darboux theorem for linear spaces [4].
Additionally, we will give a new proof about the existence of a complementary
for a polarisation that is isotropic relative to the differential two-forms of
a $k$-symplectic structure. On the other hand, Darboux theorems give rise to
the hereafter called flat $k$-symplectic and $k$-presymplectic connections,
which, in turn, lead to other proofs of respective Darboux theorems. It is
worth noting that an alternative, somehow different, development of these
ideas for the $k$-symplectic case can be found in [14]. Moreover, some new
structures will arise in our approach and our results concerning
$k$-presymplectic manifolds seem to be absolutely new.

######  Definition 3.1.

Let $M$ be an $n(k{+}1)$-dimensional manifold. A $k$-symplectic structure on
$M$ is a family $(\omega^{1},\dotsc,\omega^{k},V)$, where $V$ is an integrable
distribution on $M$ of rank $nk$, and $\omega^{1},\dotsc,\omega^{k}$ are
closed differential $2$-forms on $M$ satisfying that

  1. (1)

${\left.\kern-1.2pt\omega^{\alpha}\right|_{V\times V}}=0$, for
$1\leq\alpha\leq k$,

  2. (2)

$\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}=\\{0\\}$.

Under the above hypotheses, $(M,\omega^{1},\dotsc,\omega^{k},V)$ is called a
$k$-symplectic manifold. We call $V$ a polarisation of the $k$-symplectic
manifold.

Our notion of $k$-symplectic manifold matches the one given by A. Awane [4,
6]. Moreover, it is equivalent to the concepts of standard polysymplectic
structure of C. Günther [39] and integrable $p$-almost cotangent structure
introduced by M. de León et al [24, 25]. In the case $k=1$, Awane’s definition
reduces to the notion of polarised symplectic manifold, that is a symplectic
manifold with a Lagrangian foliation. We will illustrate in forthcoming
examples that the distribution $V$ is needed to ensure the existence of a
particular type of Darboux coordinates.

In fact, Günther calls polysymplectic manifolds the differential geometric
structures obtained from our definition by removing the existence of the
distribution $V$. Meanwhile, a _standard_ polysymplectic manifold in Günther’s
paper is a polysymplectic manifold admitting an atlas of Darboux coordinates.
Note that a polysymplectic manifold may have atlas of Darboux coordinates
without a distribution $V$. In a particular case, if we think of symplectic
manifolds as a one-symplectic manifold, then it is clear that it has local
Darboux coordinates, but the standard symplectic manifold on the sphere does
not have a polarisation [5]. Then, Günther’s definition is more general than
ours, while it is equivalent to our definition if the compatibility of two
charts Darboux coordinates $\\{y^{i},p^{\alpha}_{i}\\}$ and
$\\{x^{i},\pi^{\alpha}_{i}\\}$ involves that $x=x(y)$ and the momenta are
transformed accordingly, namely $\pi^{\alpha}=\pi^{\alpha}(y,p)$ are the
momenta of the $\\{x^{i}\\}$. Otherwise, the equivalence is only local.

Let us provide a Darboux theorem at the tangent space of a point of a
$k$-symplectic manifold. Since every $k$-symplectic manifold
$(M,\omega^{1},\dotsc,\omega^{k},V)$ induces at every $\mathrm{T}_{x}M$ for
$x\in M$ a so-called $k$-symplectic vector space, Theorem 3.2 can be
understood as a Darboux theorem for $k$-symplectic vector spaces.

######  Theorem 3.2.

($k$-symplectic linear Darboux theorem) Assume that
$(M,\omega^{1},\ldots,\omega^{k},V)$ is a $k$-symplectic manifold. For every
$x\in M$, there exists a basis
$\\{e^{1},\ldots,e^{n};e_{1}^{\beta},\ldots,e_{n}^{\beta}\\}_{\beta=1,\ldots,k}$
of $\mathrm{T}^{\ast}_{x}M$ such that

$\omega^{\beta}=\sum_{i=1}^{n}e^{i}\wedge e_{i}^{\beta}\,,\qquad
V=\bigoplus_{\alpha=1}^{k}V_{\alpha},\qquad V_{\beta}=\langle
e^{1}_{\beta},\ldots,e^{n}_{\beta}\rangle,\qquad\beta=1,\ldots,k\,.$

Note that $\\{e_{1},\ldots,e_{n},e_{\beta}^{1},\ldots,e_{\beta}^{n}\\}$ is the
dual basis in $\mathrm{T}_{x}M$.

###### Proof.

The result amounts to the Darboux theorem for symplectic linear spaces for
$k=1$. Hence, let us assume $k>1$. Since
$\\{0\\}=\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}$, one has that

$\mathrm{T}^{*}_{x}M=\operatorname{Im}\omega^{1}_{x}+\dotsb+\operatorname{Im}\omega^{k}_{x}\,,\qquad\forall
x\in M\,.$ (4)

Although all posterior structures in this proof refer to the point $x$, the
point will be omitted to simplify the notation. Since
$\omega^{\beta}|_{V\times V}=0$, one has that $\omega^{\beta}(V)\subset
V^{\circ}$ for $\beta=1,\ldots,k$. If $W$ is a regular distribution
supplementary to $V$, then $\operatorname{rank}W=n$, and
$\operatorname{rank}\omega^{\beta}\,(W)\leq n$. Note that

$\omega^{1}(V)+\dotsb+\omega^{k}(V)\subset V^{\circ}\,.$

Due to (4) and the above discussion, one has that

$\omega^{1}(W)+\dotsb+\omega^{k}(W)$

is a distribution of rank $nk$, at least. This implies that
$\operatorname{rank}\omega^{\beta}(W)=n$ and

$\omega^{1}(W)\oplus\dotsb\oplus\omega^{k}(W)\oplus
V^{\circ}=\mathrm{T}^{*}M\,,\qquad
V^{\circ}=\omega^{1}(V)+\ldots+\omega^{k}(V)\,.$

If $\omega^{\alpha}(v+w)=0$, where $v\in V$ and $w\in W$, then
$\omega^{\alpha}(v)=-\omega^{\alpha}(w)$. Since
$\omega^{\alpha}(W)\cap\omega^{\alpha}(V)=0$ and
$\operatorname{rank}\omega^{\alpha}|_{W}=n$, then $\omega^{\alpha}(v)=0$ and
$\omega^{\alpha}(w)=0$, which implies that $w=0$ and
$v\in\ker\omega^{\alpha}$. Hence, $\ker\omega^{\alpha}\subset V$. We can
consider the distributions

$V_{\beta}=\bigcap_{\begin{subarray}{c}1\leq\alpha\leq k\\\
\alpha\neq\beta\end{subarray}}\ker\omega^{\alpha}\,,\qquad\beta=1,\ldots,k\neq
1\,,\qquad\text{or}\qquad V^{1}=V\ \ (k=1)\,.$

Note that $\omega^{\beta}(V_{\alpha})=0$ for $\alpha\neq\beta$ and for every
$\alpha,\beta=1,\ldots,k$.

Let $\\{w_{1},\ldots,w_{n}\\}$ be a basis of $W$. Since $\omega^{\alpha}(W)$
has rank $n$ and its elements do not belong to $V^{\circ}$, then the
restrictions of $\omega^{\alpha}(w_{1}),\ldots,\omega^{\alpha}(w_{n})$ to $V$
are linearly independent and there exist $v_{1},\ldots,v_{n}$ in $V$ such that
$\omega^{\alpha}(v_{1}),\ldots,\omega^{\alpha}(v_{n})$ are linearly
independent on $W$, e.g. $\omega^{\alpha}(w_{i},v_{j})=\delta_{ij}$ for
$i,j=1,\ldots,n$. Hence, $\operatorname{rank}\omega^{\alpha}(V)\geq n$. Since
$\omega^{\alpha}(V)\subset V^{\circ}$, then
$\operatorname{rank}\omega^{\alpha}(V)=n$. In particular,
$\operatorname{rank}\omega^{\alpha}(V)=n$ for every $\alpha=1,\ldots,k$. Since
$\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}=0$ and ${\rm
Im}\,\omega^{\alpha}(V)\subset V^{\alpha}$ for $\alpha=1,\ldots,k$, it follows
that $\phi:v\in V\mapsto(\omega^{1}(v),\ldots,\omega^{k}(v))\in
V^{\circ}\oplus\overset{(k)}{\dotsb}\oplus V^{\circ}$, where $\oplus$ stands
for a Whitney sum of vector bundles in the natural way, is injective. Hence,
$\phi$ becomes an isomorphism and $V\simeq\bigoplus_{\alpha=1}^{k}V_{\alpha}$.
Indeed, $v=\sum_{\alpha=1}^{k}\phi^{-1}({\rm pr}_{\alpha}(v))$, where ${\rm
pr}_{\alpha}:(w_{1},\ldots,w_{k})\in V^{\circ}\oplus\ldots\oplus
V^{\circ}\mapsto(0,\ldots,0,w_{\alpha},0,\ldots,0)\mapsto
V^{\circ}\oplus\ldots\oplus V^{\circ}$, is the corresponding decomposition.

Since $V=\bigoplus_{\alpha=1}^{k}V_{\alpha}$ and
$\omega^{\beta}(V_{\beta})\subset V^{\circ}$ has the same rank as $V^{\beta}$,
it follows that $\operatorname{rank}V_{\beta}=n$. Hence, one can consider a
basis $\\{e^{1},\ldots,e^{n}\\}$ of $V^{\circ}$. There exists a basis
$f_{\beta}^{1},\ldots,f_{\beta}^{n}$ of each $V^{\beta}$ such that
$\omega^{\alpha}(f_{\beta}^{i})=-e^{i}\delta_{\alpha}^{\beta}$ for
$i=1,\ldots,n$ and $\alpha,\beta=1\,\ldots,k$. Considering a dual basis
$\\{f^{\beta}_{i},e^{i}\\}$ of $\mathrm{T}_{x}^{*}M$, one has that

$\omega^{\beta}=e^{i}\wedge f^{\beta}_{i}+c_{ij}^{\beta}e^{i}\wedge
e^{j}\,,\qquad\beta=1,\ldots,k\,.$ (5)

If $e^{\beta}_{i}=f^{\beta}_{i}+c_{ij}^{\beta}e^{j}$, then

$\omega^{\beta}=\sum_{i=2}^{n}e^{i}\wedge
e^{\beta}_{i}\,,\qquad\beta=1,\ldots,k\,.$

Note that the change on the covectors $e^{\beta}_{i}$ implies that, in the
dual bases to the bases $\\{e^{i},e_{i}^{\alpha}\\}$ and
$\\{e^{i},f_{i}^{\alpha}\\}$ in $\mathrm{T}_{x}^{*}M$, one has that
$f_{\alpha}^{i}=e_{\alpha}^{i}$ for $\alpha=1,\ldots,k$ and $i=1,\ldots,n$.
Hence, $V_{\alpha}=\langle f_{\alpha}^{1},\ldots,f^{n}_{\alpha}\rangle=\langle
e_{\alpha}^{1},\ldots,e_{\alpha}^{n}\rangle$ for $\alpha=1,\ldots,k$. ∎

It stems from Theorem 3.2 that $\omega^{1},\ldots,\omega^{k}$ have constant
rank. This fact comes from the definition of $k$-symplectic structure, the
dimension of $M$, and the rank of $V$. Note also that the last paragraph in
the proof of Theorem 3.2 can be almost straightforwardly changed to put a
symplectic linear form and a Lagrangian subspace into a canonical form.

######  Definition 3.3.

Given a $k$-symplectic manifold $(M,\omega^{1},\ldots,\omega^{k},V)$ with
$k\neq 1$, we set

$V_{\beta}=\bigcap_{\begin{subarray}{c}\alpha=1\\\
\alpha\neq\beta\end{subarray}}^{k}\ker\omega^{\alpha}\,,\qquad\beta=1,\ldots,k.$

######  Lemma 3.4.

For a $k$-symplectic manifold $(M,\omega^{1},\ldots,\omega^{k},V)$, the
distributions $V^{1},\ldots,V^{k}$ satisfy that every $x\in M$ admits a
coordinate system
$\\{y^{1},\ldots,y^{n};y^{\alpha}_{1},\dotsc,y_{n}^{\alpha}\\}$ on a
neighbourhood of $x$ so that

$V_{\alpha}=\left\langle\frac{\partial}{\partial
y^{\alpha}_{1}},\ldots,\frac{\partial}{\partial
y^{\alpha}_{n}}\right\rangle,\qquad\alpha=1,\ldots,k.$

###### Proof.

Let $y^{1},\ldots,y^{n}$ be common functionally independent first-integrals
for all vector fields taking values in $V$. If $k=1$, the result follows
trivially, so we assume $k>1$. Given different
$\alpha_{1},\ldots,\alpha_{k-1}\in\\{1,\ldots,k\\}$, one has that

$V_{\alpha_{1}}\oplus\dotsb\oplus V_{\alpha_{k-1}}=\ker\omega^{\beta},$

where $\beta$ is the only number in $\\{1,\ldots,k\\}$ not included in
$\\{\alpha_{1},\ldots,\alpha_{k-1}\\}$. Hence, the distribution
$V_{\alpha_{1}}\oplus\ldots\oplus V_{\alpha_{k-1}}$ has rank $n(k-1)$, it is
integrable because $\omega^{\beta}$ is closed, and the vector fields taking
values in it have $n$ common local first-integrals
$y^{\beta}_{1},\ldots,y^{\beta}_{n}$ such that
$\mathrm{d}y^{\beta}_{1}\wedge\ldots\wedge\mathrm{d}y^{\beta}_{n}\wedge\mathrm{d}y^{1}\wedge\ldots\wedge\mathrm{d}y^{n}\neq
0$. By construction,
$\\{y^{1}_{1},\ldots,y^{1}_{n},\ldots,y^{k}_{1},\ldots,y^{k}_{n},y^{1},\ldots,y^{n}\\}$
becomes a local coordinate system on $M$ and

$V_{\alpha}=\left(\bigcap_{i=1}^{n}\ker\mathrm{d}y^{i}\right)\cap\left(\bigcap_{\begin{subarray}{c}\beta\neq\alpha\\\
i=1,\ldots,n\end{subarray}}\ker\mathrm{d}y^{\beta}_{i}\,\right).$

Moreover, $\dfrac{\partial}{\partial
y^{\beta}_{1}},\ldots,\dfrac{\partial}{\partial y^{\beta}_{n}}$ vanish on all
coordinates $y^{\alpha}_{1},\ldots,y^{\alpha}_{n}$ with $\alpha\neq\beta$.
Hence,

$V_{\beta}=\left\langle\frac{\partial}{\partial
y^{\beta}_{1}},\ldots,\frac{\partial}{\partial
y^{\beta}_{n}}\right\rangle=\bigcap_{\alpha\neq\beta=1}^{k}\ker\omega^{\alpha},\qquad\beta=1,\ldots,k.$

∎

######  Theorem 3.5 (Darboux theorem for $k$-symplectic manifolds).

Let $(M,\omega^{1},\dotsc,\omega^{k},V)$ be a $k$-symplectic manifold. Around
every point $x\in M$, there exist local coordinates
$\\{q^{i},p^{\alpha}_{i}\\}$, with $1\leq i\leq n$ and $1\leq\alpha\leq k$,
such that

$\omega^{\alpha}=\sum_{i=1}^{n}\mathrm{d}q^{i}\wedge\mathrm{d}p^{\alpha}_{i}\,,\quad
V=\left\langle\frac{\partial}{\partial
p^{\alpha}_{i}}\right\rangle_{\tiny\begin{gathered}i=1,\ldots,n,\\\
\alpha=1,\ldots,k\end{gathered}}.$ (6)

###### Proof.

By our Darboux theorem for $k$-symplectic vector spaces, namely Theorem 3.2,
there exists a basis
$\\{e^{1},\dotsc,e^{n};e_{1}^{\alpha},\dotsc,e_{n}^{\alpha}\\}_{\alpha=1,\dotsc,k}$
of $\mathrm{T}^{\ast}_{x}M$ such that
$\omega^{\alpha}_{x}=\sum_{i=1}^{n}e^{i}\wedge e_{i}^{\alpha}$ for
$\alpha=1,\dotsc,k$. The basis is chosen so that the dual basis
$\\{e_{1},\dotsc,e_{n},e^{1}_{\alpha},\dotsc,e^{n}_{\alpha}\\}$ , with
$\alpha=1,\ldots,k$, is such that $V=\langle
e_{\alpha}^{i}\rangle_{\begin{subarray}{c}\alpha=1,\dotsc,k\\\
i=1,\dotsc,n\end{subarray}}$. Recall that the subspaces in $\mathrm{T}_{x}M$
of the form

$V_{\beta x}=\bigcap_{\begin{subarray}{c}\alpha=1\\\
\alpha\neq\beta\end{subarray}}^{k}\ker\omega_{x}^{\alpha}=\left\langle
e_{1}^{\beta},\dotsc,e_{n}^{\beta}\right\rangle,\quad\beta=1,\dotsc,k\,,$

satisfy that $V_{x}=\bigoplus_{\beta=1}^{k}V_{\beta x}$. By Lemma 3.4, there
exist variables $\\{y^{j},y_{j}^{\beta}\\}$, with $j=1,\dotsc,n$ and
$\beta=1,\dotsc,k$, such that, locally, $\displaystyle
V^{\beta}=\Big{\langle}\frac{\partial}{\partial
y^{\beta}_{1}},\dotsc,\frac{\partial}{\partial y^{\beta}_{n}}\Big{\rangle}$,
with $\beta=1,\dotsc,k$. Moreover,
$\ker\omega^{\beta}=\bigoplus_{\alpha\neq\beta}V^{\alpha}$. Using previous
results and since ${\left.\kern-1.2pt\omega^{\beta}\right|_{V\times V}}=0$, we
have
$\omega^{\beta}=f_{i}^{j\beta}\mathrm{d}y^{i}\wedge\mathrm{d}y^{\beta}_{j}+g_{ij}\mathrm{d}y^{i}\wedge\mathrm{d}y^{j}$
for certain functions $g_{ij},f_{i}^{j\beta}$, with $i,j=1,\dotsc,n$ and
$\beta=1,\dotsc,k$. Since $\mathrm{d}\omega^{\beta}=0$, it follows that
$f^{j\beta}_{i}=f^{j\beta}_{i}(y^{l},y_{l}^{\beta})$ and
$g_{ij}=g_{ij}(y^{l},y_{l}^{\beta})$ for $i,j,l=1,\ldots,n$. Therefore, each
$\omega^{\beta}$ can be considered as a differential two-form on
$\mathbb{R}^{2n}$. Moreover, each $V^{\beta}$ can be then considered as a
Lagrangian distribution of a symplectic two-form $\omega^{\beta}$, when it is
considered as a differential two-form on $\mathbb{R}^{2n}$. Consequently, for
a fixed $\beta$, one has

$0=-\frac{\partial}{\partial
y^{\beta}_{j}}y^{i}=\operatorname{\iota}_{X^{\beta}_{y^{i}}}\operatorname{\iota}_{\partial/\partial
y^{\beta}_{j}}\omega^{\beta},\quad\quad
i,j=1,\dotsc,n\quad\Longrightarrow\quad
X^{\beta}_{y^{i}}\in(V_{\beta})^{\perp}=V_{\beta}\,,\quad i=1,\dotsc,n\,.$

Note that the orthogonal is relative to the restriction of $\omega^{\beta}$ to
$\mathbb{R}^{2n}$. Whatever, by additionally considering
$\iota_{X_{y^{i}}^{\beta}}\omega^{\alpha}=0$ for $\alpha\neq\beta$, one can
also see that $X^{\beta}_{y^{i}}$ becomes a vector field taking values in
$V^{\beta}$. What follows is an adaptation of the Liouville–Mineur–Arnold
theorem (see also [15]). Since $V^{\alpha}$ is integrable, we can consider a
leaf $F$ of $V^{\alpha}$ and its canonical inclusion
$\jmath_{F}:F\hookrightarrow M$. Let us define the map $\zeta\colon x\in
M\mapsto(y^{1}(x),\ldots,y^{n}(x))\in\mathbb{R}^{n}$. Consider a regular point
$x^{\prime}\in M$ of $\zeta$. Since the map $\zeta$ is regular in an open
neighbourhood of $x^{\prime}$, there exist vector fields $Y_{1},\dotsc,Y_{n}$
on a neighbourhood of $x^{\prime}$ such that $Y_{i}$ and
$\displaystyle\frac{\partial}{\partial y^{i}}$ on $\mathbb{R}^{n}$ are
$\zeta$-related for $i=1,\ldots,n$. Consider the inner contractions
$\Theta_{i}^{\alpha}=\operatorname{\iota}_{Y_{i}}\omega^{\alpha}$ for
$i=1,\ldots,n$ on a neighbourhood of $x^{\prime}$ in $M$ and the vector fields
$X^{\alpha}_{y^{i}}$, which take values in $V_{\alpha}$. Then,

$\operatorname{\iota}_{X^{\alpha}_{y^{i}}}{\Theta_{j}^{\alpha}}=\operatorname{\iota}_{X^{\alpha}_{y^{i}}}\operatorname{\iota}_{Y_{j}}\omega^{\alpha}=\omega^{\alpha}(Y_{j},X^{\alpha}_{y^{i}})=-\omega^{\alpha}(X^{\alpha}_{y^{i}},Y_{j})=-Y_{j}y^{i}=-\delta^{i}_{j},\qquad
i,j=1,\ldots,n.$

Hence, given two vector fields $X^{\alpha}_{y^{i}},X^{\alpha}_{y^{j}}$, one
has

$(\mathrm{d}\Theta_{\ell}^{\alpha})(X^{\alpha}_{y^{i}},X^{\alpha}_{y^{j}})=X^{\alpha}_{y^{i}}\Theta^{\alpha}_{\ell}(X^{\alpha}_{y^{j}})-X^{\alpha}_{y^{j}}\Theta^{\alpha}_{\ell}(X^{\alpha}_{y^{i}})-\Theta^{\alpha}_{\ell}([X^{\alpha}_{y^{i}},X^{\alpha}_{y^{j}}])=0\,.$

The latter is due to the fact that $[X^{\alpha}_{y^{i}},X^{\alpha}_{y^{j}}]$
is the Hamiltonian vector field of
$\\{y^{i},y^{j}\\}=X^{\alpha}_{y^{j}}y^{i}=0$ because $X_{y^{j}}$ takes values
in $V_{\alpha}$. Thus, $\jmath_{F}^{*}\Theta^{\alpha}$ is closed and there
exists a potential
$\jmath_{F}^{*}\Theta_{i}^{\alpha}=\mathrm{d}p^{\alpha}_{i}$. And recalling
that $\omega^{\alpha}|_{V_{\alpha}\times V_{\alpha}}=0$, it follows that
$\omega^{\alpha}=\mathrm{d}y^{i}\wedge\mathrm{d}p^{\alpha}_{i}$. Moreover, it
follows that

$V_{\alpha}=\left\langle\frac{\partial}{\partial
p^{\alpha}_{i}}\right\rangle,\qquad\alpha=1,\ldots,k,$

and $V$ takes the proposed form. ∎

Let us recall that the above proof could have been cut by the half by
referring straightforwardly to the Liouville–Mineur–Arnold theorem, as since
$\\{y_{i},y_{j}\\}=0$, with $i,j=1,\ldots,n$, that theorem implies that there
are functions $p^{\beta}_{1},\dotsc,p^{\beta}_{n}$ with $\beta=1,\ldots,k$
such that $\omega^{\beta}=\mathrm{d}y^{i}\wedge\mathrm{d}p^{\beta}_{i}$ for
each $\beta=1,\ldots,k$. Instead, we decided to give a complete, self-
contained proof. Without this full explanation, Theorem 3.5 would probably be
the shortest direct proof of the Darboux theorem for $k$-symplectic manifolds
in the literature. Although Theorem 3.5 relies on Lemma 3.4 and the
$k$-symplectic linear Darboux theorem, Lemma 3.4 is a rather straightforward
geometric result, which was described carefully to verify all the details, and
only the fact that $V=\bigoplus_{\alpha=1}^{k}V^{\alpha}$ is needed from the
$k$-symplectic linear Darboux theorem to prove our full $k$-symplectic Darboux
theorem.

Moreover, note that one could have assumed that Darboux coordinates only are
concerned with the canonical expressions of $\omega^{1},\ldots,\omega^{k}$. It
turns out that given the conditions on the distribution, once we put
$\omega^{1},\ldots,\omega^{k}$ in a canonical manner, we also put a basis of
$V$ in the desired form. We will see in next section that this is not the case
for Darboux coordinates for other structures.

######  Definition 3.6.

Given a $k$-symplectic manifold $(M,\omega^{1},\ldots,\omega^{k},V)$, we call
$k$-symplectic Darboux coordinates the coordinates allowing us to write
$\omega^{1},\ldots,\omega^{k}$ and $V$ in the form (6).

The $k$-symplectic Darboux coordinates will be called just Darboux coordinates
when it does not lead to any misunderstanding. Note that the proof of Theorem
3.5 shows that $k$-symplectic Darboux coordinates induce the existence of a
distribution $V^{\prime}=\langle\partial/\partial
y^{1},\ldots,\partial/\partial y^{n}\rangle$ that allows us to state the
following result.

######  Corollary 3.7.

Every $k$-symplectic manifold $(M,\omega^{1},\ldots,\omega^{k},V)$ admits,
locally, a supplementary integrable distribution $V^{\prime}$ on $M$ such that
$V\oplus V^{\prime}=\mathrm{T}M$ and $\omega^{\alpha}|_{V^{\prime}\times
V^{\prime}}=0$ for $\alpha=1,\ldots,k$.

The canonical model of a $k$-symplectic manifold is the cotangent bundle of
$k^{1}$-covelocities, namely
$\bigoplus^{k}\mathrm{T}^{\ast}Q=\mathrm{T}^{\ast}Q\oplus\overset{k}{\dotsb}\oplus\mathrm{T}^{\ast}Q$
(the Whitney sum of $k$ copies of the cotangent bundle of a manifold $Q$),
equipped with the distribution $V=\ker\mathrm{T}\pi$, where
$\pi^{\alpha}\colon\bigoplus^{k}\mathrm{T}^{\ast}Q\to\mathrm{T}^{\ast}Q$ and
$\pi:\bigoplus^{k}\mathrm{T}^{\ast}Q\to Q$ are the canonical projections onto
the $\alpha$-th component and $Q$ respectively, and the canonical
presymplectic two-forms $\omega^{\alpha}=(\pi^{\alpha})^{*}\omega$ with
$\alpha=1,\ldots,k$, where $\omega$ stands for the canonical symplectic two-
form in $\mathrm{T}^{\ast}Q$. In this model, natural coordinates are Darboux
coordinates, and the $k$-symplectic Darboux theorem states that $k$-symplectic
manifolds are locally diffeomorphic to a cotangent bundle of
$k^{1}$-covelocities. Meanwhile, the distribution $V^{\prime}$ is a
distribution in $\bigoplus_{\alpha=1}^{k}\mathrm{T}^{\ast}Q$ whose leaves
project diffeomorphically onto $Q$.

As in the previous sections, one can introduce the notion of compatible
connection with a $k$-symplectic manifold [8, 14].

######  Definition 3.8.

A $k$-symplectic connection on a $k$-symplectic manifold
$(M,\omega^{\alpha},V)$ is a connection $\nabla$ on $M$ such that
$\nabla\omega^{\alpha}=0$ for every $\alpha=1,\dotsc,k$.

Again, Darboux coordinates allow us to define, locally, a connection,
$\nabla$, such that $\nabla\omega^{\alpha}=0$ for $\alpha=1,\ldots,k$. And
vice versa, the $k$-symplectic linear Darboux Theorem allows us to put
$\omega^{1},\ldots,\omega^{k}$ and the distribution $V$ into a canonical form
on the tangent space at a point and, a flat connection compatible with the
$k$-symplectic manifold enables us to expand this canonical form to an open
neighbourhood of the initial point where $\omega^{1},\ldots,\omega^{k}$ and
$V$ take the form (6).

It is worth recalling the interesting work [14], where connections compatible
with $k$-symplectic structures are studied. These connections depend on the
existence of certain foliations and are canonical once such foliations are
given. By using such foliations and distributions, the Darboux theorem can be
proved. We find that our approach here is more direct than that in [14] and
the Darboux theorem is given in our work more geometrically. Note that a
$k$-symplectic Darboux theorem also appears as a particular case of the
multisymplectic theory in [29].

Now, let us study Darboux theorems for $k$-presymplectic manifolds (see [30,
37] for some previous results on this case). This case poses several
fundamental problems. First, there exist several possible definitions of
$k$-presymplectic manifolds depending on their possible applications or
representative cases. Some possible definition of $k$-presymplectic manifold
can be found in [30]. Meanwhile, [37] defines a $k$-presymplectic manifold as
a manifold equipped with $k$ closed two-forms. It is clear that we will not
have Darboux coordinates with such a general definition. As shown next, a
direct analogue of the Darboux coordinates is not available in some of the
possible definitions of $k$-presymplectic structure, while cases that admit
Darboux coordinates may not be of physical interest. Let us give a brief
analysis of this matter.

######  Definition 3.9.

Let $\bigoplus_{\alpha=1}^{k}\mathrm{T}^{*}Q$ be endowed with its canonical
$k$-symplectic structure $\omega^{1},\ldots,\omega^{k}$ and let
$\pi\colon\bigoplus_{\alpha=1}^{k}\mathrm{T}^{*}Q\rightarrow Q$ be the
canonical projection onto $Q$. A canonical foliated $k$-presymplectic manifold
is a tuple $(S,\omega_{S}^{1},\ldots,\omega_{S}^{k})$ given by a submanifold
$S\subset\bigoplus_{\alpha=1}^{k}\mathrm{T}^{*}Q$ such that $\pi|_{S}\colon
S\rightarrow Q$ is a fibre bundle and $S$ is endowed, for $\jmath_{S}\colon
S\rightarrow\bigoplus_{\alpha=1}^{k}\mathrm{T}^{*}Q$ being the canonical
inclusion, with the $k$ differential two-forms
$\omega^{\alpha}_{S}=\jmath^{*}_{S}\omega^{\alpha}$, for $\alpha=1,\ldots,k$.
The rank of the fibration $\pi|_{S}:S\rightarrow Q$ is called the rank of
$(S,\omega^{1}_{S},\ldots,\omega^{k}_{S})$ while
$\omega^{1}_{S},\ldots,\omega^{k}_{S}$ are called a canonical foliated
$k$-presymplectic structure.

More generally, the above gives rise to the following definition.

######  Definition 3.10.

A foliated $k$-presymplectic manifold is a tuple
$(M,\omega^{1},\ldots,\omega^{k})$ such that there exists a canonical foliated
$k$-presymplectic manifold $(S,\omega^{1}_{S},\ldots,\omega^{k}_{S})$ and a
global diffeomorphism $\phi:M\rightarrow S$ such that
$\phi^{*}\omega_{S}^{\alpha}=\omega^{\alpha}$ for $\alpha=1,\ldots,k$. A
foliated $k$-presymplectic manifold $(M,\omega^{1},\ldots,\omega^{k})$ is
exact if $\omega^{1},\ldots,\omega^{k}$ are exact.

It is worth noting that the previous definition also makes sense for $\phi$
being, only, a local diffeomorphism. In that case, the main results to be
displayed afterward remain valid, but many more technical details are to be
considered to prove them. To keep our presentation simple and highlight the
main ideas about Darboux coordinates, which are generically local, we have
defined $\phi$ to be a global diffeomorphism.

Definition 3.9 implies that $\omega_{S}^{1},\ldots,\omega_{S}^{k}$ admit a
natural distribution $V=\ker\mathrm{T}\pi\cap\mathrm{T}S$ of rank $\dim S-\dim
Q$ such that $\omega_{S}^{\alpha}|_{V\times V}=0$ for $\alpha=1,\ldots,k$. If
$S=\bigoplus_{\alpha=1}^{k}\mathrm{T}^{\ast}Q$, then $V=\ker\mathrm{T}\pi$ and
$S$ gives rise to a $k$-symplectic structure admitting Darboux coordinates.

Let us illustrate by means of a simple example why a Darboux $k$-presymplectic
theorem does not exist for general foliated $k$-presymplectic manifolds. It is
worth noting that Darboux coordinates for families of closed differential
forms are, at the very last instance, a way of writing them in a coordinate
system so that their associated coordinates are constant. The following
theorem shows that this is impossible for general $k$-presymplectic manifolds.

######  Theorem 3.11.

Every rank-zero exact canonical foliated $k$-presymplectic structure is
equivalent to $k$ exact differential two-forms on $Q$.

###### Proof.

An exact canonical foliated $k$-presymplectic manifold
$(S\subset\bigoplus_{\alpha=1}^{k}\mathrm{T}^{\ast}Q,\omega_{S}^{1},\ldots,\omega_{S}^{k})$
gives rise, as $S$ is diffeomorphic to $Q$ via
$\pi|_{S}:S\subset\bigoplus_{\alpha=1}^{k}\mathrm{T}^{\ast}Q\rightarrow Q$, to
a unique family of exact differential two-forms,
$\omega^{1}_{Q},\ldots,\omega^{k}_{Q}$, on $Q$ satisfying that
$\pi|_{S}^{*}\omega_{Q}^{1}=\omega_{S}^{1},\ldots,\pi|_{S}^{*}\omega_{Q}^{k}=\omega_{S}^{k}$.

Conversely, $k$ exact presymplectic two-forms
$\omega^{1}_{Q},\ldots,\omega^{k}_{Q}$ on $Q$ with potentials
$\theta^{1},\ldots,\theta^{k}$ give rise to a section
$S=\\{(q,\theta^{1}(q),\ldots,\theta^{k}(q))\mid q\in Q\\}$ of
$\pi\colon\bigoplus_{\alpha=1}^{k}\mathrm{T}^{*}Q\rightarrow Q$. Note that

$\jmath_{S}^{*}\omega^{\alpha}=-\jmath_{S}^{*}\mathrm{d}(p^{\alpha}_{i}\mathrm{d}y^{i})=-\mathrm{d}\theta^{\alpha}|_{S}=\pi|_{S}^{*}\omega^{\alpha}_{Q}\,,\qquad\alpha=1,\ldots,k\,.$

Then, $\omega^{1}_{Q},\ldots,\omega^{k}_{Q}$ are exact and equivalent to a
rank-zero canonical foliated $k$-presymplectic structure. ∎

Since there is no way to put $k$ arbitrary closed differential two-forms on
$Q$ into a coordinate system so that all of them will have constant
coefficients, there will be no general Darboux theorem for foliated
$k$-presymplectic manifolds, and thus there is no Darboux theorem for
$k$-presymplectic manifolds in general.

Theorem 3.11 can be considered as an extreme case of canonical foliated
$k$-presymplectic manifold. For the case of a fibration $\pi|_{S}:S\rightarrow
Q$ of rank one, it is simple to find new examples where there will be no
Darboux coordinates. Assume the simple case of a fibration of rank one given
by a submanifold
$S\subset\bigoplus_{\alpha=1}^{2}\mathrm{T}^{\ast}\mathbb{R}^{2}$ onto
$\mathbb{R}^{2}$. Since $S$ has dimension three, the two differential forms
$\omega_{S}^{1},\omega_{S}^{2}$ can be assumed to have rank two and non-
trivial common intersection of their kernels. In such a case, they are
proportional. One of them can always be put into canonical form for certain
variables, because is presymplectic. Since they are proportional and due to
the closeness condition, they depend only on two variables. Hence, to put them
in canonical form with some Darboux variables amounts to putting two different
volume forms on $\mathbb{R}^{2}$ in canonical form for the same Darboux
variables, which is impossible.

######  Example 3.12.

Let us describe in more detail a more complex example of a foliated
2-presymplectic manifold that does not admit Darboux coordinates. Consider
$\bigoplus_{\alpha=1}^{2}\mathrm{T}^{\ast}\mathbb{R}^{2}$ and the fibration of
the submanifold $S$ onto $\mathbb{R}^{2}$ with rank one of the form

$S=\\{(p^{(1)}_{1}(\lambda,y^{1},y^{2})\mathrm{d}y^{1}+p^{(1)}_{2}(\lambda,y^{1},y^{2})\mathrm{d}y^{2},p^{(2)}_{1}(\lambda,y^{1},y^{2})\mathrm{d}y^{1}+p^{(2)}_{2}(\lambda,y^{1},y^{2})\mathrm{d}y^{2}):\lambda,y^{1},y^{2}\in\mathbb{R}\\}.$

In particular, consider

$p_{1}^{(1)}=\lambda\,,\qquad p_{2}^{(1)}=0\,,\qquad
p_{1}^{(2)}=f(\lambda,y^{1})\,,\qquad p_{2}^{(2)}=0\,,$

for a certain function $f(\lambda,y^{1})$ such that $\partial
f/\partial\lambda$ is different from the constant functions zero and one.
Hence, $\omega^{1}_{S}=\mathrm{d}\lambda\wedge\mathrm{d}y^{1}$ and
$\displaystyle\omega^{2}_{S}=\frac{\partial
f}{\partial\lambda}(\lambda,y^{1})\mathrm{d}\lambda\wedge\mathrm{d}y^{1}$,
which are closed, proportional, have rank-one kernel and cannot be put into a
canonical form for canonical coordinates because
$\omega^{1}_{S},\omega^{2}_{S}$ amount to two different volume forms on
$\mathbb{R}^{2}$.

There are several manners of defining a $k$-presymplectic manifold. The
following one offers a possibility.

######  Definition 3.13.

Let $M$ be a $(n(k+1)-m)$-dimensional manifold, with $0\leq m\leq nk$. A
$k$-presymplectic structure on $M$ is a family
$(\omega^{1},\dotsc,\omega^{k},V)$, where $V$ is an $r$-dimensional integrable
distribution and $\omega^{1},\dotsc,\omega^{k}$ are closed differential two-
forms on $M$ with $\operatorname{rank}\omega^{\alpha}=2r_{\alpha}$ and
$r=\sum_{\alpha=1}^{k}r_{\alpha}$, where $1\leq r_{\alpha}\leq n$, satisfying
that

${\left.\kern-1.2pt\omega^{\alpha}\right|_{V\times
V}}=0\,,\qquad\alpha=1,\ldots,k\,.$

A manifold $M$ endowed with a $k$-presymplectic structure is called a
$k$-presymplectic manifold.

We would expect to obtain for every $k$-presymplectic structure
$(M,\omega^{1},\ldots,\omega^{k},V)$, where
$\operatorname{rank}\omega^{\alpha}=2r_{\alpha}$, with $1\leq r_{\alpha}\leq
n$, and every $x\in M$ a local coordinate system $\\{y^{i},p_{i}^{\alpha}\\}$
around $x$ so that

$\omega^{\alpha}=\mathrm{d}y^{i^{\alpha}_{j}}\wedge\mathrm{d}p^{\alpha}_{i^{\alpha}_{j}}\,,\qquad\alpha=1,\ldots,k\,,$

for certain $i^{\alpha}_{j}\in\\{1,\ldots,n\\}$ for $j=1,\ldots,r_{\alpha}$
for every $\alpha=1,\ldots,k$. Nevertheless, Example 3.12 represents a
counterexample for the existence of Darboux coordinate system for
$k$-presymplectic structures.

Contrary to previous examples, we will give conditions ensuring that a
$k$-presymplectic manifold admits Darboux coordinates. Indeed, the manifold
$S$ is three-dimensional, while $k=2$. The associated presymplectic forms have
rank two. The distribution $V$ is then two-dimensional and generated, for
instance, by the vector fields
$\langle\partial/\partial\lambda,\partial/\partial y^{2}\rangle$. Then, $n$
and $m$ can be fixed to be two and three.

It is worth noting that for a $k$-presymplectic structure on $M$, any
Riemannian metric $g$ on $M$ allows one to obtain a decomposition of a
subspace $E\subset\mathrm{T}_{x}M$ as a direct sum of subspaces

$E^{\kappa_{1},\ldots,\kappa_{k}}=E\cap\left(\bigcap_{\alpha=1}^{k}(\ker\omega_{x}^{\alpha})^{\kappa_{\alpha}}\right)\,,$
(7)

where $\kappa_{\alpha}\in\\{0,1\\}$, while
$(\ker\omega_{x}^{\alpha})^{0}=\ker\omega_{x}^{\alpha}$ and
$(\ker\omega_{x}^{\alpha})^{1}=(\ker\omega_{x}^{\alpha})^{\perp_{g}}$, where
$\perp_{g}$ is the orthogonal relative to the introduced metric $g$. The main
aim of this decomposition is to divide $\mathrm{T}_{x}M$ into two subspaces,
$V,S$, given by direct sums of the subspaces in (7) in such a manner that
$\omega^{\alpha}(V)$ and $\omega^{\alpha}(S)$ have rank $r_{\alpha}$, while
$\omega^{\alpha}(V)\cap\omega^{\alpha}(S)=0$ for $\alpha=1,\ldots,k$. As in
the case of $k$-symplectic linear spaces, one can now prove a
$k$-presymplectic linear space Darboux theorem.

######  Lemma 3.14.

($k$-presymplectic linear Darboux theorem) Given a $k$-presymplectic structure
$(\omega^{1},\ldots,\omega^{k},V)$ on $M$, where
$\operatorname{rank}\omega^{\alpha}=2r^{\alpha}$ for $\alpha=1,\ldots,r$. Let
$D=\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}$ have rank $d$ and let
$\operatorname{rank}V=r+d=\sum_{\alpha=1}^{k}r_{\alpha}+d$ be so that

$\dim V_{\alpha}=r_{\alpha},\quad
V=D\oplus\bigoplus_{\beta=1}^{k}V_{\beta}\,,\quad
D+V_{\alpha}=V\cap\left(\bigcap_{\beta\neq\alpha}\ker\omega^{\beta}\right)\,,\,\,(k\neq
1)\quad\alpha=1,\ldots,k\ ,$ (8)

and $\dim M=n+r+d$. Then, at every $\mathrm{T}^{\ast}_{x}M$, for $x\in M$, one
can set a basis of the form
$\\{e^{1},\ldots,e^{n};e^{\alpha}_{\mu^{\alpha}_{j}},v^{1},\dotsc,v^{d}\\}$,
with $\mu^{\alpha}_{j}\in I_{\alpha}\subset\\{1,\dotsc,n\\}$ and
$\absolutevalue{I_{\alpha}}=r_{\alpha}$ with $\alpha=1,\ldots,k$, of
$\mathrm{T}^{\ast}_{x}M$ such that

$\omega_{x}^{\alpha}=\sum_{j=1}^{r_{\alpha}}e^{\mu^{\alpha}_{j}}\wedge
e^{\alpha}_{\mu^{\alpha}_{j}}\,,\qquad\alpha=1,\ldots,k\,,\qquad D_{x}=\langle
v_{1},\ldots,v_{d}\rangle,\qquad V_{\alpha x}=\left\langle
e_{\alpha}^{\mu^{\alpha}_{j}}\right\rangle.$

###### Proof.

Note that $\dim M=n+r+d$. Since $D=\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}$
has rank $d$, one has that

$D_{x}^{\circ}=\operatorname{Im}\omega^{1}_{x}+\ldots+\operatorname{Im}\omega^{k}_{x}\,,\qquad\forall
x\in M\,,$ (9)

is such that $\operatorname{rank}D^{\circ}=n+r$. Since
$\omega^{\beta}|_{V\times V}=0$, it follows that $\omega^{\beta}(V)\subset
V^{\circ}$ for every $\beta$. We have

$\omega^{1}(V)+\ldots+\omega^{k}(V)\subset V^{\circ}.$

Note that $\operatorname{rank}V^{\circ}=n$. From the second and third
condition in (8), it follows that $V_{\alpha}\cap\ker\omega^{\alpha}=0$.
Moreover, one has that
$\operatorname{rank}\omega^{\alpha}(V_{\alpha})=r_{\alpha}=\operatorname{rank}\omega^{\alpha}(V)$
for $\alpha=1,\ldots,k$. Consider the supplementary $S=V^{\perp_{g}}$ to $V$.
Then, $\operatorname{rank}V^{\perp_{g}}=\dim M-r-d=n$ and
$\operatorname{rank}\omega_{x}^{\beta}(S_{x})\leq n$ for every $x\in M$. Due
to (9) and the above, one has that

$\omega^{1}(V^{\perp_{g}})+\ldots+\omega^{k}(V^{\perp_{g}})$

is a distribution of rank $r$ at least. By our decomposition, every $\alpha$
allows us to divide $V^{\perp_{g}}$ into two spaces in the form
$V^{\perp_{g}}=\Upsilon_{\alpha}\oplus\left(\ker\omega^{\alpha}\cap
V^{\perp_{g}}\right)$, where $\Upsilon_{\alpha}$ has rank $r_{\alpha}$ because
$\ker\omega^{\alpha}=n+r+d-2r_{\alpha}$ and $\ker\omega^{\alpha}\cap
V=r+d-r_{\alpha}$. Then, $\omega^{\alpha}(V^{\perp_{g}})$ is equal to the
image of a subspace of rank $r_{\alpha}$ of $V^{\perp_{g}}$ and it therefore
has rank $r_{\alpha}$ and $\omega^{\alpha}(S)\cap\omega^{\alpha}(V)=0$. Then,

$\rank(\omega^{1}(V^{\perp_{g}})+\ldots+\omega^{k}(V^{\perp_{g}}))=r\,,\quad\omega^{1}(V)+\ldots+\omega^{k}(V)=V^{\circ}\,.$
(10)

Note that $\ker\omega^{\alpha}=n+d+r-2r_{\alpha}$ and
$\omega^{\alpha}(V)=\omega^{\alpha}(V_{\alpha}\oplus(\ker\omega^{\alpha}\cap
V))$. Due to the second expression in (10), the sum of the codistributions
$S_{*}^{\alpha}=\omega^{\alpha}(V)$ of $\mathrm{T}^{\ast}M$ for
$\alpha=1,\ldots,k$ has rank $n$, but they do not need to be in direct sum. A
non-degenerate contravariant symmetric tensor, $g^{*}$. on
$S_{*}=S_{*}^{1}+\ldots+S_{*}^{k}$ can be used to give a decomposition of it
into subspaces in direct sum of the form

$S_{*}^{\kappa_{1},\ldots,\kappa_{k}}=\bigcap_{\alpha=1}^{k}S^{\kappa_{\alpha}}_{\alpha*},$

where $\kappa_{\alpha}\in\\{0,1\\}$ for $\alpha=1,\ldots,k$, while
$S^{1}_{\alpha*}=S^{\alpha}_{*}$ and
$S^{0}_{\alpha}=(S^{\alpha}_{*})^{\perp_{g^{*}}}$, namely the orthogonal in
$S_{*}$ of $S_{*}^{\alpha}$ relative to $g^{*}$. Take a basis of $S_{*}$
associated with our decomposition. For the elements of such a basis spanning
$S_{*}^{\alpha}$, there will be unique elements in $V_{\alpha}$ whose image
under $\omega^{\alpha}$ give minus the corresponding basis in
$S_{*}^{\alpha}$. Take a supplementary to $S_{*}$ in $\mathrm{T}^{\ast}M$, of
dimension $d+r$, dual to a basis adapted to the decomposition of $V$ and
vanishing on $V^{\perp_{g}}$. It is worth noting that we have a decomposition

$\mathrm{T}M=\left[\Upsilon_{\alpha}\oplus(\ker\omega^{\alpha}\cap
V^{\perp_{g}})\right]\oplus\left[V_{\alpha}\oplus\left(\bigoplus_{\beta\neq\alpha}V_{\beta}\right)\oplus
D\right]$

and a dual one in $S_{*}\oplus(V^{\perp_{g}})^{\circ}.$ In such a basis, the
form of $\omega^{\alpha}$ goes back to (5) and the same technique in Theorem
3.2 gives the canonical form for every $\omega^{\alpha}$ with
$\alpha=1,\ldots,k$. Finally, if $w_{1},\ldots,w_{d}$ is a basis of $D$ dual
to the one chosen in $\mathrm{T}^{\ast}_{x}M$, one has that

$\omega^{\beta}=\sum_{j=1}^{r_{\beta}}e^{\mu_{j}^{\beta}}\wedge
e^{\beta}_{\mu_{j}^{\beta}}\,,\qquad V_{\beta}=\left\langle
e_{\beta}^{\mu^{\beta}_{j}}\right\rangle,\qquad\beta=1,\ldots,k.$

∎

As proved above, depending on their exact definition, $k$-presymplectic
manifolds need not have a Darboux theorem (whatever this means, because we can
have different ways of defining such an object). That is why we hereafter a
definition of $k$-presymplectic manifold ensuring the existence of a
particular case of $k$-presymplectic Darboux theorem. This is done by assuming
the existence of certain integrable distribution with particular properties.

######  Definition 3.15.

A $k$-presymplectic manifold $(M,\omega^{1},\ldots,\omega^{k},V)$ is a
$k$-presymplectic manifold such that $\dim M=n+r+d$ where
$d=\operatorname{rank}\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}$ and
$\operatorname{rank}\omega^{\alpha}=2r^{\alpha}$, the $V$ is an integrable
distribution such that $\omega|_{V\times V}=0$ of rank $r+d$ and there are
integrable distributions $\bigoplus_{\alpha=1}^{k}V_{\alpha}$,
$V_{1},\ldots,V_{k},D$ so that

$V=\bigoplus_{\alpha=1}^{k}V_{\alpha}\oplus D,\qquad
D=\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha},\qquad
D+V_{\beta}=\bigcap_{\beta\neq\alpha}\ker\omega^{\alpha}\cap V\ \ (k\neq
1)\,,\qquad\beta=1,\ldots k\,.$

######  Lemma 3.16.

Given a $k$-presymplectic manifold $(M,\omega^{1},\ldots,\omega^{k},V)$, the
distributions $V^{\alpha}$, with $\alpha=1,\ldots,k$, satisfy that every $x\in
M$ admits a coordinated neighbourhood with coordinates

$\\{y^{1},\ldots,y^{n},z^{1},\ldots,z^{d},y^{\alpha}_{1},\dotsc,y_{r_{\alpha}}^{\alpha}\\}\,,\qquad\alpha=1,\ldots,k\,,$

on a neighbourhood of $x$ so that

$V_{\alpha}=\left\langle\frac{\partial}{\partial
y^{\alpha}_{1}},\ldots,\frac{\partial}{\partial
y^{\alpha}_{r_{\alpha}}}\right\rangle\,,\qquad\alpha=1,\ldots,k\,,\qquad
D=\left\langle\frac{\partial}{\partial z^{1}},\ldots,\frac{\partial}{\partial
z^{d}}\right\rangle\,.$

###### Proof.

Let $y^{1},\ldots,y^{n}$ be common functionally independent first-integrals
for all vector fields taking values in $V$. Since $D$ is a regular
distribution of rank $d$ given by the intersection of kernels of the closed
forms $\omega^{1},\ldots,\omega^{k}$, it is integrable. It is assumed that
$\bigoplus_{\alpha=1}^{k}V_{\alpha}$ is integrable. Hence,
$V_{1}\oplus\ldots\oplus V_{k}$ has common first-integrals
$z^{1},\ldots,z^{d}$ such that
$\mathrm{d}z^{1}\wedge\ldots\wedge\mathrm{d}z^{d}\wedge\mathrm{d}y^{1}\wedge\ldots\wedge\mathrm{d}y^{n}\neq
0$. If $k=1$, the result of our lemma easily follows. Assume that $k>1$. Given
different integers $\alpha_{1},\ldots,\alpha_{k-1}\in\\{1,\ldots,k\\}$, one
has that,

$V_{\alpha_{1}}\oplus\dotsb\oplus V_{\alpha_{k-1}}\oplus
D=\ker\omega^{\beta}\cap V\,,$

where $\beta$ is the only number in $\\{1,\ldots,k\\}$ not included in
$\\{\alpha_{1},\ldots,\alpha_{k-1}\\}$. Hence, the distribution
$V_{\alpha_{1}}\oplus\ldots\oplus V_{\alpha_{k-1}}\oplus D$ has corank
$r_{\beta}$, it is integrable, and the vector fields taking values in it have
$r_{\beta}$ common local first-integrals
$y^{\beta}_{1},\ldots,y^{\beta}_{r_{\beta}}$ such that

$\mathrm{d}z_{1}\wedge\ldots\wedge\mathrm{d}z_{d}\wedge\mathrm{d}y^{\beta}_{1}\wedge\ldots\wedge\mathrm{d}y^{\beta}_{r_{\beta}}\wedge\mathrm{d}y^{1}\wedge\ldots\wedge\mathrm{d}y^{n}\neq
0.$

By construction,
$\\{y^{1}_{1},\ldots,y^{1}_{r_{1}},\ldots,y^{k}_{1},\ldots,y^{k}_{r_{k}},z^{1},\ldots,z^{d},y^{1},\ldots,y^{n}\\}$
becomes a local coordinate system on $M$ and

$V_{\alpha}=\left(\bigcap_{i=1}^{d}\ker\mathrm{d}z^{i}\right)\cap\left(\bigcap_{i=1}^{n}\ker\mathrm{d}y^{i}\right)\cap\bigcap_{\begin{subarray}{c}\beta\neq\alpha\\\
i=1,\ldots,r_{\beta}\end{subarray}}\ker\mathrm{d}y^{\beta}_{i}\,.$

Moreover, $\dfrac{\partial}{\partial y^{\beta}_{i}}$ with
$i=1,\ldots,r_{\beta}$ vanish on all coordinates $y^{\alpha}_{j}$ with
$\alpha\neq\beta$ and $j=1,\ldots,r_{\alpha}$. Hence,

$\left\langle\frac{\partial}{\partial
y^{\beta}_{1}},\ldots,\frac{\partial}{\partial
y^{\beta}_{r_{\beta}}}\right\rangle=V_{\beta}\,,\qquad\beta=1,\ldots,k\,,$

and

$\left\langle\frac{\partial}{\partial z^{1}},\ldots,\frac{\partial}{\partial
z^{d}}\right\rangle=D\,.$

∎

Once the above is proved, the following theorem is immediate. One only has to
slightly adapt Theorem 3.5 by considering that
$\operatorname{rank}V_{\alpha}=r_{\alpha}$ for $\alpha=1,\ldots,k$ and to
restrict $\omega^{\alpha}$ to the integral submanifolds of
$V_{\alpha}\oplus\Upsilon_{\alpha}$, which have dimension $2r^{\alpha}$, where
$\omega^{\alpha}$ becomes symplectic.

######  Theorem 3.17 ($k$-presymplectic Darboux theorem).

Let $(M,\omega^{1},\dotsc,\omega^{k},V)$ be a $k$-presymplectic manifold such
that $\rank\omega^{\alpha}=2r_{\alpha}$, with $1\leq r_{\alpha}\leq n$. The
dimension of $M$ is $n+r+d$. For every point $x\in M$, there exist local
coordinates $\\{y^{i},y^{\alpha}_{\mu^{\alpha}_{j}},z^{j}\\}$, with $1\leq
i\leq n$, $\mu^{\alpha}_{j}\in I_{\alpha}\subseteq\\{1,\dotsc,n\\}$,
$|I_{\alpha}|=r_{\alpha}$, $1\leq j\leq r_{\alpha}$ and $1\leq\alpha\leq k$,
such that

$\omega^{\alpha}=\sum_{j=1}^{r_{\alpha}}\mathrm{d}y^{\mu^{\alpha}_{j}}\wedge\mathrm{d}y^{\alpha}_{\mu^{\alpha}_{j}},\quad
V_{\alpha}=\left\langle\displaystyle\frac{\partial}{\partial
y^{\alpha}_{\mu^{\alpha}_{j}}}\right\rangle\,,\quad\alpha=1,\ldots,k\,,\quad\displaystyle\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}=\left\langle\displaystyle\frac{\partial}{\partial
z^{j}}\right\rangle\,.$

##  4 $k$-cosymplectic and $k$-precosymplectic manifolds

Similarly to previous sections, let us study $k$-cosymplectic and
$k$-precosymplectic manifolds. Our investigation will introduce relevant
technical issues to be addressed that were not present in previous sections.
One of its main differences with respect to previous Darboux theorems relies
on the fact that Reeb vector fields are not uniquely defined in the case of
$k$-precosymplectic manifolds. This suggests that Darboux coordinates for
$k$-precosymplectic manifolds should not assume a canonical form form Reeb
vector fields. Moreover, additional conditions will be needed to assume so as
to obtain canonical bases for the distributions after having the corresponding
differential forms written in a canonical manner.

######  Definition 4.1.

Let $M$ be an $(n(k+1)+k)$-dimensional manifold. A $k$-cosymplectic structure
in $M$ is a family $(\eta^{\alpha},\omega^{\alpha},V)$, with $1\leq\alpha\leq
k$, where $\eta^{1},\ldots,\eta^{k}$ are closed one-forms on $M$, while
$\omega^{1},\ldots,\omega^{k}$ are closed two-forms in $M$, and $V$ is an
integrable $nk$-dimensional integrable distribution in $M$ satisfying that

  1. (1)

$\eta^{1}\wedge\dots\wedge\eta^{k}\neq 0\,$,
$\eta^{\alpha}|_{V}=0\,,\quad\omega^{\alpha}|_{V\times V}=0\,,$

  2. (2)

$\displaystyle{\bigcap_{\alpha=1}^{k}}\left(\ker\eta^{\alpha}\cap\ker\omega^{\alpha}\right)=\\{0\\}\,$,
$\displaystyle\operatorname{rank}\,{\bigcap_{\alpha=1}^{k}}\ker\omega^{\alpha}=k\,.$

A manifold $M$ endowed with a $k$-cosymplectic structure is said to be a
$k$-cosymplectic manifold.

Every $k$-cosymplectic structure $(\eta^{\alpha},\omega^{\alpha},V)$ in $M$
admits a unique family of vector fields $R_{1},\ldots,R_{k}$ on $M$, called
Reeb vector fields, such that

$\operatorname{\iota}_{R_{\alpha}}\eta^{\beta}=\delta^{\beta}_{\alpha}\,,\qquad\operatorname{\iota}_{R_{\alpha}}\omega^{\beta}=0\,,\qquad\alpha,\beta=1,\ldots,k\,.$
(11)

Note that the existence of Reeb vector fields is independent of the existence
or not of the distribution $V$.

Given a one-cosymplectic manifold $(M,\eta,\omega,V)$, the pair
$(\eta,\omega)$ is a special type of cosymplectic structure in $M$ that
additionally admits the distribution $V$. Not every cosymplectic structure
admits such a $V$. In fact, consider
$(M=\mathbb{R}\times\mathbb{S}^{2},\eta,\omega)$, where $\eta$ is the one-form
on $M$ obtained by pulling-back the one form $\mathrm{d}t$ on $\mathbb{R}$,
and $\omega$ is the pull-back to $M$ of the standard symplectic form on
$\mathbb{S}^{2}$. Then, $(M=\mathbb{R}\times\mathbb{S}^{2},\eta,\omega)$ is
not a one-cosymplectic manifold because the standard symplectic form on
$\mathbb{S}^{2}$ does not admit a distribution as commented previously in this
paper.

######  Theorem 4.2 ($k$-cosymplectic Darboux theorem).

Given a $k$-cosymplectic manifold of the form
$(M,\eta^{1},\ldots,\eta^{k},\omega^{1},\ldots,\omega^{k},V)$, every point
$x\in M$ admits a neighbourhood with local coordinates
$\\{x^{\alpha},y^{i},y^{\alpha}_{i}\\}$, with $1\leq\alpha\leq k$, $1\leq
i\leq n$, such that

$\eta^{\alpha}=\mathrm{d}x^{\alpha}\,,\qquad\omega^{\alpha}=\sum_{i=1}^{n}\mathrm{d}y^{i}\wedge\mathrm{d}y^{\alpha}_{i}\,,\qquad\alpha=1,\ldots,k.$

In these coordinates, $R_{\alpha}=\dfrac{\partial}{\partial x^{\alpha}}$ for
$\alpha=1,\ldots,k$. If $k\neq 1$, then
$V=\left\langle\dfrac{\partial}{\partial y^{\alpha}_{i}}\right\rangle$ where
$\alpha=1,\ldots,k$ and $i=1,\ldots,n$. If $k=1$ and
$[\ker\omega,V]\subset\ker\omega\oplus V$, then
$V=\left\langle\dfrac{\partial}{\partial y^{\alpha}_{i}}\right\rangle$.

###### Proof.

Since $\eta^{1},\ldots,\eta^{k}$ are closed and
$\eta^{1}\wedge\ldots\wedge\eta^{k}$ does not vanish at any point of $M$, one
has that $H=\bigcap_{\alpha=1}^{k}\ker\eta^{\alpha}$ is an integrable
distribution of corank $k$. Moreover, $V$ is contained in $H$ by the
definition of $k$-cosymplectic manifolds. Consider one of the integral leaves,
$\mathcal{S}$, of $H$, and the natural local immersion
$\jmath_{\mathcal{S}}:\mathcal{S}\hookrightarrow M$. The
$\jmath_{\mathcal{S}}^{*}\omega^{\alpha}$ along with the restriction of $V$ to
$\mathcal{S}$ give rise to a $k$-symplectic manifold since a vector field
taking values in $H$ that is orthogonal to $H$ relative to
$\omega^{1},\ldots,\omega^{k}$ belongs to
$\bigcap_{\alpha=1}^{k}(\ker\eta^{\alpha}\cap\ker\omega^{\alpha})=0$. Hence,
$\jmath_{\mathcal{S}}^{*}\omega^{1},\ldots,\jmath_{\mathcal{S}}^{*}\omega^{k}$
admit $k$-symplectic Darboux coordinates. Doing the same along different
leaves of $H$ and gluing the results, we obtain that
$\omega^{1},\ldots,\omega^{k},\eta^{1},\ldots,\eta^{k}$ have their canonical
form. Let us explain this in detail. The differential forms
$\omega^{1},\ldots,\omega^{k},\eta^{1},\ldots,\eta^{k}$ are invariant relative
to the Reeb vector fields of the $k$-cosymplectic manifold and their value in
$M$ can be understood as the extension to $M$ obtained from their value on
$\mathcal{S}$ by the extension by one-parametric groups of diffeomorphisms of
the vector fields $R_{1},\ldots,R_{k}$. Consider coordinates
$x^{1},\ldots,x^{k}$ rectifying simultaneously the vector fields
$R_{1},\ldots,R_{k}$. If one consider the coordinate system in $M$ given by
the coordinates $x^{\alpha},y^{i},y_{i}^{\alpha}$ on $M$, where
$y^{i},y^{\alpha}_{i}$ are invariant under the flows of $R_{1},\ldots,R_{k}$
and match the $k$-symplectic Darboux coordinates on $\mathcal{S}$, one gets
that the $x^{\alpha}$ are functionally independent of the
$y^{i},y^{\alpha}_{i}$. Moreover, since $R_{1},\ldots,R_{k}$ are in the
kernels of $\omega^{1},\ldots,\omega^{k}$ and they are invariant relative to
$R_{1},\ldots,R_{k}$, it follows that their form on $M$ is the same as in
$\mathcal{S}$. Meanwhile, $\eta^{\alpha}=dx^{\alpha}$ for $\alpha=1,\ldots,k$.
and the forms $\omega^{1},\ldots,\omega^{k},\eta^{1},\ldots,\eta^{k}$ on $M$
take a canonical form.

To obtain a canonical basis of the distribution $V$, additional conditions
must be added for $k=1$. On the other hand, if $k>1$, then each distribution
$V_{\alpha}$ is the intersection of the kernels of $\omega^{\beta}$ for
$\beta\neq\alpha$ along with the intersection with
$\bigcap_{\beta=1}^{k}\ker\eta^{\beta}$. They are therefore invariant relative
to the Reeb vector fields. So, they can be put in canonical form on
$\mathcal{S}$ and extended as previously from $\mathcal{S}$ to vector fields
on $M$ with a canonical form. On the other hand, if $k=1$, one has that $V$
may not be the kernel of a closed form invariant relative to the associated
Reeb vector field and the previous method fails. To ensure this, one has to
assume $[\ker\omega,V]\subset\ker\omega\oplus V$. Equivalently, $[R,V]\subset
V$ for the unique Reeb vector field of the one-cosymplectic manifold. ∎

The conditions given in [49, Lemma 5.1.1] and [21] for the Darboux theorem for
$k$-cosymplectic manifolds may be a little bit misleading since a necessary
condition in the case $k=1$, namely $V$ must be invariant relative to the
action of the Reeb vector field, is not given in [21, Theorem II.4] and [49,
Lemma 5.1.1], but in [21, Remark II.5] or [49, Note 5.2.1], respectively,
after them. Moreover, the above-mentioned condition in [49, 21, Lemma 5.1.1]
can be rewritten in a new way, namely $[R_{\alpha},V]\subset V$, with
$\alpha=1,\ldots,k$, can be rewritten by saying that the distributions
$\ker\omega$ and $V$ are integrable and their direct sum is integrable. This
is also commented in [32].

As shown in the previous theorem, the condition $[\ker\omega,V]\subset
V\oplus\ker\omega$ is necessary in order to ensure a canonical form for the
elements of a basis of $V$. Notwithstanding, if one is mainly concerned with
the canonical form of the
$\eta^{1},\ldots,\eta^{k},\omega^{1},\ldots,\omega^{k}$, this condition can be
avoided. This is the reason why we skipped $[\ker\omega,V]\subset
V\oplus\ker\omega$ in our definition of $k$-cosymplectic manifolds.

######  Example 4.3.

Let $\\{x^{1},\ldots,x^{k}\\}$ be a linear coordinate system on
$\mathbb{R}^{k}$. Given the canonical projections
$\bar{\pi}_{1}\colon\mathbb{R}^{k}\times(\mathrm{T}^{1}_{k})^{*}Q\to\mathbb{R}^{k}$,
$\bar{\pi}_{2}\colon\mathbb{R}^{k}\times(\mathrm{T}^{1}_{k})^{*}Q\to(\mathrm{T}^{1}_{k})^{*}Q$,
$\bar{\pi}_{0}\colon\mathbb{R}^{k}\times(\mathrm{T}^{1}_{k})^{*}Q\to\mathbb{R}^{k}\times
Q$. The canonical model for $k$-cosymplectic structures is

$(\mathbb{R}^{k}\times(\mathrm{T}^{1}_{k})^{*}Q,(\bar{\pi}_{1})^{*}\mathrm{d}x^{\alpha},(\bar{\pi}_{2})^{*}\omega^{\alpha},V=\ker(\bar{\pi}_{0})_{*})\,,$

where $\omega^{1},\ldots,\omega^{k}$ are the two-forms of the canonical
$k$-symplectic structure on $(\mathrm{T}^{1}_{k})^{*}Q$.

More generally, one has the following construction.

######  Example 4.4.

Let $(N,\varpi^{\alpha},\mathcal{V})$ be an arbitrary $k$-symplectic manifold.
Given the canonical projections

$\pi_{\mathbb{R}^{k}}\colon\mathbb{R}^{k}\times
N\longrightarrow\mathbb{R}^{k}\,,\qquad\pi_{N}\colon\mathbb{R}^{k}\times
N\longrightarrow N$

define the differential forms

$\eta^{\alpha}=\pi_{\mathbb{R}^{k}}^{\ast}(\mathrm{d}x^{\alpha})\,,\quad\omega^{\alpha}=\pi_{N}^{\ast}\varpi^{\alpha}\,,\qquad\alpha=1,\ldots,k\,.$

The distribution $\mathcal{V}$ in $N$ defines a distribution $V$ in
$M=\mathbb{R}^{k}\times N$ by considering the vector fields on $N$ as vector
fields in $M$ in the natural way via the isomorphism
$\mathrm{T}M=\mathrm{T}\mathbb{R}^{k}\oplus\mathrm{T}N$. All conditions given
in Definition 4.1 are verified, and hence $(M=\mathbb{R}^{k}\times
N,\eta^{\alpha},\omega^{\alpha},V)$ is a $k$-cosymplectic manifold.

As in the case of $k$-presymplectic manifolds, there are many ways of defining
a $k$-precosymplectic structure. Note that in the $k$-precosymplectic case,
one cannot, in general, extend the notion of Reeb vector fields to give an
object that is uniquely defined. Hence, one may wonder about the necessity of
putting them into a canonical form in Darboux coordinates, since they are not
unique. Taking this into account, let us give one of the possible definitions
for $k$-precosymplectic manifolds. No condition for the determination of the
canonical form of the Reeb vector fields will be assumed.

######  Definition 4.5.

Let $M$ be a manifold of dimension $n(k+1)+k-m$, with $0\leq m\leq nk$. A
$k$-precosymplectic structure in $M$ is a family
$(\eta^{\alpha},\omega^{\alpha},V)$, with $1\leq\alpha\leq k$, where
$\eta^{\alpha}$ are closed one-forms in $M$, while $\omega^{\alpha}$ are
closed two-forms in $M$ such that
$\operatorname{rank}\omega^{\alpha}=2r_{\alpha}$, with $1\leq r_{\alpha}\leq
n$, and $V$ is an integrable distribution in $M$ of corank $n+k$ satisfying
that

  1. 1.

$\eta^{1}\wedge\dots\wedge\eta^{k}\neq
0\,,\quad\eta^{\alpha}|_{V}=0\,,\quad\omega^{\alpha}|_{V\times
V}=0\,,\qquad\alpha=1,\ldots,k,$

  2. 2.

$\displaystyle\operatorname{rank}\bigcap_{\alpha=1}^{k}\ker\omega^{\alpha}=k+d\,,$

  3. 3.

$\displaystyle\operatorname{rank}\bigcap_{\alpha=1}^{k}\big{(}\ker\omega^{\alpha}\cap\ker\eta^{\alpha}\big{)}=d\,,$

  4. 4.

one has that $V$ is an integrable distribution admitting a decomposition into
integrable distributions $V=\bigoplus_{\alpha=1}^{k}V_{\alpha}\oplus D$ such
that
$D+V_{\beta}=\left(\bigcap_{\beta\neq\alpha}\ker\omega^{\alpha}\right)\cap V$
for $\beta=1,\ldots,k$ and $k\neq 1$ for $\dim V_{\beta}=r_{\beta}$.

A manifold $M$ endowed with a $k$-precosymplectic structure is called a
$k$-precosymplectic manifold. We here after define
$r=\sum_{\alpha=1}^{k}r_{\alpha}$.

######  Example 4.6.

Consider a $k$-presymplectic manifold $(P,\varpi^{\alpha},V)$. Let us
construct a $k$-precosymplectic structure on $\mathbb{R}^{k}\times P$. First,
consider the canonical projections

$\mathbb{R}^{k}\times
P\overset{\pi}{\longrightarrow}P\,,\qquad\mathbb{R}^{k}\times
P\overset{\tau}{\longrightarrow}\mathbb{R}^{k}\,.$

Then, define $\eta^{\alpha}=\tau^{\ast}\mathrm{d}x^{\alpha}$, where
$x^{1},\ldots,x^{k}$ are linear coordinates in $\mathbb{R}^{k}$, and
$\omega^{\alpha}=\pi^{\ast}\varpi^{\alpha}$ for $\alpha=1,\ldots,k$. Then,
$(\mathbb{R}^{k}\times P,\eta^{\alpha},\omega^{\alpha})$ is a
$k$-precosymplectic manifold.

Let us prove a technical result that is necessary to asses the role played by
the distribution $\bigcap_{\alpha=1}^{k}\ker\eta^{\alpha}$ in
$k$-precosymplectic manifolds.

######  Lemma 4.7.

Given a $k$-precosymplectic manifold
$(M,\eta^{1},\ldots,\eta^{k},\omega^{1},\ldots,\omega^{k},V)$, every $x\in M$
admits a coordinated neighbourhood with coordinates

$\\{x^{1},\ldots,x^{k},y^{1},\ldots,y^{n},z^{1},\ldots,z^{d},y^{\alpha}_{1},\dotsc,y_{r_{\alpha}}^{\alpha}\\}\,,\qquad\alpha=1,\ldots,k\,,$

on a neighbourhood of $x$ so that

$V_{\alpha}=\left\langle\frac{\partial}{\partial
y^{\alpha}_{1}},\ldots,\frac{\partial}{\partial
y^{\alpha}_{r_{\alpha}}}\right\rangle\,,\qquad\alpha=1,\ldots,k\,,\qquad
D=\left\langle\frac{\partial}{\partial z^{1}},\ldots,\frac{\partial}{\partial
z^{d}}\right\rangle\,.$

###### Proof.

Since $\eta^{1},\ldots,\eta^{k}$ are closed, they admit potentials
$x^{1},\ldots,x^{k}$, respectively. Let $y^{1},\ldots,y^{n}$ be common
functionally independent first integrals for all vector fields taking values
in the integrable distribution $V$ such that

$\mathrm{d}x^{1}\wedge\ldots\wedge\mathrm{d}x^{k}\wedge\mathrm{d}y^{1}\wedge\ldots\wedge\mathrm{d}y^{n}\neq
0.$

It is assumed that $\bigoplus_{\alpha=1}^{k}V_{\alpha}$ is integrable. Hence,
$V_{1}\oplus\ldots\oplus V_{k}$ has common first integrals
$z^{1},\ldots,z^{d}$ such that

$\mu=\mathrm{d}x^{1}\wedge\ldots\wedge\mathrm{d}x^{k}\wedge\mathrm{d}z^{1}\wedge\ldots\wedge\mathrm{d}z^{d}\wedge\mathrm{d}y^{1}\wedge\ldots\wedge\mathrm{d}y^{n}\neq
0.$

Given different integers $\alpha_{1},\ldots,\alpha_{k-1}\in\\{1,\ldots,k\\}$,
and $k>1$, one has that,

$V_{\alpha_{1}}\oplus\dotsb\oplus V_{\alpha_{k-1}}\oplus
D=\ker\omega^{\beta}\cap V\,,$

where $\beta$ is the only number in $\\{1,\ldots,k\\}$ not included in
$\\{\alpha_{1},\ldots,\alpha_{k-1}\\}$. Hence, the distribution
$V_{\alpha_{1}}\oplus\ldots\oplus V_{\alpha_{k-1}}\oplus D$ has corank
$r_{\beta}$ in $V$, it is integrable, and the vector fields taking values in
it have $r_{\beta}$ common local first-integrals
$y^{\beta}_{1},\ldots,y^{\beta}_{r_{\beta}}$ such that
$\mathrm{d}y^{\beta}_{1}\wedge\ldots\wedge\mathrm{d}y^{\beta}_{r_{\beta}}\wedge\mu\neq
0$. Note that, if $k=1$, a similar result can be obtained by considering
$V=V_{1}\oplus D$ and some $r_{1}$ functionally independent integrals of $D$.
By construction,
$\\{x^{1},\ldots,x^{k},y^{1}_{1},\ldots,y^{1}_{r_{1}},\ldots,y^{k}_{1},\ldots,y^{k}_{r_{k}},z^{1},\ldots,z^{d},y^{1},\ldots,y^{n}\\}$
becomes a local coordinate system on $M$ and

$V_{\alpha}=\left(\bigcap_{i=1}^{d}\ker\mathrm{d}z^{i}\right)\cap\left(\bigcap_{i=1}^{n}\ker\mathrm{d}y^{i}\right)\cap\Bigg{(}\bigcap_{\begin{subarray}{c}\beta\neq\alpha\\\
i=1,\ldots,r_{\beta}\end{subarray}}\ker\mathrm{d}y^{\beta}_{i}\,\Bigg{)}\cap\Bigg{(}\bigcap_{\beta=1}^{k}\ker\mathrm{d}x^{\beta}\Bigg{)}.$

for $k>1$. For $k=1$, a similar expression is obtained by skipping the kernels
of the $\mathrm{d}y^{1}_{i}$. Moreover, $\dfrac{\partial}{\partial
y^{\beta}_{i}}$ with $i=1,\ldots,r_{\beta}$ vanish on all coordinates
$y^{\alpha}_{j},y^{i}$, with $\alpha\neq\beta$ and $j=1,\ldots,r_{\alpha}$,
and the $z^{1},\ldots,z^{d}$. Hence,

$\left\langle\frac{\partial}{\partial
y^{\beta}_{1}},\ldots,\frac{\partial}{\partial
y^{\beta}_{r_{\beta}}}\right\rangle=V_{\beta}\,,\qquad\beta=1,\ldots,k\,,$

and

$\left\langle\frac{\partial}{\partial z^{1}},\ldots,\frac{\partial}{\partial
z^{d}}\right\rangle=D\,.$

∎

The corresponding Darboux theorem for $k$-precosymplectic manifolds reads as
follows.

######  Theorem 4.8 ($k$-precosymplectic Darboux Theorem).

Let $M$ be a $k$-precosymplectic manifold such that $\dim M=n+d+r+k$, while
$\operatorname{rank}\omega^{\alpha}=2r_{\alpha}$, with $1\leq r_{\alpha}\leq
n$. Let us assume the existence of $k$ Reeb vector fields $R_{1},\ldots,R_{k}$
spanning an integrable $k$-dimensional distribution and such that they commute
among themselves. For every $x\in M$, there exists a local chart of
coordinates

$\\{x^{\alpha},y^{i},y^{\alpha}_{\mu_{\alpha}},z^{j}\\}\,,\quad
1\leq\alpha\leq k\,,\quad 1\leq i\leq n\,,\quad\mu_{\alpha}\in
I_{\alpha}\subseteq\\{1,\dots,n\\}\,,\quad|I_{\alpha}|=r_{\alpha}\,,\quad
1\leq j\leq d\,,$

such that

$\eta^{\alpha}=\mathrm{d}x^{\alpha}\,,\quad\omega^{\alpha}=\sum_{\mu_{\alpha}\in
I_{\alpha}}\mathrm{d}y^{\mu_{\alpha}}\wedge\mathrm{d}y^{\alpha}_{\mu_{\alpha}}\,\qquad\alpha=1,\ldots,k\,.$

If additionally $[R_{i},V]\subset V$, then

$V=\left\langle\displaystyle\frac{\partial}{\partial
y^{\alpha}_{\mu_{\alpha}}},\frac{\partial}{\partial
z^{j}}\right\rangle\,,\qquad\bigcap_{\alpha=1}^{k}(\ker\eta^{\alpha}\cap\ker\omega^{\alpha})=\left\langle\displaystyle\frac{\partial}{\partial
z^{j}}\right\rangle\,.$

###### Proof.

Consider the distribution $\Upsilon=\bigcap_{\alpha=1}^{k}\ker\eta^{\alpha}$,
which is integrable of rank $n+d+r$. One can define a leaf $S_{\lambda}$ of
$\Upsilon$. Then, one has the immersion
$\jmath_{\lambda}:S_{\lambda}\hookrightarrow M$. Since $V$ is included in
$\Upsilon$, one has that
$\jmath^{*}_{\lambda}\omega^{1},\ldots,\jmath^{*}_{\lambda}\omega^{k}$ allow
us to define a $k$-presymplectic manifold with Darboux coordinates for the
$\eta^{\alpha}$ and the $\omega^{\alpha}$ which depend smoothly on $\lambda$.
Note that the $\omega^{\alpha},\eta^{\alpha}$ are invariant relative to some
Reeb vector fields $R_{1},\ldots,R_{k}$ spanning an involutive distribution
and commuting among themselves. Using this fact and proceeding as in the
Darboux $k$-cosymplectic manifold structure, we obtain our Darboux coordinates
for the $\eta^{\alpha}$ and the $\omega^{\alpha}$. Note that the same applies
to the canonical basis for
$\bigcap_{\alpha=1}^{k}(\ker\eta^{\alpha}\cap\ker\omega^{\alpha})$ even for
$k=1$.

Notwithstanding, the form of the basis for the distribution $V$ needs the
additional condition about its invariance relative to $R_{1},\ldots,R_{k}$.
Then, gluing together as in Theorem 4.2, the result follows.

∎

######  Remark 4.9.

Note that $k$-precosymplectic manifolds admit Reeb vector fields, but they are
not uniquely defined by conditions (11). One must impose some additional
condition on $M$ to determine them uniquely. For instance, let us restrict
ourselves to a $k$-precosymplectic structure on $\mathbb{R}^{k}\times M$,
where $M$ is a $k$-presymplectic manifold. Then, if we ask the Reeb vectors
fields to be vertical with respect to the projection $\mathbb{R}^{k}\times
M\to\mathbb{R}^{k}$, the system of equations (11) determines univocally Reeb
vector fields. An equivalent way of obtaining this same family is taking the
vector fields $\displaystyle\left\\{\frac{\partial}{\partial
x^{\alpha}}\right\\}$ on $\mathbb{R}^{k}$ and lifting them to
$\mathbb{R}^{k}\times M$ with the trivial connection
$\displaystyle\mathrm{d}x^{\alpha}\otimes\frac{\partial}{\partial
x^{\alpha}}$. As it is obvious, in Darboux coordinates we have that these
vector fields are $\displaystyle R_{\alpha}=\frac{\partial}{\partial
x^{\alpha}}$. Note that every $k$-presymplectic structure in this case will
also satisfy the conditions established in our Darboux theorem.

##  5 Multisymplectic and premultisymplectic structures

Let us now comment on certain results on Darboux coordinates for
multisymplectic forms [13, 29, 55]. First, let us detail some results on
(pre)multisymplectic geometry (see [12, 13, 28] for further references).

In the context of (pre)multisymplectic geometry, the standard kernel of a
differential form is called the one-kernel.

######  Definition 5.1.

Let $M$ be an $n$-dimensional differentiable manifold. A closed form
$\Omega\in\Omega^{k}(M)$ whose one-kernel is a distribution of constant rank
is called a premultisymplectic form. Additionally, if
$\operatorname{\iota}_{X}\Omega=0$ for a vector field $X\in\mathfrak{X}(M)$
implies that $X=0$, then $\Omega$ is said to be one-nondegenerate and it
becomes a multisymplectic form. The pair $(M,\Omega)$ is said to be a
premultisymplectic or a multisymplectic manifold of degree $k$, if the one-
kernel of $\Omega$ is one-degenerate or one-nondegenerate, respectively.

First examples of multisymplectic manifolds are symplectic manifolds, i.e.
multisymplectic manifolds of degree $2$, and orientable manifolds, namely
multisymplectic manifolds with a volume form.

The following is a linear analogue of (pre)multisymplectic manifolds.

######  Definition 5.2.

A $k$-covector $\Omega$ on $\mathbb{R}^{n}$ is called a premultisymplectic
linear form. If $\operatorname{\iota}_{v}\Omega=0$ for some
$x\in\mathbb{R}^{n}$ implies that $v=0$, then $\Omega$ is said to be one-
nondegenerate and it becomes a multisymplectic linear form or $k$-plectic
linear form. The pair $(\mathbb{R}^{n},\Omega)$ is said to be a
premultisymplectic linear space or a linear multisymplectic linear space of
degree $k$, respectively. Multisymplectic linear spaces given by a
$k$-covector are also called $k$-plectic vector spaces.

Other typical examples of multisymplectic manifolds are given by the so-called
bundles of forms, which, in addition, are the canonical models of
multisymplectic manifolds. These canonical models are constructed as follows.

  * •

Let $Q$ be a manifold. Consider the bundle
$\rho\colon\Lambda^{k}(\mathrm{T}^{*}Q)\to Q$, i.e. the bundle of $k$-forms in
$Q$ (also called the $k$-multicotangent bundle of $Q$). This bundle is endowed
with a canonical structure called the tautological or canonical form
$\Theta_{Q}\in\Omega^{k}(\Lambda^{k}(\mathrm{T}^{*}Q))$ given by

$\Theta_{Q_{\widehat{\mu}}}(V_{1},\ldots,V_{k})=\operatorname{\iota}(\rho_{*}V_{1}\wedge\ldots\wedge\rho_{*}V_{k})\widehat{\mu},$

for every $\widehat{\mu}\in\Lambda^{k}(\mathrm{T}^{*}Q)$ and
$V_{1},\ldots,V_{k}\in\mathrm{T}_{\widehat{\mu}}(\Lambda^{k}(\mathrm{T}^{*}Q))$.
Then,
$\Omega_{Q}=\mathrm{d}\Theta_{Q}\in\Omega^{k+1}(\Lambda^{k}(\mathrm{T}^{*}Q))$
is a one-nondegenerate form and hence
$(\Lambda^{k}(\mathrm{T}^{*}Q),\Omega_{Q})$ is a multisymplectic manifold of
degree $k+1$. Furthermore, denoting by $\\{x^{i},p_{i_{1}\ldots i_{k}}\\}$ the
charts of natural coordinates in $\Lambda^{k}(\mathrm{T}^{*}Q)$, these
canonical forms read locally as

$\Theta_{Q}=p_{i_{1}\ldots
i_{k}}\mathrm{d}x^{i_{1}}\wedge\ldots\wedge\mathrm{d}x^{i_{k}}\,,\quad\Omega_{Q}=\mathrm{d}p_{i_{1}\ldots
i_{k}}\wedge\mathrm{d}x^{i_{1}}\wedge\ldots\wedge\mathrm{d}x^{i_{k}}\ .$

Such coordinates are Darboux coordinates in $\Lambda^{k}(\mathrm{T}^{*}Q)$.

  * •

If $\pi\colon Q\to M$ is a fibre bundle, let
$\rho_{r}\colon\Lambda^{k}_{r}(\mathrm{T}^{*}Q)\to Q$ be the subbundle of
$\Lambda^{k}(\mathrm{T}^{*}Q)$ made of the $r$-horizontal $k$-forms in $Q$
with respect to the projection $\pi$, namely the $k$-forms in $Q$ vanishing
when applied to $r$ vector fields that in $Q$ that are $\pi$-vertical. If
$\rho^{k}_{r}\colon\Lambda_{r}^{k}(\mathrm{T}^{*}Q)\to\Lambda^{k}(\mathrm{T}^{*}Q)$
is the canonical injection, then
$\Theta^{r}_{Q}=(\rho^{k}_{r})^{*}\Theta_{Q}\in\Omega^{k}(\Lambda^{k}_{r}(\mathrm{T}^{*}Q))$
is the tautological $k$-form in $\Lambda^{k}_{r}(\mathrm{T}^{*}Q)$, and then,
taking
$\Omega^{r}_{Q}=\mathrm{d}\Theta^{r}_{Q}\in\Omega^{k+1}(\Lambda^{k}_{r}(\mathrm{T}^{*}Q))$,
we have that $(\Lambda^{k}_{r}(\mathrm{T}^{*}Q),\Omega^{r}_{Q})$ is a
multisymplectic manifold of degree $k+1$. As above, the charts of natural
coordinates in $\Lambda^{k}_{r}(\mathrm{T}^{*}Q)$ are also charts of Darboux
coordinates, on which these canonical forms have local expressions similar to
the above ones.

Nevertheless, in general, multisymplectic manifolds are not (locally)
diffeomorphic to these canonical models.

Note that a multisymplectic form with Darboux coordinates admits a local flat
connection compatible with it. Furthermore, if the multisymplectic form has a
compatible flat connection, it admits coordinates in which the multisymplectic
form has constant coordinates, but it does not need to be of the previous
form. In particular, if a multisymplectic form has kernels of higher order to
those of $\Omega_{Q}$, then there is no Darboux theorem in the above senses.
In particular, this is a typical problem for Darboux coordinates: differential
forms can be put into a form with constants coefficients in many manners and
Darboux theorems use to stress one particular form over others, although
others may be of interest too.

In general, multisymplectic manifolds do not need to have a coordinate system
that makes the multisymplectic form to have constant coordinates, which is the
very most basic condition for the existence of a Darboux theorem. Indeed,
multisymplectic manifolds of this type are called flat in the literature [55].
The exact definition is given next.

######  Definition 5.3.

A multisymplectic manifold $(M,\omega)$ is called flat near $x\in M$, if there
exists a mapping $\phi:U\subset M\rightarrow\mathrm{T}_{x}M$ such that
$\phi(x)=0$ and $\phi^{*}\omega_{x}=\omega$ for $\omega_{x}$ being a constant-
coefficient non-degenerate multilinear-form on $\mathrm{T}_{x}M$.

######  Definition 5.4.

An $(n+1)$-plectic vector space $(V,\omega)$ is called standard if there
exists a linear subspace $W\subset V$ such that $\operatorname{\iota}_{u\wedge
v}\omega=0$ for all $u,v\in W$, and

$\omega^{\sharp}:w\in W\mapsto\omega^{\sharp}(w)\in\Lambda^{n}(V/W)^{*}$

such that
$\omega^{\sharp}(w)(v_{1}+W,\dotsc,v_{n}+W)=\omega(w,v_{1},\dotsc,v_{n})$ for
every $v_{1},\ldots,v_{n}\in V$, is an isomorphism.

In the above situation, $W$ is unique if $n\geq 2$ and then often denoted
$W_{\omega}$. From [13, 47, 55], the following result can easily be derived.

######  Theorem 5.5.

Let $n\geq 2$ and let $(M,\omega)$ be a standard $(n+1)$-plectic manifold,
i.e. $(M,\omega)$ has as constant linear type fixed standard $(n+1)$-plectic
vector space. Then, $W_{\omega}=\bigsqcup_{x\in M}W_{\omega_{x}}\subset TM$ is
a smooth distribution. Furthermore, $(M,\omega)$ is flat if and only if
$W_{\omega}$ is integrable.

Let us just recall that our definition of $(n+1)$-plectic symplectic manifold
is sometimes called a $n$-plectic manifold in the literature [55].

Let us now turn to a type of multisymplectic manifold for which we will obtain
Darboux coordinates.

######  Definition 5.6.

A special multisymplectic manifold is a multisymplectic manifold $(M,\Omega)$
of degree $k$ such that $\Omega=\mathrm{d}\Theta$, for some
$\Theta\in\Omega^{k-1}(M)$, and there is a diffeomorphism $\phi\colon
M\to\Lambda^{k-1}(\mathrm{T}^{*}Q)$, $\dim\,Q=n\geq k-1$, (or $\phi\colon
M\to\Lambda^{k-1}_{r}(\mathrm{T}^{*}Q)$), and a fibration $\pi\colon M\to Q$
such that $\rho\circ\phi=\pi$ (resp. $\rho_{r}\circ\phi=\pi$), and
$\phi^{*}\Theta_{Q}=\Theta$ (resp. $\phi^{*}\Theta_{Q}^{r}=\Theta$).

And, as a result of the above discussion, we state the following result.

######  Theorem 5.7 (Restricted multisymplectic Darboux Theorem).

Special multisymplectic manifolds $(M,\Omega)$ are multisymplectomorphic to
bundles of forms. Therefore, there is a local chart of Darboux coordinates
around every point $x\in M$.

Like in the $k$-symplectic and $k$-cosymplectic cases, some additional
properties are needed to assure the existence of Darboux-type coordinates [47]
and then to have multisymplectic manifolds that locally behave as the
canonical models. To state these additional conditions, we need to introduce
some generalisations of concepts of symplectic geometry. So, if $(M,\Omega)$
is a multisymplectic manifold of degree $k$ and ${\mathcal{W}}$ a distribution
on $M$, we define [13, 19] the $r$-orthogonal multisymplectic vector space at
$p\in M$ of $\mathcal{W}$ as

${\mathcal{W}}_{x}^{\perp,r}=\\{v\in\mathrm{T}_{x}M\mid\operatorname{\iota}(v\wedge
w_{1}\wedge\ldots\wedge w_{r})\Omega_{p}=0\,,\ \forall
w_{1},\ldots,w_{r}\in{\mathcal{W}}_{x}\\}\,.$

Then, the $r$-orthogonal multisymplectic complement of ${\cal W}$ is the
distribution ${\mathcal{W}}^{\perp,r}=\displaystyle\bigsqcup_{x\in
M}{\mathcal{W}}_{x}^{\perp,r}$, and we say that ${\mathcal{W}}$ is an
$r$-coisotropic or an $r$-isotropic distribution if
${\mathcal{W}}^{\perp,r}\subset{\mathcal{W}}$ or
${\mathcal{W}}\subset{\mathcal{W}}^{\perp,r}$, respectively (if
${\mathcal{W}}={\mathcal{W}}^{\perp,r}$ then ${\mathcal{W}}$ is an
$r$-Lagrangian distribution). Let us use previous notions.

######  Definition 5.8.

Let $(M,\Omega)$ be a multisymplectic manifold of degree $k$, and let
${\mathcal{W}}$ be a regular one-isotropic involutive distribution in
$(M,\Omega)$.

  1. 1.

A multisymplectic manifold of type $(k,0)$ is a triple
$(M,\Omega,{\mathcal{W}})$ such that, for every $x\in M$,

    1. (a)

$\dim{\mathcal{W}}(x)=\dim\Lambda^{k-1}(\mathrm{T}_{x}M/{\mathcal{W}}(x))^{*}$.

    2. (b)

$\dim(\mathrm{T}_{x}M/{\mathcal{W}}(x))>k-1$.

  2. 2.

A multisymplectic manifold of type $(k,r)$ ($1\leq r\leq k-1$) is a quadruple
$(M,\Omega,{\mathcal{W}},{\mathcal{E}})$, where ${\mathcal{E}}$ is a
distribution on $M$ such that, for every $x\in M$, one has that
${\mathcal{E}}(x)$ is a vector subspace of $\mathrm{T}_{x}M/{\mathcal{W}}(x)$
satisfying the following properties:

    1. (a)

If $\pi_{x}\colon\mathrm{T}_{x}M\to\mathrm{T}_{x}M/{\mathcal{W}}(x)$ is the
canonical projection, then $\operatorname{\iota}(v_{1}\wedge\ldots\wedge
v_{r})\Omega_{x}=0$, for every $v_{i}\in\mathrm{T}_{x}M$ such that
$\pi_{x}(v_{i})\in{\mathcal{E}}(x)$ ($i=1,\ldots,r$).

    2. (b)

$\dim{\mathcal{W}}(x)=\dim\Lambda_{r}^{k-1}(\mathrm{T}_{x}M/{\mathcal{W}}(x))^{*}$,
where the horizontal forms are considered with respect to the subspace
${\mathcal{E}}(x)$.

    3. (c)

$\dim(\mathrm{T}_{x}M/{\mathcal{W}}(x))>k-1$.

Then, the fundamental result is the following [19, Corollary 3.31].

######  Theorem 5.9 (Generalised multisymplectic Darboux Theorem).

Every multisymplectic manifold $(M,\Omega)$ of type $(k,0)$ (resp. of type
$(k,r)$) is locally multisymplectomorphic to a bundle of $(k-1)$-forms
$\Lambda^{k-1}(\mathrm{T}^{*}Q)$ (resp. $\Lambda^{k-1}_{r}(\mathrm{T}^{*}Q)$),
for some manifold $Q$; that is, to a canonical multisymplectic manifold.
Therefore, there is a local chart of Darboux coordinates around every point
$x\in M$.

######  Definition 5.10.

Multisymplectic manifolds that are locally multisymplectomorphic to bundles of
forms are called locally special multisymplectic manifolds.

As a relevant example, if $\pi\colon E\to M$ is a fiber bundle (where $M$ is
an $m$-dimensional oriented manifold), $J^{1}\pi$ is the corresponding first-
order jet bundle, and ${\cal L}$ is a first-order regular or hyperregular
Lagrangian density, then the Poincaré–Cartan form $\Omega_{\cal
L}\in\Omega^{m+1}(J^{1}\pi)$ is a multisymplectic form and
$(J^{1}\pi,\Omega_{\cal L})$ is a (locally) special multisymplectic manifold.
If ${\cal L}$ is a singular Lagrangian, then $(J^{1}\pi,\Omega_{\cal L})$ is a
premultisymplectic manifold.

######  Definition 5.11.

A special premultisymplectic manifold is a premultisymplectic manifold
$(M,\Omega)$ of degree $k$ such that $M/\ker\Omega$ is a manifold and the
unique multisymplectic form $\Omega^{\prime}$ on $M/\ker\Omega$ such that
$\pi^{*}\Omega^{\prime}=\Omega$ is a special multisymplectic form.

The following naturally follows.

######  Definition 5.12.

Let $(M,\Omega)$ be a premultisymplectic manifold of degree $k$, and
${\mathcal{W}}$ a regular one-isotropic involutive distribution in
$(M,\Omega)$ such that $\ker\Omega\subset\mathcal{W}$ and $d=\dim\ker\Omega$.

  1. 1.

A premultisymplectic manifold of type $(d,k,0)$ is a triple
$(M,\Omega,{\mathcal{W}})$ such that, for every $x\in M$,

    1. (a)

$\dim{\mathcal{W}}(x)-d=\dim\Lambda^{k-1}(\mathrm{T}_{x}M/{\mathcal{W}}(x))^{*}$.

    2. (b)

$\dim(\mathrm{T}_{x}M/{\mathcal{W}}(x))>k-1$.

  2. 2.

A premultisymplectic manifold of type $(d,k,r)$ ($1\leq r\leq k-1$) is a
quadruple $(M,\Omega,{\mathcal{W}},{\mathcal{E}})$, where ${\mathcal{E}}$ is a
distribution on $M$ such that, for every $x\in M$, the space
${\mathcal{E}}(x)$ is a vector subspace of $\mathrm{T}_{x}M/{\mathcal{W}}(x)$
with the following properties:

    1. (a)

If $\pi_{x}\colon\mathrm{T}_{x}M\to\mathrm{T}_{x}M/{\mathcal{W}}(x)$ is the
canonical projection, then $\operatorname{\iota}(v_{1}\wedge\ldots\wedge
v_{r})\Omega_{p}=0$, for every $v_{i}\in\mathrm{T}_{x}M$ such that
$\pi_{x}(v_{i})\in{\mathcal{E}}(x)$, $i=1,\ldots,r$.

    2. (b)

$\dim{\mathcal{W}}(x)-d=\dim\Lambda_{r}^{k-1}(\mathrm{T}_{x}M/{\mathcal{W}}(x))^{*}$,
where the horizontal forms are considered with respect to the subspace
${\mathcal{E}}(x)$.

    3. (c)

$\dim(\mathrm{T}_{x}M/{\mathcal{W}}(x))>k-1$.

######  Theorem 5.13 (Generalised premultisymplectic Darboux Theorem).

Every premultisymplectic manifold $(M,\Omega)$ of type $(d,k,0)$ (resp. of
type $(d,k,r)$) is locally premultisymplectomorphic to a canonical
premultisymplectic manifold of type $(d,k,0)$ (resp. of type $(d,k,r)$).
Therefore, there is a local chart of Darboux coordinates around every point
$x\in M$.

As in previous structures, analogous claims can be done concerning the
existence of connected compatible connections with premultisymplectic
manifolds.

##  6 Conclusions and outlook

The focus of this research is the exploration of Darboux-type theorems
concerning geometric structures defined by closed differential forms. The
initial section of this study entails an examination of the Darboux theorem
for symplectic, presymplectic, and cosymplectic manifolds. By imposing minimal
regularity conditions, we have successfully established a proof for a Darboux
theorem applicable to precosymplectic manifolds. Within the realm of geometric
mechanics, these manifolds serve as the phase spaces for both regular and
singular autonomous and non-autonomous dynamical systems.

We have presented novel proofs for the Darboux theorem concerning
$k$-symplectic and $k$-cosymplectic manifolds. These proofs appear to be
simpler compared to the previously known ones. Additionally, we have
introduced and demonstrated new Darboux theorems for specific families of
$k$-presymplectic and $k$-precosymplectic manifolds. Furthermore, we have
provided a counterexample illustrating that a general Darboux-type theorem
does not hold for $k$-presymplectic manifolds. We have conducted a thorough
review of previous findings regarding the existence of Darboux coordinates for
certain types of multisymplectic manifolds. Lastly, we have presented fresh
results that establish the existence of Darboux coordinates for particular
cases of premultisymplectic manifolds. All of these structures play a vital
role in the geometric representation of both regular and singular classical
field theories. The relations of Darboux theorems with flat connections have
been studied, which provides new viewpoints and gathers previous scattered
results in the literature.

The ideas of this paper can be extended to other geometric structures related
with closed one or two-forms of different types. Notwithstanding, the
formalism on flat compatible connections does not apply to geometric
structures related to families of different forms that do not allow for a
locally constant form and therefore closed, e.g. for contact and precontact
structures, and their extensions (which appear, for instance, in the geometric
description of dissipative and action-dependent systems in physics). It would
be interesting to find an analogue of our formalism for such theories. In
particular, note that non-closed differential forms may have flat compatible
connections provided torsion different from zero is allowed. For instance,
consider the manifold $M=\mathbb{R}^{3}$ with natural coordinates
$\\{t,x,p\\}$, the one-form $\eta=\mathrm{d}t-p\mathrm{d}x$ and the connection
$\nabla$ in $M$ whose only non-vanishing Christoffel symbol is
$\Gamma_{px}^{t}=-1$. It is easy to check that $\eta$ is a contact one-form on
$M$, and parallel relative to the connection $\nabla$, namely $\nabla\eta=0$.
However, the connection $\nabla$ is not torsion-free: its torsion has local
expression $T=\mathrm{d}x\otimes\mathrm{d}p\otimes\dfrac{\partial}{\partial
t}-\mathrm{d}p\otimes\mathrm{d}x\otimes\dfrac{\partial}{\partial t}$. This
torsion takes account of the non-integrability of the contact distribution
$D=\ker\eta$. Meanwhile, $\nabla$ is flat. The relation between integrability
of a geometric structure and the torsion of compatible connections will be
investigated in a future work.

Moreover, this work has studied conditions for Darboux theorems of various
types. We believe that there is still room to provide more types of Darboux
coordinates, and that more research in the study of necessary and sufficient
conditions for their existence is needed. In particular, this specially
applies to $k$-pre(co)symplectic manifolds.

### Acknowledgments

We thank M. de León and J. Gaset for fruitful discussions and comments. We
acknowledge partial financial support from the Spanish Ministry of Science and
Innovation, grants PID2021-125515NB-C21, PID2021-125515NB-C22, and
RED2022-134301-T of AEI, and of the Ministry of Research and Universities of
the Catalan Government, project 2021 SGR 00603 Geometry of Manifolds and
Applications, GEOMVAP. J. de Lucas and X. Rivas acknowledge partial financial
support from project IDUB with number PSP: 501-D111-20-2004310. X. Rivas would
like to thank the cordiality shown during his stays at the Faculty of Physics
of the University of Warsaw financed from the above mentioned IDUB project.

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