# A rigidity result for coisotropic submanifolds in contact geometry

Stephane Geudens  Department of Mathematics, University College London, 25
Gordon St, London WC1H0AY, UK<EMAIL_ADDRESS>and  Alfonso G. Tortorella
DipMat, Università degli Studi di Salerno, Via Giovanni Paolo II n°132, 84084
Fisciano, Italy<EMAIL_ADDRESS>
###### Abstract.

We study coisotropic deformations of a compact regular coisotropic submanifold
$C$ in a contact manifold $(M,\xi)$. Our main result states that $C$ is rigid
among nearby coisotropic submanifolds whose characteristic foliation is
diffeomorphic to that of $C$. When combined with a classical rigidity result
for foliations, this yields conditions under which $C$ is rigid among all
nearby coisotropic submanifolds.

###### Contents

  1. 1 Background and setup
  2. 2 First order deformations with fixed characteristic foliation
  3. 3 Infinitesimal rigidity
  4. 4 Rigidity
  5. 5 Application to the classical coisotropic deformation problem
  6. A The Atiyah algebroid of a line bundle

## Introduction

This paper is devoted to deformations of a compact coisotropic submanifold $C$
in a contact manifold $(M,\xi)$. We will additionally assume that the
coisotropic submanifold $C$ is generic, in the sense that $C$ is transverse to
the hyperplane distribution $\xi$. Such coisotropic submanifolds are called
regular. The coisotropic deformation problem of a regular coisotropic
submanifold $C\subset(M,\xi)$ is known to be quite involved. First, it is
highly non-linear, because deformations are given by Maurer-Cartan elements of
a suitable $L_{\infty}$-algebra [9]. Second, the deformation problem is
obstructed [12], meaning that there may exist first order deformations which
are not tangent to any path of deformations. All of this is in sharp contrast
with the deformation problem of a compact Legendrian submanifold
$L\subset(M,\xi)$. The latter is linear hence unobstructed, and moreover
compact Legendrians are rigid, in the sense that smooth Legendrian
deformations are generated by contact isotopies.

A regular coisotropic submanifold $C\subset(M,\xi)$ carries a characteristic
foliation $\mathcal{F}$. Its properties, especially its instability, seem to
play an important role in the coisotropic deformation problem of $C$, as
demonstrated by the following results:

  * •

In [12, §3.1] the second author constructed an obstructed example, featuring a
compact regular coisotropic submanifold $C$ which has arbitrarily
$\mathcal{C}^{1}$-small deformations whose characteristic foliation is not
diffeomorphic to that of $C$.

  * •

A compact regular coisotropic submanifold $C$ is called integral if its
characteristic foliation $\mathcal{F}$ is simple, i.e. given by the fibers of
a fiber bundle $C\rightarrow C/\mathcal{F}$. When deformed within the class of
integral coisotopic submanifolds, $C$ is rigid under contact isotopies, as
proved by the second author [12]. In particular, this deformation problem is
unobstructed.

These results are our motivation for considering a restricted version of the
coisotropic deformation problem of $C$. Instead of allowing all
$\mathcal{C}^{1}$-small coisotropic deformations of $C$, we will only consider
those whose characteristic foliation is diffeomorphic to that of $C$. In other
words, we deform $C$ within the class

$\text{Def}_{\mathcal{F}}(C):=\big{\\{}C^{\prime}\subset(M,\xi)\ \text{regular
coisotropic}:\ \exists\ \text{foliated diffeomorphism}\
(C,\mathcal{F})\overset{\sim}{\rightarrow}(C^{\prime},\mathcal{F}^{\prime})\big{\\}}.$

Our main result (see Thm. 4.1) states that $C$ is rigid, when deformed inside
$\text{Def}_{\mathcal{F}}(C)$.

######  Main Theorem.

Let $C\subset(M,\xi)$ be a compact regular coisotropic submanifold, and let
$C_{t}$ be a smooth path in $\text{Def}_{\mathcal{F}}(C)$ with $C_{0}=C$. Then
there exists an isotopy of contactomorphisms
$\psi_{t}\colon(M,\xi)\to(M,\xi)$, locally defined around $C$, such that
$C_{t}=\psi_{t}(C)$ for small enough $t$.

Clearly, for a path of regular coisotropic submanifolds to be generated by a
contact isotopy, the diffeomorphism type of the characteristic foliation needs
to be constant along that path. Our result shows that constancy of the
foliation is in fact the only obstruction. When the characteristic foliation
$\mathcal{F}$ is simple, our theorem reduces to the rigidity result for
integral coisotropic submanifolds from [12] mentioned above.

Our Main Theorem is a rigidity result for the _restricted_ deformation problem
of $C$ inside $\text{Def}_{\mathcal{F}}(C)$. It can be combined with a well-
known rigidity theorem for foliations to obtain a rigidity result for the
classical coisotropic deformation problem, which allows _all_ coisotropic
deformations of $C$ (see Cor. 5.1). The rigidity result for foliations we are
referring to – which appeared in various versions [7],[2],[3] – concerns
Hausdorff foliations, i.e. those $\mathcal{F}$ for which the leaf space
$C/\mathcal{F}$ is Hausdorff when endowed with the quotient topology. Such a
foliation has a generic leaf $L$, to which all leaves in an open dense subset
are diffeomorphic.

######  Corollary.

Let $C\subset(M,\xi)$ be a compact, connected regular coisotropic submanifold
with characteristic foliation $\mathcal{F}$. Assume that $\mathcal{F}$ is
Hausdorff and that its generic leaf $L$ satisfies $H^{1}(L)=0$. If $C_{t}$ is
a smooth path of coisotropic submanifolds with $C_{0}=C$, then there is an
isotopy $\psi_{t}$ of locally defined contactomorphisms such that
$C_{t}=\psi_{t}(C)$ for small $t$.

Overview of the paper.

In Section 1, we collect some background information concerning coisotropic
submanifolds in contact geometry. We recall that first order deformations of a
regular coisotropic submanifold $C\subset(M,\xi)$, modulo those induced by
contact isotopies, are given by the cohomology group
$H^{1}(\mathcal{F},\ell)$. Here $\ell:=TC/(TC\cap\xi)$ is the quotient line
bundle on $C$, which carries a canonical $T\mathcal{F}$-representation
$\nabla^{\ell}$. With the aim of this note in mind, we also recall that first
order deformations of the foliation $\mathcal{F}$, modulo those induced by
isotopies, are given by the first foliated cohomology group with values in the
normal bundle $H^{1}(\mathcal{F},N\mathcal{F})$.

In Sections 2 and 3, we study the deformation problem of $C$ inside
$\text{Def}_{\mathcal{F}}(C)$ at the infinitesimal level. We first argue what
are the appropriate first order deformations when deforming $C$ inside
$\text{Def}_{\mathcal{F}}(C)$. By the above, these should be closed elements
in $\Omega^{1}(\mathcal{F},\ell)$ which in some way give rise to a trivial
class in $H^{1}(\mathcal{F},N\mathcal{F})$. To achieve this assignment, we
construct a non-canonical chain map (see Def. 2.4 and Thm. 2.14)

$\Phi_{G}\colon\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}\longrightarrow\big{(}\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),-d_{\nabla^{N\mathcal{F}}}\big{)},$

which depends on a choice of complement $G$ to $T\mathcal{F}\subset TC$. We
then show that first order deformations of $C$ are closed elements in
$\Omega^{1}(\mathcal{F},\ell)$ whose image under $\Phi_{G}$ is exact (see Thm.
2.6). It turns out that the map induced in cohomology
$[\Phi]:H^{1}(\mathcal{F},\ell)\rightarrow H^{1}(\mathcal{F},N\mathcal{F})$ is
canonical (see Thm. 2.19) and injective (see Prop. 3.1), hence first order
deformations of $C\subset\text{Def}_{\mathcal{F}}(C)$ are in fact trivial in
$H^{1}(\mathcal{F},\ell)$. This means exactly that $C$ is infinitesimally
rigid under contact isotopies, when deformed inside
$\text{Def}_{\mathcal{F}}(C)$. In particular, this deformation problem is
unobstructed (see Cor. 3.4). This is the contact analog of an unobstructedness
result for coisotropic submanifolds in symplectic geometry due to the first
author [5].

The infinitesimal rigidity result proved in Section 3 raises the question
whether $C$ is actually rigid inside $\text{Def}_{\mathcal{F}}(C)$. We show in
Section 4 that this is indeed the case, by establishing our Main Theorem
mentioned above. The proof relies on an extension of Gray stability for
contact structures to the case of pre-contact structures (see Prop. 4.3) and a
uniqueness result concerning coisotropic embeddings in contact manifolds (see
Prop. 4.5). In Section 5, we prove the above Corollary concerning coisotropic
submanfolds with Hausdorff foliation.

Throughout the paper, we make frequent use of the fact that (pre-)contact
structures can be viewed as (pre-)symplectic forms on a certain Lie algebroid,
called Atiyah algebroid [13]. This point of view allows one to study
(pre-)contact structures using (pre-)symplectic geometry. We recall the
details of this correspondence in the Appendix.

######  Acknowledgements.

S.G. acknowledges support from the UCL Institute for Mathematics & Statistical
Science (IMSS). A.G.T. is a member of the National Group for Algebraic and
Geometric Structures, and their Applications (GNSAGA - INdAM).

##  1\. Background and setup

###  1.1. Coisotropic submanifolds in contact geometry

We recall some background information about contact manifolds and their
coisotropic submanifolds, following [9, §5]. Let $M$ be a manifold with a
hyperplane distribution $\xi$. Denoting the quotient line bundle by
$L:=TM/\xi$, the distribution $\xi$ is the kernel of the structure form
$\theta\in\Gamma(T^{*}M\otimes L)$ of $(M,\xi)$, given by

$\theta:TM\rightarrow L:X\mapsto X\ \text{mod}\ \xi.$

The curvature form $\omega\in\Gamma(\wedge^{2}\xi^{*}\otimes L)$ of $(M,\xi)$
is defined by

$\omega(X,Y)=\theta([X,Y]),\hskip 28.45274ptX,Y\in\Gamma(\xi),$

and we denote the associated vector bundle morphism by

$\omega^{\flat}:\xi\rightarrow\xi^{*}\otimes L:X\mapsto\omega(X,-).$

The characteristic distribution of $(M,\xi)$ is given by $\ker\omega^{\flat}$.
It is an involutive distribution which is singular in general.

######  Definition 1.1.

We call $(M,\xi)$ a pre-contact manifold if the characteristic distribution
$\ker\omega^{\flat}$ has constant rank. In that case, we refer to
$\text{rk}(\ker\omega^{\flat})$ as the rank of $(M,\xi)$. The foliation
$\mathcal{F}$ integrating $\ker\omega^{\flat}$ is the characteristic foliation
of $(M,\xi)$.

The curvature form $\omega$ of $(M,\xi)$ vanishes exactly when $\xi$ is
integrable. Consequently, we say that $\xi$ is maximally non-integrable if
$\omega$ is non-degenerate, i.e. $\ker\omega^{\flat}$ is trivial.

######  Definition 1.2.

We call $(M,\xi)$ a contact manifold if $\xi$ is maximally non-integrable.

Let $C$ be a submanifold of a contact manifold $(M,\xi)$. The intersection
$\xi_{C}:=TC\cap\xi$ is a generically singular distribution on $C$ – if $C$ is
transverse to $\xi$, then $\xi_{C}$ is a hyperplane distribution on $C$.
Denote by $\xi_{C}^{\perp_{\omega}}$ the bundle (possibly of varying rank) of
$\omega$-orthogonals of $\xi_{C}$ in $(\xi,\omega)$, i.e.

$\xi_{C}^{\perp_{\omega}}=\\{v\in\xi:\ \omega(v,w)=0\ \ \forall
w\in\xi_{C}\\}.$

######  Definition 1.3.

A submanifold $C$ of a contact manifold $(M,\xi)$ is coisotropic if
$(\xi_{C})_{p}$ is a coisotropic subspace of $(\xi_{p},\omega_{p})$ for all
$p\in C$, i.e. $\xi_{C}^{\perp_{\omega}}\subset\xi_{C}$. We call $C$ regular
coisotropic if $C$ is coisotropic and transverse to $\xi$.

In this note, we focus on regular coisotropic submanifolds.

######  Remark 1.4.

A submanifold $C^{k+1}\subset(M^{2n+1},\xi)$ is regular coisotropic exactly
when $\xi_{C}$ is a pre-contact structure of rank $2n-k$. Indeed, because
$\ker(\omega|_{\xi_{C}}^{\flat})=\xi_{C}\cap\xi_{C}^{\perp_{\omega}}$, we have
the following equivalences:

$\displaystyle C^{k+1}\subset(M^{2n+1},\xi)\ \text{is regular coisotropic}$
$\displaystyle\hskip 85.35826pt\Leftrightarrow\dim(\xi_{C})_{p}=k\ \text{and}\
\big{(}\xi_{C}\cap\xi_{C}^{\perp_{\omega}}\big{)}_{p}=\big{(}\xi_{C}^{\perp_{\omega}}\big{)}_{p}\
\ \ \forall p\in C,$ $\displaystyle\hskip
85.35826pt\Leftrightarrow\dim(\xi_{C})_{p}=k\ \text{and}\
\dim\big{(}\xi_{C}\cap\xi_{C}^{\perp_{\omega}}\big{)}_{p}=2n-k\ \ \ \forall
p\in C,$ $\displaystyle\hskip 85.35826pt\Leftrightarrow\xi_{C}\ \text{is pre-
contact of rank}\ 2n-k.$

Hence, a regular coisotropic submanifold $C^{k+1}\subset(M^{2n+1},\xi)$
carries a characteristic foliation $\mathcal{F}$, which integrates the
distribution $\xi_{C}^{\perp_{\omega}}$ and has $(2n-k)$-dimensional leaves.

We also mention an alternative characterization of regular coisotropic
submanifolds, which will be important in Section 4. It is reminiscent of the
definition of coisotropic submanifolds in symplectic geometry, relying on the
fact that contact structures can be viewed as symplectic Atiyah forms. For the
necessary background and notation concerning Atiyah forms, we refer to the
Appendix.

######  Proposition 1.5 ([12]).

Let $(M,\xi)$ be a contact manifold with quotient line bundle $L$, and denote
by $\Omega\in\Omega^{2}_{D}(L)$ its symplectic Atiyah form. Let
$C\subset(M,\xi)$ be a submanifold transverse to $\xi$, with restricted line
bundle $L_{C}:=L|_{C}$. Then the following are equivalent:

  1. (1)

$C$ is a regular coisotropic submanifold of $(M,\xi)$,

  2. (2)

the pullback $i^{*}\Omega\in\Omega^{2}_{D}(L_{C})$ is a pre-symplectic Atiyah
form with kernel $(DL_{C})^{\perp_{\Omega}}$. Here $(DL_{C})^{\perp_{\Omega}}$
denotes the $\Omega$-orthogonal of $DL_{C}$ inside $DL$.

A special case of [10, Thm. 3] provides a normal form for a contact manifold
$(M,\xi)$ around a regular coisotropic submanifold $C\subset(M,\xi)$, similar
to Gotay’s theorem in symplectic geometry [6]. Before introducing the local
model, we have to fix some notation. Let $\mathcal{F}$ be the characteristic
foliation of the pre-contact manifold $(C,\xi_{C})$. Denote by
$\ell:=TC/\xi_{C}$ its line bundle, and by $\theta_{C}\in\Omega^{1}(C,\ell)$
its structure form. The local model lives on a neighborhood of the zero
section of the vector bundle $p:T^{*}\mathcal{F}\otimes\ell\rightarrow C$. Its
construction depends on a choice of complement $G$ to the characteristic
distribution $T\mathcal{F}$, i.e. $TC=T\mathcal{F}\oplus G$. The model
involves the following elements of
$\Omega^{1}\big{(}T^{*}\mathcal{F}\otimes\ell,p^{*}\ell\big{)}$:

  1. i)

We pull back the structure form $\theta_{C}\in\Omega^{1}(C,\ell)$ to
$p^{*}\theta_{C}\in\Omega^{1}\big{(}T^{*}\mathcal{F}\otimes\ell,p^{*}\ell\big{)}$,

  2. ii)

We define
$\theta_{G}\in\Omega^{1}\big{(}T^{*}\mathcal{F}\otimes\ell,p^{*}\ell\big{)}$
as follows. For $\alpha\in(T^{*}\mathcal{F}\otimes\ell)_{x}$ and $v\in
T_{\alpha}(T^{*}\mathcal{F}\otimes\ell)$,

$(\theta_{G})_{\alpha}(v)=\big{\langle}\alpha,\text{pr}_{T\mathcal{F}}(dp(v))\big{\rangle}.$

Here $\text{pr}_{T\mathcal{F}}:TC\rightarrow T\mathcal{F}$ is the projection
in the splitting $TC=T\mathcal{F}\oplus G$.

It turns out that the kernel of

$\gamma_{G}:=p^{*}\theta_{C}+\theta_{G}\in\Omega^{1}\big{(}T^{*}\mathcal{F}\otimes\ell,p^{*}\ell\big{)}$

defines a contact structure on a neighborhood $U$ of $C\subset
T^{*}\mathcal{F}\otimes\ell$. We call $(U,\ker\gamma_{G})$ the contact
thickening of $(C,\xi_{C})$. This is the local model for $(M,\xi)$ around $C$.

######  Theorem 1.6.

[10] If $C\subset(M,\xi)$ is a regular coisotropic submanifold, then a
neighborhood of $C$ in $M$ can be identified with the contact thickening
$(U,\ker\gamma_{G})$.

By Thm. 1.6, studying $\mathcal{C}^{1}$-small coisotropic deformations of a
regular coisotropic submanifold $C\subset(M,\xi)$ amounts to studying sections
of the contact thickening $(U,\ker\gamma_{G})$ whose graph is coisotropic.
Note here that small enough coisotropic deformations of $C$ are automatically
regular. The following result describes regular coisotropic sections of
$(U,\ker\gamma_{G})$.

######  Lemma 1.7.

Let $C^{k+1}\subset(M^{2n+1},\xi)$ be a regular coisotropic submanifold with
local model $(U,\ker\gamma_{G})$. For any section $\eta\in\Gamma(U)$, the
following are equivalent:

  1. i)

$\text{Graph}(\eta)\subset(U,\ker\gamma_{G})$ is regular coisotropic,

  2. ii)

$\ker\big{(}\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}$ is a
pre-contact structure of rank $(2n-k)$ on $C$.

Recall that we fixed a complement $TC=T\mathcal{F}\oplus G$. The map
$\operatorname{pr}_{T\mathcal{F}}^{*}$ appearing in $ii)$ above is the dual of
the projection $\operatorname{pr}_{T\mathcal{F}}:TC\rightarrow T\mathcal{F}$
along $G$.

###### Proof.

The section $\eta\in\Gamma(U)$ defines a diffeomorphism

$\sigma_{\eta}:T^{*}\mathcal{F}\otimes\ell\rightarrow
T^{*}\mathcal{F}\otimes\ell:(x,\alpha)\mapsto(x,\alpha+\eta(x)),$

which is just the translation by $\eta$. Note that $\sigma_{\eta}$ preserves
the fibers of $p:T^{*}\mathcal{F}\otimes\ell\rightarrow C$, hence it induces a
well-defined pullback on $\Omega^{1}(T^{*}\mathcal{F}\otimes\ell,p^{*}\ell)$.
Since $\sigma_{\eta}$ takes $C$ to $\text{graph}(\eta)$, we have that
$\text{graph}(\eta)$ is regular coisotropic for $\ker\gamma_{G}$ exactly when
$C$ is regular coisotropic for
$(\sigma_{-\eta})_{*}\big{(}\ker\gamma_{G}\big{)}=\ker\big{(}\sigma_{\eta}^{*}\gamma_{G}\big{)}$.
We claim to have the following expression for $\sigma_{\eta}^{*}\gamma_{G}$.

Claim:
$\sigma_{\eta}^{*}\gamma_{G}=\gamma_{G}+p^{*}\big{(}\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}$.

To prove the claim, we only have to check that
$\sigma_{\eta}^{*}\theta_{G}=\theta_{G}+p^{*}\big{(}\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}$.
To this end, we compute for $\alpha\in(T^{*}\mathcal{F}\otimes\ell)_{x}$ and
$v\in T_{\alpha}(T^{*}\mathcal{F}\otimes\ell)$,

$\displaystyle\big{(}\sigma_{\eta}^{*}\theta_{G}\big{)}_{\alpha}(v)$
$\displaystyle=\big{(}\theta_{G}\big{)}_{\alpha+\eta(x)}((\sigma_{\eta})_{*}v)$
$\displaystyle=\big{\langle}\alpha+\eta(x),\text{pr}_{T\mathcal{F}}(p_{*}(\sigma_{\eta})_{*}v)\big{\rangle}$
$\displaystyle=\big{(}\theta_{G}\big{)}_{\alpha}(v)+\big{\langle}\eta(x),\text{pr}_{T\mathcal{F}}(p_{*}v)\big{\rangle}$
$\displaystyle=\big{(}\theta_{G}\big{)}_{\alpha}(v)+\big{(}p^{*}\big{(}\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}\big{)}_{\alpha}(v).$

The result now follows from Rem. 1.4, since the pullback of
$\sigma_{\eta}^{*}\gamma_{G}$ to $C$ reads

$i^{*}(\sigma_{\eta}^{*}\gamma_{G})=\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta.\qed$

######  Remark 1.8.

The proof of Lemma 1.7 shows that the graph of any section $\eta\in\Gamma(U)$
is automatically transverse to the hyperplane distribution $\ker\gamma_{G}$.
Therefore, the regularity requirement is automatically met. To see why
$\text{graph}(\eta)\pitchfork\ker\gamma_{G}$, note that this is equivalent
with
$C\pitchfork\ker\big{(}\gamma_{G}+p^{*}\big{(}\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}\big{)}$,
which in turn is equivalent with
$\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta$ being nowhere zero. The
latter statement holds, for if
$\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta$ would vanish at some
point, then $\theta_{C}$ would vanish there, which is impossible.

Lemma 1.7 has the advantage that it describes regular coisotropic deformations
of $C$ in terms of information that lives on the manifold $C$ itself. It shows
in particular that small coisotropic deformations of $C$ induce deformations
of the pre-contact structure $\xi_{C}$ with the same rank, i.e. there is an
assignment

$\text{Def}_{coiso}(C)\rightarrow\text{Def}_{pre-
contact}(\xi_{C}):\eta\mapsto\ker\big{(}\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}.$

###  1.2. Coisotropic deformations with fixed characteristic foliation

Assume we are given a regular coisotropic submanifold $C\subset(M,\xi)$ with
characteristic foliation $\mathcal{F}$. We aim to study coisotropic
deformations $C^{\prime}$ of $C$, which are small enough so that they remain
regular and whose characteristic foliation $\mathcal{F}^{\prime}$ is
diffeomorphic to $\mathcal{F}$. More precisely, we are interested in
$\mathcal{C}^{1}$-small elements of the following space.

######  Definition 1.9.

We define $\text{Def}_{\mathcal{F}}(C)$ to be the space of regular coisotropic
submanifolds $C^{\prime}\subset(M,\xi)$ for which there exists a foliated
diffeomorphism
$(C,\mathcal{F})\overset{\sim}{\rightarrow}(C^{\prime},\mathcal{F}^{\prime})$.

Thanks to Thm. 1.6, we can pass from $(M,\xi)$ to the contact thickening
$(U,\gamma_{G})$. Doing so turns the problem into studying coisotropic
sections $\eta\in\Gamma(U,\gamma_{G})$ whose characteristic foliation is
diffeomorphic to $\mathcal{F}$. In this respect, we remark the following.

######  Lemma 1.10.

For any coisotropic section $\eta\in\Gamma(U,\gamma_{G})$, the following are
equivalent:

  1. (1)

There is a diffeomorphism $\psi:C\rightarrow\text{graph}(\eta)$ which takes
$\mathcal{F}$ to the characteristic foliation of the pre-contact structure
$\ker(i^{*}\gamma_{G})$.

  2. (2)

There is a diffeomorphism $\phi:C\rightarrow C$ which takes $\mathcal{F}$ to
to the characteristic foliation of the pre-contact structure
$\ker\big{(}\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}$.

###### Proof.

The section $\eta$ defines a diffeomorphism
$\tau_{\eta}:C\rightarrow\text{graph}(\eta)$ which, as shown in the proof of
Lemma 1.7, pulls back
$i^{*}\gamma_{G}\in\Omega^{1}(\text{graph}(\eta),p^{*}\ell|_{\text{graph}(\eta)})$
to $\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta\in\Omega^{1}(C,\ell)$.
The inverse of $\tau_{\eta}$ is just the restriction of the bundle projection
$p|_{\text{graph}(\eta)}$. Hence, given $\psi:C\rightarrow\text{graph}(\eta)$
as in $(1)$ above, we can set $\phi:=p|_{\text{graph}(\eta)}\circ\psi$.
Conversely, given given $\phi$ as in $(2)$ above, we can set
$\psi=\tau_{\eta}\circ\phi$. ∎

With Lemma 1.10 in mind, we now introduce a space
$\text{Def}^{U}_{\mathcal{F}}(C)$ which serves as a local model for
$\text{Def}_{\mathcal{F}}(C)$. The space $\text{Def}^{U}_{\mathcal{F}}(C)$ is
the main object of study in this note.

######  Definition 1.11.

Let $C\subset(M,\xi)$ be a regular coisotropic submanifold with characteristic
foliation $\mathcal{F}$ and contact thickening $(U,\gamma_{G})$. We define
$\text{Def}^{U}_{\mathcal{F}}(C)$ to be the space of sections
$\eta\in\Gamma(U)$ such that

$\begin{cases}\text{Graph}(\eta)\ \text{is coisotropic in}\ (U,\gamma_{G})\\\
\text{There is a foliated diffeomorphism}\
(C,\mathcal{F})\overset{\sim}{\rightarrow}\big{(}C,\mathcal{F}_{\eta}\big{)}\end{cases}.$

Here $\mathcal{F}_{\eta}$ is the characteristic foliation of the pre-contact
structure
$\ker\big{(}\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{*}\eta\big{)}$.

###  1.3. First order deformations of coisotropic submanifolds and foliations

There are two classes of geometric structures whose deformation theory is
important in this note, namely coisotropic submanifolds of contact manifolds
and foliations. We will now recall the cochain complexes underlying first
order deformations of these objects.

####  1.3.1. First order deformations of coisotropic submanifolds

Let $C^{k+1}\subset(M^{2n+1},\xi)$ be a regular coisotropic submanifold with
characteristic foliation $\mathcal{F}$. First order deformations of $C$ as a
coisotropic submanifold – or equivalently, as a regular coisotropic
submanifold (see Rem. 1.8)– are $1$-cocycles in the complex
$\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}$, which is
defined as follows. Note that the line bundle $\ell=TC/\xi_{C}$ carries a flat
$T\mathcal{F}$-connection $\nabla^{\ell}$, given by

$\nabla^{\ell}_{X}\theta_{C}(Y)=\theta_{C}([X,Y]),\hskip
14.22636ptX\in\Gamma(T\mathcal{F}),Y\in\mathfrak{X}(C),$ (1)

where $\theta_{C}\in\Omega^{1}(C,\ell)$ is the structure form of $\xi_{C}$.
This connection induces a differential $d_{\nabla^{\ell}}$ on
$\Gamma(\wedge^{\bullet}T^{*}\mathcal{F}\otimes\ell)$, defined by the usual
Koszul formula

$\displaystyle d_{\nabla^{\ell}}\eta(X_{0},\ldots,X_{m})$
$\displaystyle=\sum_{i=0}^{m}(-1)^{i}\nabla^{\ell}_{X_{i}}\big{(}\eta(X_{0},\ldots,X_{i-1},\widehat{X_{i}},X_{i+1},\ldots,X_{m})\big{)}$
$\displaystyle\hskip
14.22636pt+\sum_{i<j}(-1)^{i+j}\eta\big{(}[X_{i},X_{j}],X_{0},\ldots,\widehat{X_{i}},\ldots,\widehat{X_{j}},\ldots,X_{m}\big{)}.$
(2)

The resulting complex
$\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}$ is the
one that governs first order deformations of $C$. This follows in particular
from the main result in [9, §5], which shows that coisotropic deformations of
$C$ correspond with Maurer-Cartan elements of a certain $L_{\infty}$-algebra
structure on $\Omega^{\bullet}(\mathcal{F},\ell)[1]$ whose unary bracket
coincides with the differential $d_{\nabla^{\ell}}$. Below, we present a
simple geometric proof for this fact which avoids the use of
$L_{\infty}$-algebras.

######  Proposition 1.12.

Let $C^{k+1}\subset(M^{2n+1},\xi)$ be a regular coisotropic submanifold with
characteristic foliation $\mathcal{F}$ and local model $(U,\ker\gamma_{G})$.
If $(\eta_{t})$ is a smooth path of coisotropic sections of $U$ starting at
the zero section $\eta_{0}=0$, then $\accentset{\mbox{\large.}}{\eta_{0}}$ is
closed with respect to $d_{\nabla^{\ell}}$.

###### Proof.

By Lemma 1.7, we get a smooth family of pre-contact forms
$\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t}\in\Omega^{1}(C,\ell)$.
Denoting the characteristic foliation of
$\ker(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t})$ by
$\mathcal{F}_{t}$, we have a smooth family of rank $2n-k$ distributions
$T\mathcal{F}_{t}$ on $C$. Choose a local frame
$X_{1}^{t},\ldots,X_{2n-k}^{t}$ of $T\mathcal{F}_{t}$ that is smoothly
depending on $t$. In particular, we have

$\displaystyle(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t})(X^{t}_{i}$
$\displaystyle)=0,$ (3)
$\displaystyle(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t})([X^{t}_{i}$
$\displaystyle,X^{t}_{j}])=0,$ (4)

for all $i,j=1,\ldots,2n-k$. Differentiating the equation (3) at time $t=0$
and using the hypothesis $\eta_{0}=0$, one gets that

$\theta_{C}(\accentset{\mbox{\large.}}{X}^{0}_{i})+\accentset{\mbox{\large.}}{\eta}_{0}(X^{0}_{i})=0$
(5)

for all $i=1,\ldots,2n-k$. Similarly, differentiating the equation (4) at time
$t=0$ and using $\eta_{0}=0$ as well as the equality (5), we get

$\displaystyle 0$
$\displaystyle=\theta_{C}([\accentset{\mbox{\large.}}{X}^{0}_{i},X^{0}_{j}])+\theta_{C}([X^{0}_{i},\accentset{\mbox{\large.}}{X}^{0}_{j}])+\accentset{\mbox{\large.}}{\eta}_{0}([X^{0}_{i},X^{0}_{j}])$
$\displaystyle=-\nabla^{\ell}_{X^{0}_{j}}(\theta_{C}(\accentset{\mbox{\large.}}{X}^{0}_{i}))+\nabla^{\ell}_{X^{0}_{i}}(\theta_{C}(\accentset{\mbox{\large.}}{X}^{0}_{j}))+\accentset{\mbox{\large.}}{\eta}_{0}([X^{0}_{i},X^{0}_{j}])$
$\displaystyle=\nabla^{\ell}_{X^{0}_{j}}(\accentset{\mbox{\large.}}{\eta}_{0}(X^{0}_{i}))-\nabla^{\ell}_{X^{0}_{i}}(\accentset{\mbox{\large.}}{\eta}_{0}(X^{0}_{j}))+\accentset{\mbox{\large.}}{\eta}_{0}([X^{0}_{i},X^{0}_{j}])$
$\displaystyle=(d_{\nabla^{\ell}}\accentset{\mbox{\large.}}{\eta}_{0})(X^{0}_{j},X^{0}_{i}),$

for all $i,j=1,\ldots,2n-k$. This proves that
$d_{\nabla^{\ell}}\accentset{\mbox{\large.}}{\eta}_{0}$ vanishes. ∎

We now consider first order deformations of $C\subset(M,\xi)$ that are
geometrically trivial, in the sense that they arise from applying a contact
isotopy of $(M,\xi)$ to $C$. Recall that a contact isotopy is the flow of a
time-dependent contact vector field on $(M,\xi)$.

######  Remark 1.13.

Given a contact manifold $(M,\xi)$ with quotient line bundle $L=TM/\xi$ and
structure form $\theta\in\Omega^{1}(M,L)$, the space of contact vector fields
$\mathfrak{X}_{\xi}(M)$ fits in a canonical direct sum decomposition [9, Prop.
5.5]

$\mathfrak{X}(M)=\mathfrak{X}_{\xi}(M)\oplus\Gamma(\xi).$

It follows that there is an isomorphism of vector spaces

$X_{(-)}:\Gamma(L)\rightarrow\mathfrak{X}_{\xi}(M):\lambda\mapsto
X_{\lambda},$ (6)

which is defined by setting $X_{\lambda}$ to be the unique contact vector
field satisfying $\theta(X_{\lambda})=\lambda$.

It turns out that first order deformations of $C$ arising from contact
isotopies are cohomologically trivial, i.e. one-coboundaries in the complex
$\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}$. This
also follows from the main result in [9], since the notion of gauge
equivalence in the $L_{\infty}$-algebra structure on
$\Omega^{\bullet}(\mathcal{F},\ell)[1]$ described there agrees with the
equivalence relation by contact isotopies. Again, we believe it is instructive
to give a geometric proof which avoids the use of $L_{\infty}$-algebras.

######  Proposition 1.14.

Let $C\subset(M,\xi)$ be a regular coisotropic submanifold with characteristic
foliation $\mathcal{F}$ and local model $(U,\ker\gamma_{G})$. If a smooth path
$(\eta_{t})$ of coisotropic sections of $U$, starting at $\eta_{0}=0$, is
generated by a contact isotopy of $(U,\ker\gamma_{G})$, then
$\accentset{\mbox{\large.}}{\eta}_{0}$ is a $1$-coboundary in the complex
$(\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}})$.

###### Proof.

Let $\varphi_{t}$ be a contact isotopy of $(U,\ker\gamma_{G})$ generating
$\eta_{t}$, i.e. $\varphi_{t}(C)=\eta_{t}(C).$ It yields an isotopy
$\psi_{t}:=p\circ\varphi_{t}|_{C}$ of $C$ satisfying

$\varphi_{t}|_{C}=\eta_{t}\circ\psi_{t}.$ (7)

We claim that $(\psi_{t})_{*}$ matches the pre-contact distributions
$\ker\theta_{C}$ and
$\ker(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t})$. To prove
this, note that

$\ker\theta_{C}=TC\cap\ker\gamma_{G}=\big{(}(\varphi_{t}|_{C})_{*}\big{)}^{-1}\ker\gamma_{G},$

where we used that $\varphi_{t}$ is a contactomorphism. On the other hand,

$\ker(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t})=\big{(}(\eta_{t})_{*}\big{)}^{-1}\ker\gamma_{G}.$

Along with the relation (7), this proves the claim. It follows that there is a
unique isotopy $\Psi_{t}\colon\ell\to\ell$ of line bundle automorphisms
covering $\psi_{t}\colon C\to C$, such that

$\theta_{C}=\Psi_{t}^{\ast}(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t}).$
(8)

Denote by $\Delta_{t}\in\Gamma(\mathcal{D}\ell)$ the time-dependent derivation
generating $\Psi_{t}$; its symbol is the time dependent vector field
$X_{t}:=\sigma(\Delta_{t})\in\mathfrak{X}(C)$ generating $\psi_{t}$. This
means in particular that, for all $\lambda\in\Gamma(\ell)$ and
$Y\in\mathfrak{X}(C)$,

$\left.\frac{d}{dt}\right|_{t=0}\Psi_{t}^{\ast}\lambda=\Delta_{0}\lambda\quad\text{and}\quad\left.\frac{d}{dt}\right|_{t=0}\psi_{t}^{\ast}Y=[X_{0},Y].$

Consequently, we have for all $\alpha\in\Omega^{1}(C,\ell)$ and
$Y\in\mathfrak{X}(C)$,

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}(\Psi_{t}^{\ast}\alpha)(Y)=\left.\frac{d}{dt}\right|_{t=0}\Psi_{t}^{\ast}\left(\alpha\big{(}(\psi_{t}^{-1})^{*}Y\big{)}\right)$
$\displaystyle=\Delta_{0}(\alpha(Y))-\alpha([X_{0},Y]).$ (9)

If we evaluate both sides of (8) on arbitrary $Y\in\Gamma(T\mathcal{F})$ and
then differentiate at time $t=0$ using (9), we get

$\displaystyle 0$ $\displaystyle=\left.\frac{d}{dt}\right|_{t=0}\theta_{C}(Y)$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0}(\Psi_{t}^{\ast}(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t}))(Y)$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0}(\Psi_{t}^{\ast}\theta_{C})(Y)+\left.\frac{d}{dt}\right|_{t=0}\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t}(Y)$
$\displaystyle=\Delta_{0}(\theta_{C}(Y))-\theta_{C}([X_{0},Y]))+\accentset{\mbox{\large.}}{\eta}_{0}(Y)$
$\displaystyle=\nabla^{\ell}_{Y}(\theta_{C}(X_{0}))+\accentset{\mbox{\large.}}{\eta}_{0}(Y)$
$\displaystyle=(d_{\nabla^{\ell}}(\theta_{C}(X_{0}))+\accentset{\mbox{\large.}}{\eta}_{0})(Y),$

This shows that
$\accentset{\mbox{\large.}}{\eta}_{0}=-d_{\nabla^{\ell}}(\theta_{C}(X_{0}))$,
i.e. the $1$-cocycle $\accentset{\mbox{\large.}}{\eta}_{0}$ in
$(\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}})$ is actually a
coboundary, with primitive given by $-\theta_{C}(X_{0})$. This finishes the
proof. ∎

Prop. 1.12 and Prop. 1.14 motivate the following definition.

######  Definition 1.15.

Let $C\subset(M,\xi)$ be regular coisotropic with characteristic foliation
$\mathcal{F}$.

  1. i)

A first order deformation of $C$ is an element
$\eta\in\Omega^{1}(\mathcal{F},\ell)$ that is $d_{\nabla^{\ell}}$-closed.

  2. ii)

We call $C$ infinitesimally rigid if the cohomology group
$H^{1}(\mathcal{F},\ell)$ vanishes.

In this note, we are interested in (un)obstructedness of first order
deformations.

######  Definition 1.16.

Let $C\subset(M,\xi)$ be a regular coisotropic submanifold with characteristic
foliation $\mathcal{F}$ and local model $(U,\ker\gamma_{G})$. A first order
deformation $\eta$ of $C$ is said to be unobstructed if there exists a smooth
path of coisotropic sections $\zeta_{t}\in\Gamma(U,\ker\gamma_{G})$ satisfying
$\zeta_{0}=0$ and $\accentset{\mbox{\large.}}{\zeta_{0}}=\eta$. Otherwise,
$\eta$ is said to be obstructed.

It is well-known that a regular coisotropic submanifold $C\subset(M,\xi)$ has
obstructed first order deformations in general, see e.g. [12, §3.1] for such
an example. We will prove in this note that the coisotropic deformation
problem of $C\subset(M,\xi)$ becomes unobstructed if one restricts to
deformations whose characteristic foliation is diffeomorphic to $\mathcal{F}$.

####  1.3.2. First order deformations of foliations

Let $\mathcal{F}$ be a foliation on a manifold $C$. There is a flat
$T\mathcal{F}$-connection on the normal bundle
$N\mathcal{F}:=TC/T\mathcal{F}$, called the Bott connection, which is defined
by

$\nabla^{N\mathcal{F}}_{X}(Y\ \text{mod}\ T\mathcal{F})=[X,Y]\ \text{mod}\
T\mathcal{F},\quad X\in\Gamma(T\mathcal{F}),Y\in\mathfrak{X}(C).$

We obtain a differential $d_{\nabla^{N\mathcal{F}}}$ on
$\Gamma(\wedge^{\bullet}T^{*}\mathcal{F}\otimes N\mathcal{F})$ given by the
same formula as (1.3.1), and we denote by
$\big{(}\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}}\big{)}$
the resulting complex. Heitsch [8] showed that this complex and its cohomology
govern infinitesimal deformations of $\mathcal{F}$, in the following sense.

Let $\mathcal{F}_{t}$ be a smooth path of foliations with
$\mathcal{F}_{0}=\mathcal{F}$. Pick a complement $G$ to $T\mathcal{F}$ and
identify $G\cong N\mathcal{F}$. If $C$ is compact, then there exists
$\epsilon>0$ such that $T\mathcal{F}_{t}$ remains transverse to $G$ for $0\leq
t\leq\epsilon$. Hence, there exist $\eta_{t}\in\Gamma(T^{*}\mathcal{F}\otimes
N\mathcal{F})$ such that

$T\mathcal{F}_{t}=\text{graph}(\eta_{t})=\\{X+\eta_{t}(X):X\in\Gamma(T\mathcal{F})\\}.$

######  Lemma 1.17 ([8]).

In the setup described above, we have:

  1. (1)

The infinitesimal deformation $\accentset{\mbox{\large.}}{\eta_{0}}$ is closed
with respect to $d_{\nabla^{N\mathcal{F}}}$.

  2. (2)

If the path $\mathcal{F}_{t}$ is generated by an isotopy $\phi_{t}$, i.e.
$T\mathcal{F}_{t}=(\phi_{t})_{*}T\mathcal{F}$, then the corresponding
infinitesimal deformation is exact. Indeed,

$\accentset{\mbox{\large.}}{\eta_{0}}=d_{\nabla^{N\mathcal{F}}}(V_{0}\
\text{mod}\ T\mathcal{F}),$

where $V_{t}$ is the time-dependent vector field corresponding with the
isotopy $\phi_{t}$.

Lemma 1.17 motivates the following definition.

######  Definition 1.18.

Let $C$ be a manifold with a foliation $\mathcal{F}$.

  1. i)

A first order deformation of $\mathcal{F}$ is an element
$\eta\in\Omega^{1}(\mathcal{F},N\mathcal{F})$ that is
$d_{\nabla^{N\mathcal{F}}}$-closed.

  2. ii)

We call $\mathcal{F}$ infinitesimally rigid if the cohomology group
$H^{1}(\mathcal{F},N\mathcal{F})$ vanishes.

##  2\. First order deformations with fixed characteristic foliation

In this section, we argue what are first order deformations when deforming a
regular coisotropic submanifold $C$ inside $\text{Def}_{\mathcal{F}}(C)$. We
show that first order deformations of $C$ give rise to trivial first order
deformations of $\mathcal{F}$, via a chain map which we introduce below. While
the construction of this chain map involves a choice of complement to the
characteristic distribution $T\mathcal{F}$, the induced map in cohomology
turns out to be completely canonical.

###  2.1. Smooth Paths in $\text{Def}_{\mathcal{F}}(C)$

We first need to define a suitable notion of smoothness for paths in
$\text{Def}_{\mathcal{F}}(C)$ and $\text{Def}^{U}_{\mathcal{F}}(C)$.

######  Definition 2.1.

A path $C_{t}$ in $\text{Def}_{\mathcal{F}}(C)$ is smooth if there is a smooth
path of embeddings $\Phi_{t}:C\hookrightarrow M$ such that $C_{t}=\Phi_{t}(C)$
and $\Phi_{t}:(C,\mathcal{F})\rightarrow(C_{t},\mathcal{F}_{t})$ is a foliated
diffeomorphism.

It will be important that a smooth path in $\text{Def}_{\mathcal{F}}(C)$ comes
with a smooth path of foliated diffeomorphisms to $(C,\mathcal{F})$. By
passing to the contact thickening $(U,\ker\gamma_{G})$ of $C\subset(M,\xi)$,
Def. 2.1 induces a notion of smoothness for paths in
$\text{Def}^{U}_{\mathcal{F}}(C)$.

######  Definition 2.2.

A path $\eta_{t}$ in $\text{Def}^{U}_{\mathcal{F}}(C)$ is smooth if there is a
smooth path of embeddings $\Phi_{t}:C\hookrightarrow U$ such that
$\text{graph}(\eta_{t})=\Phi_{t}(C)$ and
$p\circ\Phi_{t}:(C,\mathcal{F})\rightarrow(C,\mathcal{F}_{\eta_{t}})$ is a
foliated diffeomorphism. Here $\mathcal{F}_{\eta_{t}}$ is the foliation of the
pre-contact structure
$\ker(\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t})$.

###  2.2. Description of first order deformations with fixed foliation

To argue what are first order deformations when deforming a regular
coisotropic submanifold $C$ within the class $\text{Def}_{\mathcal{F}}(C)$, we
work in the local model $\text{Def}^{U}_{\mathcal{F}}(C)$ introduced in Def.
1.11. We have to figure out what properties are satisfied by the infinitesimal
deformation $\accentset{\mbox{\large.}}{\eta_{0}}$ coming from a smooth path
$\eta_{t}$ in $\text{Def}^{U}_{\mathcal{F}}(C)$. One would expect that this
infinitesimal deformation gives rise to a trivial infinitesimal deformation of
the foliation $\mathcal{F}$, i.e. a $1$-coboundary in the Bott complex
$\big{(}\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}}\big{)}$.
We will show that this does indeed happen, by means of a certain map
$\Phi_{G}$ which is defined in terms of a complement $G$ to $T\mathcal{F}$.

We first introduce some notation. In the sequel, we will freely use concepts
related to the Atiyah algebroid and der-complex of a line bundle. See the
Appendix for more details.

Let $C$ be any manifold. • Assume that $\xi_{C}$ is a pre-contact distribution
on $C$, with line bundle $\ell=TC/\xi_{C}$ and structure form
$\theta_{C}\in\Omega^{1}(C,\ell)$. • Let $\mathcal{F}$ be the characteristic
foliation of $(C,\xi_{C})$. Fix a complement $TC=T\mathcal{F}\oplus G$. • The
pre-contact form $\theta_{C}$ corresponds with a pre-symplectic Atiyah form
$\varpi\in\Omega^{2}_{D}(\ell)$, see Prop. A.5. Denote $K=\ker\varpi$. The
symbol map $\sigma:D\ell\rightarrow TC$ restricts to a Lie algebroid
isomorphism $\sigma:K\overset{\sim}{\rightarrow}T\mathcal{F}.$

Clearly, the chosen complement $G$ of $T\mathcal{F}$ gives rise canonically to
a complement $\widetilde{G}$ of $K$.

######  Lemma 2.3.

The relation $\sigma(\widetilde{G})=G$ establishes a canonical 1-1
correspondence between:

  1. i)

vector subbundles $G\subseteq TC$ such that $TC=T\mathcal{F}\oplus G$,

  2. ii)

vector subbundles $\widetilde{G}\subseteq D\ell$ such that
$D\ell=K\oplus\widetilde{G}$ and $\mathbbm{1}\in\Gamma(\widetilde{G})$.

The definition of the map $\Phi_{G}$ involves the following ingredients:

  * •

The complement $\widetilde{G}$ yields a projection
$\text{pr}_{K}:D\ell\rightarrow K$.

  * •

The pre-symplectic Atiyah form $\varpi$ restricts to an isomorphism

$\varpi^{\flat}|_{\widetilde{G}}:\widetilde{G}\rightarrow\big{(}\widetilde{G}\big{)}^{*}\otimes\ell.$

Hence, if $\beta\in N^{*}\mathcal{F}\otimes\ell\simeq G^{*}\otimes\ell$, then
$\varpi^{\flat}|_{\widetilde{G}}^{-1}(\sigma^{*}\beta)$ is a well-defined
element of $\widetilde{G}$.

######  Definition 2.4.

We define
$\Phi_{G}\colon\Omega^{\bullet}(\mathcal{F},\ell)\longrightarrow\Omega^{\bullet}(\mathcal{F},N\mathcal{F})$
by setting

$\left\langle\beta,\Phi_{G}(\eta)\big{(}\sigma(\Delta_{1}),\ldots,\sigma(\Delta_{n})\big{)}\right\rangle=d_{D}(\operatorname{pr}_{K}^{\ast}\sigma^{*}\eta)\left(\Delta_{1},\ldots,\Delta_{n},\varpi^{\flat}|_{\widetilde{G}}^{-1}(\sigma^{\ast}\beta)\right).$
(10)

for
$\eta\in\Omega^{n}(\mathcal{F},\ell),\beta\in\Gamma(N^{*}\mathcal{F}\otimes\ell)$
and $\Delta_{1},\ldots,\Delta_{n}\in\Gamma(K)$.

######  Remark 2.5.

The action of $\Phi_{G}$ on $\eta\in\Omega^{n}(\mathcal{F},\ell)$ can be
equivalently described as follows. First apply the der-differential $d_{D}$ to
$\operatorname{pr}_{K}^{\ast}\sigma^{*}\eta\in\Omega_{D}^{n}(\ell)$, which
yields

$d_{D}(\operatorname{pr}_{K}^{\ast}\sigma^{*}\eta)\in\Gamma(\wedge^{n+1}K^{\ast}\otimes\ell)\oplus\Gamma(\wedge^{n}K^{\ast}\otimes\big{(}\widetilde{G}\big{)}^{*}\otimes\ell)\oplus\Gamma(\wedge^{n-1}K^{\ast}\otimes\wedge^{2}\big{(}\widetilde{G}\big{)}^{*}\otimes\ell).$

Then pick the component in
$\Gamma(\wedge^{n}K^{\ast}\otimes\big{(}\widetilde{G}\big{)}^{*}\otimes\ell)$
and apply to it the VB morphism

$-\operatorname{id}\otimes(\sigma\circ\varpi^{\flat}|_{\widetilde{G}}^{-1})\colon\wedge^{\bullet}K^{\ast}\otimes\big{(}\big{(}\widetilde{G}\big{)}^{*}\otimes\ell\big{)}\longrightarrow\wedge^{\bullet}K^{\ast}\otimes
G.$

The result is the element
$\sigma^{*}\left(\Phi_{G}(\eta)\right)\in\Omega^{n}(K,N\mathcal{F})$, which
determines $\Phi_{G}(\eta)\in\Omega^{n}(\mathcal{F},N\mathcal{F})$.

######  Theorem 2.6.

Let $C\subset(M,\xi)$ be a compact regular coisotropic submanifold with
characteristic foliation $\mathcal{F}$. A choice of splitting
$TC=T\mathcal{F}\oplus G$ yields the local model $(U,\ker\gamma_{G})$. Assume
that $\eta_{t}\in\Gamma(U)$ is a smooth path of coisotropic sections, with
$\eta_{0}=0$.

  1. (1)

The infinitesimal deformation $\accentset{\mbox{\large.}}{\eta_{0}}$ is closed
in $(\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}})$.

  2. (2)

Let $\mathcal{F}_{t}$ be the characteristic foliation of the pre-contact form
$\theta_{t}:=\theta_{C}+\operatorname{pr}_{T\mathcal{F}}^{\ast}\eta_{t}$. For
small enough $t$, there is a smooth path
$\zeta_{t}\in\Omega^{1}(\mathcal{F},N\mathcal{F})$ such that
$T\mathcal{F}_{t}=\text{graph}(\zeta_{t})$. Moreover,

$\Phi_{G}(\accentset{\mbox{\large.}}{\eta}_{0})=\accentset{\mbox{\large.}}{\zeta}_{0}.$

  3. (3)

If $\eta_{t}$ is a smooth path in $\text{Def}^{U}_{\mathcal{F}}(C)$, then
$\Phi_{G}(\accentset{\mbox{\large.}}{\eta_{0}})$ is exact in
$\big{(}\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}}\big{)}$.

###### Proof.

(1) This holds by Prop. 1.12.

(2) The pre-contact form $\theta_{t}$ corresponds to a pre-symplectic Atiyah
form $\varpi_{t}$, given by

$\varpi_{t}:=d_{D}(\sigma^{\ast}\theta_{t})=\varpi+d_{D}((\sigma\circ\operatorname{pr}_{K})^{\ast}\eta_{t}).$
(11)

We set $K_{t}:=\ker\varpi_{t}$. Then the following conditions are equivalent:

  1. (a)

$T\mathcal{F}_{t}$ is transverse to $G$, i.e. there exists
$\zeta_{t}\in\Omega^{1}(\mathcal{F},G)$ such that
$T\mathcal{F}_{t}=\text{graph}(\zeta_{t})$,

  2. (b)

$K_{t}$ is transverse to $\widetilde{G}$, i.e. there exists
$\mu_{t}\in\Omega^{1}(K,\widetilde{G})$ such that
$K_{t}=\text{graph}(\mu_{t})$.

Compactness of $C$ implies that there exists $\epsilon>0$ such that the
equivalent conditions (a) and (b) hold for $-\epsilon<t<\epsilon$. In this
case, $\zeta_{t}\in\Omega^{1}(\mathcal{F},G)$ and
$\mu_{t}\in\Omega^{1}(K,\widetilde{G})$ are related by

$\zeta_{t}\circ\sigma|_{K}=\sigma\circ\mu_{t}.$ (12)

We now compute for $\Delta\in\Gamma(K)$,

$0=\varpi_{t}^{\flat}(\Delta+\mu_{t}(\Delta))=\varpi^{\flat}(\mu_{t}(\Delta))+(d_{D}((\sigma\circ\operatorname{pr}_{K})^{\ast}\eta_{t}))^{\flat}(\Delta+\mu_{t}(\Delta)),$

where we used (11) and the fact that $K=\ker\varpi$. Differentiating at time
$t=0$ and using that $\eta_{0}=0$ and $\mu_{0}=0$, we get

$0=\varpi^{\flat}(\accentset{\mbox{\large.}}{\mu_{0}}(\Delta))+(d_{D}((\sigma\circ\operatorname{pr}_{K})^{\ast}\accentset{\mbox{\large.}}{\eta}_{0}))^{\flat}(\Delta).$

With Def. 2.4 in mind, this implies that for any $\Delta\in\Gamma(K)$ and
$\beta\in\Gamma(G^{*}\otimes\ell)$,

$\displaystyle 0$
$\displaystyle=\left\langle\varpi^{\flat}(\accentset{\mbox{\large.}}{\mu_{0}}(\Delta)),\varpi^{\flat}|_{\widetilde{G}}^{-1}(\sigma^{\ast}\beta)\right\rangle+\left\langle(\mathrm{d}_{D}((\sigma\circ\operatorname{pr}_{K})^{\ast}\accentset{\mbox{\large.}}{\eta}_{0}))^{\flat}(\Delta),\varpi^{\flat}|_{\widetilde{G}}^{-1}(\sigma^{\ast}\beta)\right\rangle$
$\displaystyle=-\left\langle\sigma^{\ast}\beta,\accentset{\mbox{\large.}}{\mu_{0}}(\Delta)\right\rangle+\left\langle\beta,\Phi_{G}(\accentset{\mbox{\large.}}{\eta}_{0})(\sigma(\Delta))\right\rangle$
$\displaystyle=-\left\langle\beta,\sigma(\accentset{\mbox{\large.}}{\mu_{0}}(\Delta))\right\rangle+\left\langle\beta,\Phi_{G}(\accentset{\mbox{\large.}}{\eta}_{0})(\sigma(\Delta))\right\rangle$
$\displaystyle=-\big{\langle}\beta,\accentset{\mbox{\large.}}{\zeta_{0}}(\sigma(\Delta))\big{\rangle}+\left\langle\beta,\Phi_{G}(\accentset{\mbox{\large.}}{\eta}_{0})(\sigma(\Delta))\right\rangle$
$\displaystyle=\left\langle\beta,\big{(}\Phi_{G}(\accentset{\mbox{\large.}}{\eta}_{0})-\accentset{\mbox{\large.}}{\zeta}_{0}\big{)}(\sigma(\Delta))\right\rangle,$

using (12) in the fourth equality. This shows that
$\Phi(\accentset{\mbox{\large.}}{\eta}_{0})=\accentset{\mbox{\large.}}{\zeta}_{0}$.

(3) By part $(2)$, we only have to show that
$\accentset{\mbox{\large.}}{\zeta}_{0}$ is exact in
$(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}})$.
Since $\eta_{t}$ is a smooth path in $\text{Def}_{\mathcal{F}}^{U}(C)$,
according to Def. 2.2 there exists a smooth family of foliated diffeomorphisms

$\phi_{t}\colon(C,T\mathcal{F})\longrightarrow(C,T\mathcal{F}_{t}).$

Precomposing $\phi_{t}$ with $\phi_{0}^{-1}$, we can assume that
$\phi_{0}=\operatorname{id}_{C}$, so that the smooth family of foliations
$\mathcal{F}_{t}$ is generated by the isotopy $\phi_{t}$. As
$T\mathcal{F}_{t}=\operatorname{graph}(\zeta_{t})$, Lemma 1.17 implies that
$\accentset{\mbox{\large.}}{\zeta}_{0}$ is exact in
$(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}})$. This
finishes the proof. ∎

Thm. 2.6 motivates the following (provisional) definition.

######  Definition 2.7 (Provisional).

Let $C\subset(M,\xi)$ be a compact regular coisotropic submanifold with
characteristic foliation $\mathcal{F}$. When deforming $C$ inside
$\text{Def}_{\mathcal{F}}(C)$, a first order deformation of $C$ is an element
$\eta\in\Omega^{1}(\mathcal{F},\ell)$ such that

$d_{\nabla^{\ell}}\eta=0\quad\text{and}\quad\Phi_{G}(\eta)\ \text{is exact
in}\ (\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}}).$

In what follows, we study the map $\Phi_{G}$ in more detail. Section 2.3 gives
an alternative description for it, which will allow us to prove that
$\Phi_{G}$ is a chain map from
$(\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}})$ to
$(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}})$. In
Section 2.4, we show that the induced map in cohomology is canonical, i.e.
$[\Phi_{G}]\colon H^{\bullet}(\mathcal{F},\ell)\to
H^{\bullet}(\mathcal{F},N\mathcal{F})$ does not depend on the complement $G$.
In particular, this yields an invariant description of first order
deformations which refines Def. 2.7.

###  2.3. A chain map

Assume the setup described in the box at the beginning of §2.2. To show that
$\Phi_{G}:(\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}})\to(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}})$
is a chain map, we proceed in 2 steps:

  1. (1)

Construct a canonical chain map

$\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right)\longrightarrow\big{(}\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}}\big{)}.$
(13)

Here $J^{1}_{\perp}\ell$ is a suitable subbundle of $J^{1}\ell$ to be
introduced later.

  2. (2)

Construct a chain map

$\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}\rightarrow\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right)$
(14)

which depends on the choice of complement $G$.

We then show that $\Phi_{G}$ is the composition of these two maps, hence it is
also a chain map.

####  2.3.1. Step 1 in construction of $\Phi_{G}$

To construct the map (13), we start by remarking that the Lie algebroid
isomorphism $\sigma:K\rightarrow T\mathcal{F}$ relates some natural
representations carried by these Lie algebroids. On one hand, $T\mathcal{F}$
carries the following representations.

  * •

The $T\mathcal{F}$-representation $\nabla^{\ell}$ on $\ell$ given by

$\nabla^{\ell}_{X}(\theta_{C}(Y))=\theta_{C}([X,Y]),\quad\text{for}\
X\in\Gamma(T\mathcal{F})\ \text{and}\ Y\in\mathfrak{X}(C).$

Alternatively, $\nabla^{\ell}$ can be characterized by the following
commutative diagram

$\leavevmode\hbox to91.71pt{\vbox
to46.42pt{\pgfpicture\makeatletter\hbox{\hskip
45.85309pt\lower-23.20831pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
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}\pgfsys<EMAIL_ADDRESS>(15)

To see this, we compute for $\Delta\in\Gamma(K)$ and
$\square\in\Gamma(D\ell)$,

$\displaystyle 0$ $\displaystyle=\varpi(\Delta,\square)$
$\displaystyle=(d_{D}(\sigma^{\ast}\theta))(\Delta,\square)$
$\displaystyle=\Delta(\theta(\sigma(\square)))-\cancel{\square(\theta(\sigma(\Delta)))}-\theta([\sigma(\Delta),\sigma(\square)]).$

  * •

The Bott representation $\nabla^{N\mathcal{F}}$ of $T\mathcal{F}$ on
$N\mathcal{F}$ given by

$\nabla^{N\mathcal{F}}_{X}(Y\ \text{mod}\ T\mathcal{F})=[X,Y]\ \text{mod}\
T\mathcal{F},\quad\text{for}\ X\in\Gamma(T\mathcal{F}),\ Y\in\mathfrak{X}(C).$

The Lie algebroid $K$ carries representations on $\ell$ and on vector bundles
$D_{\perp}\ell$ and $J^{1}_{\perp}\ell$ which we now introduce. We first
define $D_{\perp}\ell:=D\ell/K$. Since $\sigma:D\ell\rightarrow TC$ maps $K$
bijectively onto $T\mathcal{F}$, it descends to a VB morphism

$\sigma_{\perp}\colon D_{\perp}\ell\longrightarrow N\mathcal{F},\ \Delta\
\text{mod}\ K\longmapsto\sigma(\Delta)\ \text{mod}\ T\mathcal{F}.$

Next, we define $J^{1}_{\perp}\ell\subset
J^{1}\ell\simeq(D\ell)^{\ast}\otimes\ell$ as the annihilator of $K\subset
D\ell$, i.e.

$J^{1}_{\perp}\ell:=\sqcup_{x\in
C}\\{j^{1}_{x}\lambda\,\mid\lambda\in\Gamma(\ell)\ \text{such that}\
\delta(\lambda)=0,\ \text{for all}\ \delta\in K_{x}\\}.$

Note that we have an identification

$(D_{\perp}\ell)^{\ast}\otimes\ell\overset{\sim}{\longrightarrow}J^{1}_{\perp}\ell.$
(16)

The Lie algebroid $K$ now carries the following natural representations.

  * •

The tautological representation of $D\ell$ on $\ell$ restricts to a
representation of $K$, namely

$\nabla^{\ell}:K\rightarrow D\ell:\Delta\mapsto\Delta.$

Because of the commutative diagram (15), using the symbol $\nabla^{\ell}$ to
denote both the $T\mathcal{F}$\- and $K$-representation on $\ell$ should not
lead to any confusion.

  * •

The Bott representation $\nabla^{D_{\perp}\ell}$ of $K$ on $D_{\perp}\ell$
given by

$\nabla^{D_{\perp}\ell}_{\Delta}(\square\ \text{mod}\ K)=[\Delta,\square]\
\text{mod}\ K,\quad\text{for}\ \Delta\in\Gamma(K),\ \square\in\Gamma(D\ell).$

  * •

The $\ell$-twisted dual Bott representation of $K$ on $J^{1}_{\perp}\ell$,
which is defined by

$\left\langle\nabla^{J^{1}_{\perp}\ell}_{\Delta}\alpha,\square\ \text{mod}\
K\right\rangle=\nabla_{\Delta}^{\ell}\langle\alpha,\square\ \text{mod}\
K\rangle-\left\langle\alpha,\nabla^{D_{\perp}\ell}_{\Delta}(\square\
\text{mod}\ K)\right\rangle,$ (17)

for $\Delta\in\Gamma(K),\alpha\in\Gamma(J^{1}_{\perp}\ell)$ and
$\square\in\Gamma(D\ell)$. Here we use the identification (16). Viewing
sections of $J^{1}_{\perp}\ell$ as elements of
$\Gamma(K^{\circ}\otimes\ell)\subset\Gamma((D\ell)^{*}\otimes\ell)$, it
follows from Cartan calculus with Atiyah forms $\Omega_{D}^{\bullet}(\ell)$
that

$\nabla_{\Delta}^{J^{1}_{\perp}\ell}\alpha=\mathcal{L}_{\Delta}\alpha.$

It is clear that under the Lie algebroid isomorphism
$\sigma:K\overset{\sim}{\rightarrow}T\mathcal{F}$, the representations of
$T\mathcal{F}$ and $K$ recalled above are related as follows.

  * •

The identity map $\text{id}_{\ell}\colon\ell\to\ell$ intertwines the
representations of $K$ and $T\mathcal{F}$ on $\ell$, i.e.

$\nabla^{\ell}_{\Delta}\lambda=\nabla^{\ell}_{\sigma(\Delta)}\lambda,\quad\text{for}\
\Delta\in\Gamma(K)\ \text{and}\ \lambda\in\Gamma(\ell).$

  * •

The VB morphism $\sigma_{\perp}\colon D_{\perp}\ell\to N\mathcal{F}$
intertwines the $K$-representation on $D_{\perp}\ell$ and the
$T\mathcal{F}$-representation on $N\mathcal{F}$, i.e. for $\Delta\in\Gamma(K)$
and $\square\in\Gamma(D\ell)$,

$\sigma_{\perp}(\nabla^{D_{\perp}\ell}_{\Delta}(\square\ \text{mod}\
K))=\nabla^{N\mathcal{F}}_{\sigma(\Delta)}(\sigma(\square)\ \text{mod}\
T\mathcal{F}).$ (18)

The $K$-representations on $D_{\perp}\ell$ and $J^{1}_{\perp}\ell$ are also
related, as we show now. Recall that we are given an $\ell$-valued pre-
symplectic Atiyah form $\varpi\in\Gamma(\wedge^{2}(D\ell)^{*}\otimes\ell)$
with $\ker\varpi^{\flat}=K$. It follows that

$J^{1}_{\perp}\ell=\operatorname{im}\big{\\{}\varpi^{\flat}\colon
D\ell\to(D\ell)^{\ast}\otimes\ell\simeq J^{1}\ell\big{\\}},$

and moreover, $\varpi$ descends to a non-degenerate form
$\varpi_{\perp}\in\Gamma(\wedge^{2}(D_{\perp}\ell)^{\ast}\otimes\ell)$ which
makes the following diagram commute

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We will denote the inverse of $\varpi^{\flat}_{\perp}\colon
D_{\perp}\ell\overset{\sim}{\longrightarrow}J^{1}_{\perp}\ell$ by
$\varpi_{\perp}^{\sharp}\colon
J^{1}_{\perp}\ell\overset{\sim}{\longrightarrow}D_{\perp}\ell$.

######  Proposition 2.8.

The VB isomorphism $\varpi_{\perp}^{\flat}\colon D_{\perp}\ell\to
J^{1}_{\perp}\ell$ intertwines the $K$-representations on $D_{\perp}\ell$ and
$J^{1}_{\perp}\ell$, i.e. for $\Delta\in\Gamma(K)$ and
$\square\in\Gamma(D\ell)$ we have

$\varpi_{\perp}^{\flat}(\nabla^{D_{\perp}\ell}_{\Delta}(\square\ \text{mod}\
K))=\nabla^{J^{1}_{\perp}\ell}_{\Delta}(\varpi_{\perp}^{\flat}(\square\
\text{mod}\ K)).$

###### Proof.

Since $d_{D}\varpi=0$ and $K=\ker\varpi$, the Cartan formula implies that for
$\Delta\in\Gamma(K)$,

$\mathcal{L}_{\Delta}\varpi=\iota_{\Delta}\mathrm{d}_{D}\varpi+\mathrm{d}_{D}\iota_{\Delta}\varpi=0.$

Hence, we can compute for any $\Delta\in\Gamma(K)$ and
$\square,\square^{\prime}\in\Gamma(D\ell)$,

$\displaystyle 0$
$\displaystyle=(\mathcal{L}_{\Delta}\varpi)(\square,\square^{\prime})$
$\displaystyle=\Delta(\varpi(\square,\square^{\prime}))-\varpi([\Delta,\square],\square^{\prime})-\varpi(\square,[\Delta,\square^{\prime}])$
$\displaystyle=\nabla^{\ell}_{\Delta}\left\langle\varpi_{\perp}^{\flat}(\square\
\text{mod}\ K),\square^{\prime}\ \text{mod}\
K\right\rangle-\left\langle\varpi_{\perp}^{\flat}\big{(}\nabla^{D_{\perp}\ell}_{\Delta}(\square\
\text{mod}\ K)\big{)},\square^{\prime}\ \text{mod}\ K\right\rangle$
$\displaystyle\phantom{=}-\left\langle\varpi_{\perp}^{\flat}(\square\
\text{mod}\ K),\nabla^{D_{\perp}\ell}_{\Delta}(\square^{\prime}\ \text{mod}\
K)\right\rangle$
$\displaystyle=\left\langle\nabla^{J^{1}_{\perp}\ell}_{\Delta}(\varpi_{\perp}^{\flat}(\square\
\text{mod}\ K)),\square^{\prime}\ \text{mod}\
K\right\rangle-\left\langle\varpi_{\perp}^{\flat}\big{(}\nabla^{D_{\perp}\ell}_{\Delta}(\square\
\text{mod}\ K)\big{)},\square^{\prime}\ \text{mod}\ K\right\rangle.$

This shows that

$\nabla^{J^{1}_{\perp}\ell}_{\Delta}(\varpi_{\perp}^{\flat}(\square\
\text{mod}\
K))=\varpi_{\perp}^{\flat}\big{(}\nabla^{D_{\perp}\ell}_{\Delta}(\square\
\text{mod}\ K)\big{)}$

for all $\Delta\in\Gamma(K)$ and $\square\in\Gamma(D\ell)$, as we wanted to
prove. ∎

Combining the result (18) and Prop. 2.8, we obtain the following.

######  Corollary 2.9.

Under the Lie algebroid isomorphism $\sigma:K\overset{\sim}{\to}T\mathcal{F}$,
the VB morphism $\sigma_{\perp}\circ\varpi_{\perp}^{\sharp}\colon
J^{1}_{\perp}\ell\to N\mathcal{F}$ intertwines the $K$-represesentation on
$J^{1}_{\perp}\ell$ and the $T\mathcal{F}$-representation on $N\mathcal{F}$.
That is, for any $\Delta\in\Gamma(K)$ and
$\alpha\in\Gamma(J^{1}_{\perp}\ell)$,

$(\sigma_{\perp}\circ\varpi_{\perp}^{\sharp})\big{(}\nabla^{J^{1}_{\perp}\ell}_{\Delta}\alpha\big{)}=\nabla^{N\mathcal{F}}_{\sigma(\Delta)}((\sigma_{\perp}\circ\varpi_{\perp}^{\sharp})\alpha).$

Cor. 2.9 yields a canonical chain map which is the first step in the
construction of $\Phi_{G}$.

######  Corollary 2.10.

The VB morphism

$\wedge^{\bullet}(\sigma|_{K}^{-1})^{*}\otimes(\sigma_{\perp}\circ\varpi_{\perp}^{\sharp})\colon\wedge^{\bullet}K^{\ast}\otimes
J^{1}_{\perp}\ell\longrightarrow\wedge^{\bullet}T^{\ast}\mathcal{F}\otimes
N\mathcal{F}$

induces a chain map at the level of sections

$\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right)\longrightarrow\left(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}}\right).$

####  2.3.2. Step 2 in construction of $\Phi_{G}$

The construction of the map (14) requires a choice of complement
$TC=T\mathcal{F}\oplus G$. By Lemma 2.3, there is an induced complement
$D\ell=K\oplus\widetilde{G}$. Denoting by
$\widetilde{G}^{\dagger}:=(\widetilde{G})^{*}\otimes\ell$ the $\ell$-twisted
dual of $\widetilde{G}$, we will make the following identifications:

$D_{\perp}\ell\simeq\widetilde{G}\quad\text{and}\quad
J^{1}_{\perp}\ell\simeq\widetilde{G}^{\dagger}.$

The VB isomorphism

$\wedge^{i}(D\ell)^{\ast}\otimes\ell\simeq\bigoplus_{a+b=i}\wedge^{a}(D_{\perp}\ell)^{\ast}\otimes\wedge^{b}K^{\ast}\otimes\ell.$

induces an isomorphism of $C^{\infty}(C)$-modules

$\Omega_{D}^{i}(\ell)\simeq\bigoplus_{a+b=i}\Omega^{a,b}_{D}(\ell),\quad\text{where}\
\
\Omega_{D}^{a,b}(\ell):=\Gamma(\wedge^{a}(D_{\perp}\ell)^{\ast}\otimes\wedge^{b}K^{\ast}\otimes\ell).$
(19)

Since $K\subset D\ell$ is involutive, the Koszul formula for the der-
differential $d_{D}$ implies that

$d_{D}\Omega_{D}^{a,b}(\ell)\subseteq\Omega_{D}^{a,b+1}(\ell)\oplus\Omega_{D}^{a+1,b}(\ell)\oplus\Omega_{D}^{a+2,b-1}(\ell).$

Therefore, $d_{D}$ decomposes as

$d_{D}=d_{0,1}^{G}+d_{1,0}^{G}+d_{2,-1}^{G},$ (20)

where each summand is bi-homogeneous with respect to the bi-grading
$\Omega^{\bullet,\bullet}_{D}(\ell)$, i.e.

$d^{G}_{0,1}\colon\Omega_{D}^{\bullet,\bullet}(\ell)\to\Omega_{D}^{\bullet,\bullet+1}(\ell),\quad
d^{G}_{1,0}\colon\Omega_{D}^{\bullet,\bullet}(\ell)\to\Omega_{D}^{\bullet+1,\bullet}(\ell),\quad
d^{G}_{2,-1}\colon\Omega_{D}^{\bullet,\bullet}(\ell)\to\Omega_{D}^{\bullet+2,\bullet-1}(\ell).$

For our aims, we will only need the components $d_{0,1}^{G}$ and
$d_{1,0}^{G}$. They are explicitly given by the following expressions, similar
to the ones found in [11, Chapter 4]:

$\displaystyle(d_{1,0}^{G}\eta)(\overline{\square}_{1},$
$\displaystyle\ldots,\overline{\square}_{a+1},\Delta_{1},\ldots,\Delta_{b})$
(21)
$\displaystyle=\sum_{i=1}^{a+1}(-1)^{i+1}\square_{i}(\eta(\overline{\square}_{1},\ldots,\widehat{\overline{\square}}_{i},\ldots,\overline{\square}_{a+1},\Delta_{1},\ldots,\Delta_{b}))$
$\displaystyle\hskip 5.69046pt+\sum_{1\leq i<j\leq
a+1}(-1)^{i+j}\eta(\overline{[\square_{i},\square_{j}]},\overline{\square}_{1},\ldots,\widehat{\overline{\square}}_{i},\ldots,\widehat{\overline{\square}}_{j},\ldots,\overline{\square}_{a+1},\Delta_{1},\ldots,\Delta_{b})$
$\displaystyle\hskip
5.69046pt+\sum_{i=1}^{a+1}\sum_{j=1}^{b}(-1)^{i+j+1}\eta(\overline{\square}_{1},\ldots,\widehat{\overline{\square}}_{i},\ldots,\overline{\square}_{a+1},\operatorname{pr}_{K}[\square_{i},\Delta_{j}],\Delta_{1},\ldots,\widehat{\Delta}_{j},\ldots,\Delta_{b}),$
$\displaystyle(d_{0,1}^{G}\eta)(\overline{\square}_{1},$
$\displaystyle\ldots,\overline{\square}_{a},\Delta_{1},\ldots,\Delta_{b+1})$
(22)
$\displaystyle=\sum_{i=1}^{b+1}(-1)^{a+i+1}\Delta_{i}(\eta(\overline{\square}_{1},\ldots,\overline{\square}_{a},\Delta_{1},\dots,\widehat{\Delta}_{i},\ldots,\Delta_{b+1}))$
$\displaystyle\hskip
5.69046pt+\sum_{i=1}^{a}\sum_{j=1}^{b+1}(-1)^{i+a+j}\eta(\overline{[\square_{i},\Delta_{j}]},\overline{\square}_{1},\ldots,\widehat{\overline{\square}}_{i},\ldots,\overline{\square}_{a},\Delta_{1},\ldots,\widehat{\Delta}_{j},\ldots,\Delta_{b+1})$
$\displaystyle\hskip 5.69046pt+\sum_{1\leq i<j\leq
b+1}(-1)^{i+a+j}\eta(\overline{\square}_{1},\ldots,\overline{\square}_{a},[\Delta_{i},\Delta_{j}],\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\widehat{\Delta}_{j},\ldots,\Delta_{b+1}),$

where $\eta\in\Omega^{a,b}_{D}(\ell)$,
$\square_{1},\ldots,\square_{a+1}\in\Gamma(\widetilde{G})$ and
$\Delta_{1},\ldots,\Delta_{b+1}\in\Gamma(K)$. Above, we denoted by
$\overline{\square}_{i}$ the equivalence class $\square_{i}\
\operatorname{mod}\ K\in\Gamma(D_{\perp}\ell)$. In view of the decomposition
(20), the cohomological condition $d_{D}^{2}=0$ can be rewritten as

$\displaystyle d_{2,-1}^{G}\circ d_{2,-1}^{G}=0,$ $\displaystyle
d_{2,-1}^{G}\circ d_{1,0}^{G}+d_{1,0}^{G}\circ d_{2,-1}^{G}=0,$ $\displaystyle
d_{1,0}^{G}\circ d_{1,0}^{G}+d_{0,1}^{G}\circ d_{2,-1}^{G}+d_{2,-1}^{G}\circ
d_{0,1}^{G}=0,$ $\displaystyle d_{0,1}^{G}\circ d_{1,0}^{G}+d_{1,0}^{G}\circ
d_{0,1}^{G}=0,$ (23) $\displaystyle d_{0,1}^{G}\circ d_{0,1}^{G}=0.$ (24)

The first ingredient for the construction of the desired map (14) is the
following.

######  Lemma 2.11.

(1) For each $a\in\mathbb{N}$, we have a complex
$(\Omega_{D}^{a,\bullet}(\ell),d_{0,1}^{G})$. For $a=0$, it coincides with the
de Rham complex of $K$ with coefficients in $\ell$, i.e.

$\big{(}\Omega^{0,\bullet}_{D}(\ell),d^{G}_{0,1}\big{)}=\big{(}\Omega^{\bullet}(K,\ell),d_{\nabla^{\ell}}\big{)}.$
(25)

(2) We have a chain map

$d_{1,0}^{G}\circ(\sigma|_{K})^{*}:\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}\rightarrow\big{(}\Omega^{1,\bullet}_{D}(\ell),-d_{0,1}^{G}\big{)}.$

###### Proof.

For item $(1)$, note that $d_{0,1}^{G}$ preserves
$\Omega^{a,\bullet}_{D}(\ell)$ and squares to zero by Eq. (24). Hence
$(\Omega_{D}^{a,\bullet}(\ell),d_{0,1}^{G})$ is a complex. It is clear that
$\Omega_{D}^{0,\bullet}(\ell)=\Gamma(\wedge^{\bullet}K^{\ast}\otimes\ell)=\Omega^{\bullet}(K,\ell)$,
and Eq. (22) shows that $d_{0,1}^{G}$ acts on
$\Omega^{0,\bullet}_{D}(\ell)=\Omega^{\bullet}(K,\ell)$ exactly like
$d_{\nabla^{\ell}}$.

For item $(2)$, recall that under the Lie algebroid isomorphism
$\sigma:K\rightarrow T\mathcal{F}$, the identity map
$\text{id}_{\ell}\colon\ell\to\ell$ intertwines the representations of $K$ and
$T\mathcal{F}$ on $\ell$. Hence, we have an isomorphism of complexes

$(\sigma|_{K})^{*}:\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}\overset{\sim}{\longrightarrow}\big{(}\Omega^{\bullet}(K,\ell),d_{\nabla^{\ell}}\big{)}.$

Moreover, Eq. (23) shows that we have a chain map

$d_{1,0}^{G}\colon(\Omega^{0,\bullet}_{D}(\ell),d_{0,1}^{G})\longrightarrow(\Omega^{1,\bullet}_{D}(\ell),-d_{0,1}^{G}).$

Along with the equality of complexes (25) obtained in part $(1)$, this proves
the statement. ∎

The second ingredient is the standard degree $0$ graded
$\Omega^{\bullet}(K)$-module isomorphism

$\tau\colon\Omega^{1,\bullet}_{D}(\ell)\overset{\sim}{\longrightarrow}\Omega^{\bullet}(K,J^{1}_{\perp}\ell),$

which maps $\eta\in\Omega^{1,b}_{D}(\ell)$ to the element
$\tau(\eta)\in\Omega^{b}(K,J^{1}_{\perp}\ell)$ uniquely determined by

$\eta(\overline{\square},\Delta_{1},\ldots,\Delta_{b})=\big{\langle}\tau(\eta)(\Delta_{1},\ldots,\Delta_{b}),\overline{\square}\big{\rangle}$

for any $\square\in\Gamma(\widetilde{G})$ and
$\Delta_{1},\ldots,\Delta_{b}\in\Gamma(K)$. This map is compatible with
differentials.

######  Lemma 2.12.

The map $\tau$ is an isomorphism of complexes

$\tau\colon\big{(}\Omega^{1,\bullet}_{D}(\ell),-d_{0,1}^{G}\big{)}\longrightarrow\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right).$

###### Proof.

Fix $\eta\in\Omega^{1,b}_{D}(\ell)$. Using the Koszul formula for
$d_{\nabla^{J^{1}_{\perp}\ell}}$ and the definition (17) of
$\nabla^{J^{1}_{\perp}\ell}$, we compute for any
$\square\in\Gamma(\widetilde{G})$ and
$\Delta_{1},\ldots,\Delta_{b+1}\in\Gamma(K)$:

$\displaystyle\left\langle\left(d_{\nabla^{J^{1}_{\perp}\ell}}\tau(\eta)\right)(\Delta_{1},\ldots,\Delta_{b+1}),\overline{\square}\right\rangle$
$\displaystyle\hskip
42.67912pt=\sum_{i=1}^{b+1}(-1)^{i+1}\left\langle\nabla^{J^{1}_{\perp}\ell}_{\Delta_{i}}\big{(}\tau(\eta)(\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\Delta_{b+1})\big{)},\overline{\square}\right\rangle$
$\displaystyle\hskip
42.67912pt\phantom{=}+\sum_{i<j}(-1)^{i+j}\left\langle\tau(\eta)\big{(}[\Delta_{i},\Delta_{j}],\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\widehat{\Delta}_{j},\ldots,\Delta_{b+1}\big{)},\overline{\square}\right\rangle$
$\displaystyle\hskip
42.67912pt=\sum_{i=1}^{b+1}(-1)^{i+1}\Delta_{i}\big{(}\eta(\overline{\square},\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\Delta_{b+1})\big{)}$
$\displaystyle\hskip
42.67912pt\phantom{=}-\sum_{i=1}^{b+1}(-1)^{i+1}\eta\big{(}\overline{[\Delta_{i},\square]},\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\Delta_{b+1}\big{)}$
$\displaystyle\hskip
42.67912pt\phantom{=}+\sum_{i<j}(-1)^{i+j}\eta\big{(}\overline{\square},[\Delta_{i},\Delta_{j}],\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\widehat{\Delta}_{j},\ldots,\Delta_{b+1}\big{)}$
$\displaystyle\hskip
42.67912pt=-\big{(}d_{0,1}^{G}\eta\big{)}\big{(}\overline{\square},\Delta_{1},\ldots,\Delta_{b+1}\big{)}$
$\displaystyle\hskip
42.67912pt=-\left\langle\tau\big{(}d_{0,1}^{G}\eta\big{)}(\Delta_{1},\dots,\Delta_{b+1}),\overline{\square}\right\rangle,$

where we also used the expression for $d_{0,1}^{G}$ in Eq. (22). This proves
the statement. ∎

Combining Lemma 2.11 and Lemma 2.12, we obtain the desired chain map (14).

######  Corollary 2.13.

We have a chain map

$\tau\circ
d_{1,0}^{G}\circ(\sigma|_{K})^{*}:\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}\rightarrow\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right).$

At last, we show that $\Phi_{G}$ is the composition of the chain maps in Cor.
2.10 and Cor. 2.13.

######  Theorem 2.14.

The map $\Phi_{G}$ fits in the following commutative diagram of chain maps

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In particular, it is also a chain map

$\Phi_{G}\colon\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}\longrightarrow\big{(}\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),-d_{\nabla^{N\mathcal{F}}}\big{)}.$
(26)

###### Proof.

From Lemma 2.11 and Lemma 2.12 we already know that the left and top arrows
are chain maps. Moreover, Cor. 2.10 yields a chain map

$\wedge^{\bullet}(\sigma|_{K}^{-1})^{*}\otimes(\sigma_{\perp}\circ\varpi_{\perp}^{\sharp}):\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right)\longrightarrow\left(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),d_{\nabla^{N\mathcal{F}}}\right),$

hence changing signs shows that the right arrow is a chain map

$-\wedge^{\bullet}(-\sigma|_{K}^{-1})^{*}\otimes(\sigma_{\perp}\circ\varpi_{\perp}^{\sharp}):\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right)\longrightarrow\left(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),-d_{\nabla^{N\mathcal{F}}}\right).$

It remains to check that the diagram commutes. Pick
$\eta\in\Omega^{b}(\mathcal{F},\ell)$. For
$\Delta_{1},\ldots,\Delta_{b}\in\Gamma(K)$ and
$\beta\in\Gamma(N^{\ast}\mathcal{F}\otimes\ell)$, we can compute

$\displaystyle\left\langle\beta,\Phi_{G}(\eta)\big{(}\sigma(\Delta_{1}),\ldots,\sigma(\Delta_{b})\big{)}\right\rangle$
$\displaystyle\hskip
28.45274pt=d_{D}((\sigma\circ\operatorname{pr}_{K})^{\ast}\eta)\left(\Delta_{1},\ldots,\Delta_{b},\varpi^{\flat}|_{\widetilde{G}}^{-1}(\sigma_{\perp}^{\ast}\beta)\right)$
$\displaystyle\hskip
28.45274pt=(-1)^{b}d_{D}((\sigma\circ\operatorname{pr}_{K})^{\ast}\eta)\left(\varpi^{\flat}|_{\widetilde{G}}^{-1}(\sigma_{\perp}^{\ast}\beta),\Delta_{1},\ldots,\Delta_{b}\right)$
$\displaystyle\hskip
28.45274pt=(-1)^{b}\big{(}d_{1,0}^{G}\sigma^{*}(\eta)\big{)}\big{(}\varpi_{\perp}^{\sharp}(\sigma_{\perp}^{\ast}\beta),\Delta_{1},\ldots,\Delta_{b}\big{)}$
$\displaystyle\hskip
28.45274pt=(-1)^{b}\left\langle\tau\big{(}d_{1,0}^{G}\sigma^{*}(\eta)\big{)}(\Delta_{1},\ldots,\Delta_{b}),\varpi_{\perp}^{\sharp}(\sigma_{\perp}^{\ast}\beta)\right\rangle$
$\displaystyle\hskip
28.45274pt=-(-1)^{b}\left\langle\sigma_{\perp}^{\ast}\beta,\varpi_{\perp}^{\sharp}\left(\tau\big{(}d_{1,0}^{G}\sigma^{*}(\eta)\big{)}(\Delta_{1},\ldots,\Delta_{b})\right)\right\rangle$
$\displaystyle\hskip
28.45274pt=-(-1)^{b}\left\langle\beta,(\sigma_{\perp}\circ\varpi_{\perp}^{\sharp})\left(\tau\big{(}d_{1,0}^{G}\sigma^{*}(\eta)\big{)}(\Delta_{1},\ldots,\Delta_{b})\right)\right\rangle$
$\displaystyle\hskip
28.45274pt=\left\langle\beta,-\left(\wedge^{\bullet}(-\sigma|_{K}^{-1})^{*}\otimes(\sigma_{\perp}\circ\varpi_{\perp}^{\sharp})\circ\tau\circ
d_{1,0}^{G}\circ(\sigma|_{K})^{*}\right)(\eta)\big{(}\sigma(\Delta_{1}),\ldots,\sigma(\Delta_{b})\big{)}\right\rangle.$

This shows that the diagram commutes, hence the proof is finished. ∎

###  2.4. The induced map in cohomology

The chain map $\Phi_{G}$ depends on the choice of complement $G$. This
dependence is made explicit by Thm. 2.14, which shows that $\Phi_{G}$ is given
by the composition

$\Phi_{G}=-\wedge^{\bullet}(-\sigma|_{K}^{-1})^{*}\otimes(\sigma_{\perp}\circ\varpi_{\perp}))\circ\tau\circ
d_{1,0}^{G}\circ(\sigma|_{K})^{*}.$

The maps appearing above are all canonical, except for $d_{1,0}^{G}$. In this
section we show that, by contrast, the map induced in cohomology by $\Phi_{G}$
is canonical, i.e. independent of the complement $G$. To prove this, it
suffices to show that the map induced in cohomology by the chain map

$\tau\circ
d_{1,0}^{G}:\big{(}\Omega^{\bullet}(K,\ell),d_{\nabla^{\ell}}\big{)}\rightarrow\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right)$

is canonical. To do so, we will first construct a canonical map
$F_{\text{can}}:H^{\bullet}(K,\ell)\rightarrow
H^{\bullet}(K,J^{1}_{\perp}\ell)$, and then show that $F_{\text{can}}$ agrees
with the map induced in cohomology by $\tau\circ d_{1,0}^{G}$. The
construction of $F_{\text{can}}$ presented below is the specialization to the
Lie pair $(D\ell,K)$ of a general construction which works for any Lie pair.
In particular, for the Lie pair $(TC,T\mathcal{F})$, the construction recovers
the well-known transverse differentiation map
$d_{\nu}:H^{\bullet}(\mathcal{F})\rightarrow
H^{\bullet}(\mathcal{F},N^{*}\mathcal{F})$ in foliation theory (see e.g. [5,
§2.3]).

The map $F_{\text{can}}\colon H^{\bullet}(K,\ell)\rightarrow
H^{\bullet}(K,J^{1}_{\perp}\ell)$ will arise as the composition of two
separate canonical maps. To construct the the first component of
$F_{\text{can}}$, recall the following complexes:

  * •

We have the de Rham complex $(\Omega^{\bullet}_{D}(\ell),d_{D})$ of $D\ell$
with values in $\ell$, whose cohomology we denote by $H^{\bullet}_{D}(\ell)$.

  * •

We have the de Rham complex $(\Omega^{\bullet}(K,\ell),d_{\nabla^{\ell}})$ of
$K$ with values in $\ell$, whose cohomology we denote by
$H^{\bullet}(K,\ell)$.

  * •

We have the subcomplex
$(\Omega^{\bullet}_{D,K}(\ell),d_{D})\subset(\Omega^{\bullet}_{D}(\ell),d_{D})$
given by

$\Omega^{n}_{D,K}(\ell)=\\{\eta\in\Omega_{D}^{n}(\ell)\,\mid\,\eta(\Delta_{1},\ldots,\Delta_{n})=0,\
\text{for all}\ \Delta_{1},\ldots,\Delta_{n}\in\Gamma(K)\\},$

whose cohomology we denote by $H^{\bullet}_{D,K}(\ell)$.

These ingredients fit in a short exact sequence of complexes and chain maps

${0}$${(\Omega^{\bullet}_{D,K}(\ell),d_{D})}$${(\Omega^{\bullet}_{D}(\ell),d_{D})}$${(\Omega^{\bullet}(K,\ell),d_{\nabla^{\ell}})}$${0,}$incl$\scriptstyle{\iota_{K}^{\ast}}$

where $\iota_{K}\colon K\to D\ell$ is the inclusion of the Lie subalgebroid
$K\subset D\ell$. The corresponding long exact sequence in cohomology is

${\ldots}$${H^{n}_{D,K}(\ell)}$${H^{n}_{D}(\ell)}$${H^{n}(K,\ell)}$${H^{n+1}_{D,K}(\ell)}$${H^{n+1}_{D}(\ell)}$${H^{n+1}(K,\ell)}$${\ldots}$$\scriptstyle{\mathfrak{d}}$

where the connecting map $\mathfrak{d}\colon H^{n}(K,\ell)\to
H^{n+1}_{D,K}(\ell)$ is defined by setting

$\mathfrak{d}[\iota_{K}^{\ast}\alpha]=[\mathrm{d}_{D}\alpha],$

Since the der-complex is acyclic (see Rem. A.3), the connecting map is an
isomorphism

$\mathfrak{d}\colon
H^{\bullet}(K,\ell)\overset{\sim}{\longrightarrow}H^{\bullet+1}_{D,K}(\ell).$
(27)

This is the first component of the map $F_{\text{can}}\colon
H^{\bullet}(K,\ell)\rightarrow H^{\bullet}(K,J^{1}_{\perp}\ell)$ we aim to
construct.

We now construct the second component of $F_{\text{can}}$. Note that for any
$\alpha\in\Omega_{D,K}^{n+1}(\ell)$ there exists a unique
$p(\alpha)\in\Omega^{n}(K,J^{1}_{\perp}\ell)$ such that

$\alpha(\square,\Delta_{1},\ldots,\Delta_{n})=\left\langle
p(\alpha)(\Delta_{1},\ldots,\Delta_{n}),\overline{\square}\right\rangle,$ (28)

for $\square\in\Gamma(D\ell)$ and $\Delta_{1},\ldots,\Delta_{n}\in\Gamma(K)$.
Here we denoted $\overline{\square}:=\square\ \text{mod}\
K\in\Gamma(D_{\perp}\ell)$. The relation (28) defines a map

$p\colon\Omega^{\bullet+1}_{D,K}(\ell)\longrightarrow\Omega^{\bullet}(K,J^{1}_{\perp}\ell).$

######  Lemma 2.15.

We get a chain map

$p\colon\big{(}\Omega^{\bullet+1}_{D,K}(\ell),d_{D}\big{)}\longrightarrow\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),-d_{\nabla^{J^{1}_{\perp}\ell}}\right).$

###### Proof.

Pick $\alpha\in\Omega^{n+1}_{D,K}(\ell)$. For any
$\Delta_{1},\ldots,\Delta_{n+1}\in\Gamma(K)$ and $\square\in\Gamma(D\ell)$, we
compute

$\displaystyle\left\langle
p(d_{D}\alpha)(\Delta_{1},\ldots,\Delta_{n+1}),\overline{\square}\right\rangle$
$\displaystyle\hskip
56.9055pt=(d_{D}\alpha)(\square,\Delta_{1},\ldots,\Delta_{n+1})$
$\displaystyle\hskip
56.9055pt=\cancel{\square(\alpha(\Delta_{1},\ldots,\Delta_{n+1}))}+\sum_{i}(-1)^{i}\Delta_{i}\big{(}\alpha(\square,\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\Delta_{n+1})\big{)}$
$\displaystyle\hskip
56.9055pt\phantom{=}-\sum_{i}(-1)^{i}\alpha([\Delta_{i},\square],\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\Delta_{n+1})$
$\displaystyle\hskip
56.9055pt\phantom{=}-\sum_{i<j}(-1)^{i+j}\alpha(\square,[\Delta_{i},\Delta_{j}],\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\widehat{\Delta}_{j},\ldots,\Delta_{n+1})$
$\displaystyle\hskip
56.9055pt=-\sum_{i}(-1)^{i+1}\left\langle\nabla^{J^{1}_{\perp}\ell}_{\Delta_{i}}\big{(}p(\alpha)(\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\Delta_{n+1})\big{)},\overline{\square}\right\rangle$
$\displaystyle\hskip 56.9055pt\phantom{=}-\sum_{i<j}(-1)^{i+j}\left\langle
p(\alpha)\big{(}[\Delta_{i},\Delta_{j}],\Delta_{1},\ldots,\widehat{\Delta}_{i},\ldots,\widehat{\Delta}_{j},\ldots,\Delta_{n+1}\big{)},\overline{\square}\right\rangle$
$\displaystyle\hskip
56.9055pt=-\left\langle\left(d_{\nabla^{J^{1}_{\perp}\ell}}p(\alpha)\right)(\Delta_{1},\ldots,\Delta_{n+1}),\overline{\square}\right\rangle.$

This shows that $p\circ d_{D}=-d_{\nabla^{J^{1}_{\perp}\ell}}\circ p$, so the
proof is finished. ∎

######  Remark 2.16.

If we choose a distribution $G\subset TC$ complementary to $T\mathcal{F}$,
then $\Omega^{\bullet}_{D,K}(\ell)$ identifies with the subcomplex

$\bigoplus_{\genfrac{}{}{0.0pt}{}{i\geq 1}{j\geq
0}}\Omega_{D}^{i,j}(\ell)\subset\big{(}\Omega^{\bullet}_{D}(\ell),d_{D}\big{)},$

where we used the bi-degree notation introduced in (19). The map
$p\colon\Omega_{D,K}^{\bullet+1}\to\Omega^{\bullet}(K,J^{1}_{\perp}\ell)$ then
identifies with the following composition of chain maps

$\left(\bigoplus\nolimits_{\genfrac{}{}{0.0pt}{}{i\geq 1}{j\geq
0}}\Omega_{D}^{i,j}(\ell),d_{D}\right)\overset{\operatorname{proj}}{\longrightarrow}\big{(}\Omega_{D}^{1,\bullet}(\ell),d_{0,1}^{G}\big{)}\overset{\tau}{\longrightarrow}\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),-d_{\nabla^{J^{1}_{\perp}\ell}}\right),$

see Lemma 2.12. This provides an alternative proof for Lemma 2.15.

Since $p$ is a chain map, it induces a map in cohomology

$[p]\colon H^{\bullet+1}_{D,K}(\ell)\longrightarrow
H^{\bullet}(K,J^{1}_{\perp}\ell),$ (29)

which is the second component of the desired canonical map
$F_{\text{can}}\colon H^{\bullet}(K,\ell)\rightarrow
H^{\bullet}(K,J^{1}_{\perp}\ell)$.

######  Definition 2.17.

We define $F_{\text{can}}:=[p]\circ\mathfrak{d}$, i.e.

$F_{\text{can}}\colon H^{\bullet}(K,\ell)\longrightarrow
H^{\bullet}(K,J^{1}_{\perp}\ell),\
[\iota_{K}^{\ast}\alpha]\longmapsto[p(\mathrm{d}_{D}\alpha)].$

We think of $F_{\text{can}}$ as a kind of “transverse differentiation map” for
the Lie pair $(K,D\ell)$. Now we observe that this canonical map
$F_{\text{can}}$ arises from the (non-canonical) chain map given by $\tau\circ
d_{1,0}^{G}$, for any choice of complement $G$.

######  Lemma 2.18.

For any choice of complement $G$ to $T\mathcal{F}\subset TC$, the chain map

$\tau\circ
d_{1,0}^{G}:\big{(}\Omega^{\bullet}(K,\ell),d_{\nabla^{\ell}}\big{)}\rightarrow\left(\Omega^{\bullet}(K,J^{1}_{\perp}\ell),d_{\nabla^{J^{1}_{\perp}\ell}}\right)$

induces a map in cohomology which is canonical (i.e. independent of $G$). In
fact, we have

$[\tau\circ d_{1,0}^{G}]=F_{\text{can}}\colon
H^{\bullet}(K,\ell)\longrightarrow H^{\bullet}(K,J^{1}_{\perp}\ell).$

###### Proof.

Fix $\alpha\in\Omega^{n}(K,\ell)\simeq\Omega_{D}^{0,n}(\ell)$ with
$d_{\nabla^{\ell}}\alpha=0$. Since
$\alpha=\iota_{K}^{*}\text{pr}_{K}^{*}\alpha$ with
$\text{pr}_{K}^{*}\alpha\in\Omega^{n}_{D}(\ell)$, the definition of
$F_{\text{can}}$ shows that

$F_{\text{can}}[\alpha]=[p(d_{D}(\text{pr}_{K}^{*}\alpha))].$ (30)

For any $\Delta_{1},\ldots,\Delta_{n}\in\Gamma(K)$ and
$\square\in\Gamma(\widetilde{G})$, we compute

$\displaystyle\left\langle
p(d_{D}(\text{pr}_{K}^{*}\alpha))(\Delta_{1},\ldots,\Delta_{n}),\overline{\square}\right\rangle$
$\displaystyle=(d_{D}(\text{pr}_{K}^{*}\alpha))(\square,\Delta_{1},\ldots,\Delta_{n})$
$\displaystyle=(\mathrm{d}^{G}_{1,0}\alpha)(\overline{\square},\Delta_{1},\ldots,\Delta_{n})$
$\displaystyle=\left\langle\tau(d^{G}_{1,0}\alpha)(\Delta_{1},\ldots,\Delta_{n}),\overline{\square}\right\rangle.$

This shows that $p(d_{D}(\text{pr}_{K}^{*}\alpha))=(\tau\circ
d^{G}_{1,0})(\alpha)$. Along with (30), this shows that

$F_{\textnormal{can}}[\alpha]=[\tau\circ\mathrm{d}^{G}_{1,0}][\alpha].\qed$

Finally, we obtain the main result of this section.

######  Theorem 2.19.

For any choice of complement $G$ to $T\mathcal{F}\subset TC$, the chain map

$\Phi_{G}\colon(\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}})\longrightarrow(\Omega^{\bullet}(\mathcal{F},N\mathcal{F}),-d_{\nabla^{N\mathcal{F}}})$

induces a map in cohomology which is canonical (i.e. independent of $G$).

###### Proof.

It follows immediately from Thm. 2.14 and Lemma 2.18. ∎

Consequently, we can denote the map in cohomology unambiguously by

$[\Phi]:H^{\bullet}(\mathcal{F},\ell)\longrightarrow
H^{\bullet}(\mathcal{F},N\mathcal{F}).$

##  3\. Infinitesimal rigidity

In this section, we show that a compact regular coisotropic submanifold
$C\subset(M,\xi)$ is infinitesimally rigid inside
$\text{Def}_{\mathcal{F}}(C)$. This means that every first order deformation
of $C$ in $\text{Def}_{\mathcal{F}}(C)$ is induced by a contact isotopy of
$(M,\xi)$. Infinitesimal rigidity implies in particular that the deformation
problem of $C$ inside $\text{Def}_{\mathcal{F}}(C)$ is unobstructed.

######  Proposition 3.1.

The map $[\Phi]:H^{1}(\mathcal{F},\ell)\rightarrow
H^{1}(\mathcal{F},N\mathcal{F})$ is injective.

###### Proof.

Pick $\eta\in\Omega^{1}(\mathcal{F},\ell)$ such that
$d_{\nabla^{\ell}}\eta=0$, and assume that $[\Phi_{G}][\eta]=0$ for some
(hence any) complement $G$ to $T\mathcal{F}$. This means that there exists
$Z\in\mathfrak{X}(C)$ s.t.

$\Phi_{G}(\eta)=d_{\nabla^{N\mathcal{F}}}(Z\ \operatorname{mod}\
T\mathcal{F}).$ (31)

Claim: $\eta=-d_{\nabla^{\ell}}(\theta_{C}(Z))$, where
$\theta_{C}\in\Omega^{1}(C,\ell)$ is the pre-contact form on $C$.

Note that $\theta_{C}\in\Gamma(N^{*}\mathcal{F}\otimes\ell)$ and
$\sigma_{\perp}^{\ast}\theta_{C}=\iota_{\mathbbm{1}}\varpi\in\Gamma(J^{1}_{\perp}\ell)$,
where $\varpi$ is the pre-symplectic Atiyah form corresponding to
$\theta_{C}$. We can compute for any $\Delta\in\Gamma(K)$,

$\displaystyle\big{\langle}\theta_{C},\Phi_{G}(\eta)\big{(}\sigma(\Delta)\big{)}\big{\rangle}$
$\displaystyle=\left\langle\theta_{C},(\sigma_{\perp}\circ\varpi_{\perp}^{\sharp})\left(\tau\big{(}d_{1,0}^{G}\sigma^{*}\eta\big{)}\right)(\Delta)\right\rangle$
$\displaystyle=-\left\langle\tau\big{(}d_{1,0}^{G}\sigma^{*}\eta\big{)}(\Delta),\varpi_{\perp}^{\sharp}(\sigma_{\perp}^{\ast}\theta)\right\rangle$
$\displaystyle=-\left(d_{1,0}^{G}\sigma^{*}\eta\right)\big{(}\varpi_{\perp}^{\sharp}(\sigma_{\perp}^{\ast}\theta),\Delta\big{)}$
$\displaystyle=-\left(d_{1,0}^{G}\sigma^{*}\eta\right)\big{(}\overline{\mathbbm{1}},\Delta\big{)}$
$\displaystyle=-\mathbbm{1}(\eta(\sigma(\Delta)))-\cancel{\sigma^{*}\eta(\text{pr}_{K}[\Delta,\mathbbm{1}])},$

using Thm. 2.14 in the first equality and Eq. 21 in the last equality. On the
other hand,

$\displaystyle\left\langle\theta_{C},d_{\nabla^{N\mathcal{F}}}(Z\ \text{mod}\
T\mathcal{F})\big{(}\sigma(\Delta)\big{)}\right\rangle$
$\displaystyle=\theta_{C}\left(\nabla^{N\mathcal{F}}_{\sigma(\Delta)}Z\
\text{mod}\ T\mathcal{F}\right)$
$\displaystyle=\theta_{C}\big{(}[\sigma(\Delta),Z]\big{)}$
$\displaystyle=\nabla^{\ell}_{\sigma(\Delta)}\theta_{C}(Z)$
$\displaystyle=\big{(}d_{\nabla^{\ell}}\theta_{C}(Z)\big{)}(\sigma(\Delta)).$

In view of Eq. (31), this shows that $\eta=-d_{\nabla^{\ell}}(\theta(Z))$.
This finishes the proof. ∎

######  Remark 3.2.

We compare first order deformations of coisotropic submanifolds –under the
constraint that the diffeomorphism type of the characteristic foliation is
fixed– in symplectic and contact geometry. First, let $C$ be a compact
coisotropic submanifold of a symplectic manifold $(M,\omega)$ with
characteristic foliation $\mathcal{F}$. As shown in [5], when deforming $C$
inside the class of coisotropic submanifolds whose characteristic foliation is
diffeomorphic to $\mathcal{F}$, first order deformations are closed foliated
one-forms $\alpha\in\Omega^{1}(\mathcal{F})$ whose cohomology class lies in
the kernel of the transverse differentiation map
$d_{\nu}:H^{1}(\mathcal{F})\rightarrow
H^{1}(\mathcal{F},N^{*}\mathcal{F})\cong H^{1}(\mathcal{F},N\mathcal{F})$.
This kernel is infinite dimensional in general, see [5, Ex. 4.4]. For the same
deformation problem in contact geometry, we showed that first order
deformations are closed elements $\eta\in\Omega^{1}(\mathcal{F},\ell)$ whose
cohomology class lies in the kernel of
$[\Phi]:H^{1}(\mathcal{F},\ell)\rightarrow H^{1}(\mathcal{F},N\mathcal{F})$.
By contrast, this kernel is now trivial because of Prop. 3.1.

We can now refine Def. 2.7 as follows.

######  Corollary 3.3.

Let $C\subset(M,\xi)$ be a compact regular coisotropic submanifold with
characteristic foliation $\mathcal{F}$. When deforming $C$ inside
$\text{Def}_{\mathcal{F}}(C)$, first order deformations of $C$ are
$1$-coboundaries in
$\big{(}\Omega^{\bullet}(\mathcal{F},\ell),d_{\nabla^{\ell}}\big{)}$.

We now interpret this result in terms of the action by contact isotopies of
$(M,\xi)$ on the space of coisotropic submanifolds. It is clear that this
action restricts to $\text{Def}_{\mathcal{F}}(C)$. In view of Prop. 1.14, the
above corollary states that a compact regular coisotropic submanifold $C$ is
infinitesimally rigid with respect to the action by contact isotopies, if we
deform $C$ within the class $\text{Def}_{\mathcal{F}}(C)$. This implies
unobstructedness of the deformation problem.

######  Corollary 3.4.

Let $C\subset(M,\xi)$ be a compact regular coisotropic submanifold with
characteristic foliation $\mathcal{F}$. Then $C$ is infinitesimally rigid with
respect to the action by contact isotopies, when deforming $C$ inside
$\text{Def}_{\mathcal{F}}(C)$. In particular, the deformation problem of $C$
inside $\text{Def}_{\mathcal{F}}(C)$ is unobstructed.

###### Proof.

Denote by $(U,\ker\gamma_{G})$ the local model around $C$. Given a first order
deformation $d_{\nabla^{\ell}}\lambda$ of $C$ for some
$\lambda\in\Gamma(\ell)$, we construct a smooth path $\eta_{t}$ in
$\text{Def}_{\mathcal{F}}^{U}(C)$ such that
$\accentset{\mbox{\large.}}{\eta_{0}}=d_{\nabla^{\ell}}\lambda$.

Recall from §1.1 that $\gamma_{G}$ is a $p^{*}\ell$-valued contact form, hence
the section $-p^{*}\lambda\in\Gamma(p^{*}\ell)$ gives rise via the isomorphism
(6) to a contact vector field $X_{-p^{*}\lambda}\in\mathfrak{X}(U)$ with flow
$(\varphi_{t})$. Compactness of $C$ implies that there exists $\epsilon>0$
such that the flow lines of $X_{-p^{*}\lambda}$ starting at points of $C$
exist up to time $\epsilon$. Shrinking $\epsilon$ if necessary, we may assume
that $\varphi_{t}(C)$ is the graph of a section $\eta_{t}\in\Gamma(U)$ for
each $t\in[0,\epsilon]$. Since $\varphi_{t}$ is a contactomorphism, it takes
the characteristic foliation $\mathcal{F}$ of $C$ to that of the coisotropic
submanifold $\text{graph}(\eta_{t})$. Hence, $\eta_{t}$ is a smooth path in
$\text{Def}_{\mathcal{F}}^{U}(C)$ as required in Def. 2.2, by the proof of
Lemma 1.10.

Claim: $\accentset{\mbox{\large.}}{\eta_{0}}=d_{\nabla^{l}}\lambda$.

To prove the claim, we proceed as in the proof of Prop. 1.14. Setting
$\psi_{t}:=p\circ\varphi_{t}|_{C}$, we obtain an isotopy of $C$ satisfying

$\eta_{t}=\varphi_{t}\circ\psi_{t}^{-1}.$ (32)

Let $V_{t}\in\mathfrak{X}(C)$ be the time-dependent vector field of
$\psi_{t}$. The proof of Prop. 1.14 shows that

$\accentset{\mbox{\large.}}{\eta_{0}}=-d_{\nabla^{\ell}}(\theta_{C}(V_{0})),$

hence the claim is proved once we show that $\lambda=-\theta_{C}(V_{0})$. To
do so, note that the equation (32) implies that

$\accentset{\mbox{\large.}}{\eta_{0}}=\left.X_{-p^{*}\lambda}\right|_{C}-V_{0},$

which is an equality of elements in $\Gamma(TU|_{C})$. Evaluating
$\gamma_{G}=p^{*}\theta_{C}+\theta_{G}$ on both sides of this equality gives

$0=-(p^{*}\lambda)|_{C}-\theta_{C}(V_{0})=-\lambda-\theta_{C}(V_{0}).$

Here we use that $\theta_{G}$ vanishes along the zero section $C$, and that
$\gamma_{G}(X_{-p^{*}\lambda})=-p^{*}\lambda$ as explained in Rem. 1.13. This
shows that $\lambda=-\theta_{C}(V_{0})$, so the claim is proved. ∎

##  4\. Rigidity

In the previous section, we saw that a compact regular coisotropic submanifold
$C\subset(M,\xi)$ is infinitesimally rigid in $\text{Def}_{\mathcal{F}}(C)$.
This raises the question whether $C$ is actually rigid in
$\text{Def}_{\mathcal{F}}(C)$. We show below that this is indeed the case.
More precisely, we prove that every smooth path $C_{t}$ in
$\text{Def}_{\mathcal{F}}(C)$ deforming $C$ is generated by a contact isotopy
of $(M,\xi)$, for small enough times $t$. This extends a rigidity result due
to the second author [12], which concerns the special case in which the
characteristic foliation $\mathcal{F}$ is given by a fibration. The theorem to
which we devote this section is the following.

######  Theorem 4.1.

Let $C\subset(M,\xi)$ be a compact regular coisotropic submanifold, and let
$C_{t}$ be a smooth path in $\text{Def}_{\mathcal{F}}(C)$ with $C_{0}=C$. Then
there exists an isotopy of contactomorphisms
$\psi_{t}\colon(M,\xi)\to(M,\xi)$, locally defined around $C$, such that
$C_{t}=\psi_{t}(C)$ for small enough $t$.

The proof of this theorem relies on two results which may be interesting in
their own right. The first result is an extension of the Gray stability
theorem for contact structures [4, Thm. 2.2.2] to the case of pre-contact
structures (see Prop. 4.3). The second one is a uniqueness statement
concerning coisotropic embeddings in contact manifolds (see Prop. 4.5). To
prove the announced extension of Gray stability, we need an auxiliary lemma.

######  Lemma 4.2.

Given a line bundle $\ell\to C$, consider a smooth path of pre-contact forms
$\theta_{t}\in\Omega^{1}(C,\ell)$. Denote by $\mathcal{F}_{t}$ the
characteristic foliation of $(C,\theta_{t})$. Assume that there exists an
isotopy of foliated diffeomorphisms
$\underline{\smash{\varphi}}_{t}\colon(C,\mathcal{F}_{0})\longrightarrow(C,\mathcal{F}_{t})$
with corresponding time-dependent vector field $V_{t}\in\mathfrak{X}(C)$. If
$Z\in\Gamma(T\mathcal{F}_{t})$, then

$\left(\frac{d}{dt}\theta_{t}\right)(Z)=\theta_{t}([V_{t},Z]).$

###### Proof.

We know that $Z=(\underline{\smash{\varphi}}_{t})_{*}(X)$ for some
$X\in\Gamma(T\mathcal{F}_{0})$. Consider the family of vector fields
$W_{s}\in\Gamma(T\mathcal{F}_{s})$ for $s\in[0,1]$ defined by

$W_{s}=(\underline{\smash{\varphi}}_{s})_{*}(X).$

In particular, we have $W_{t}=Z$. Differentiating the equality
$\theta_{s}(W_{s})=0$ at time $s=t$ gives

$\displaystyle 0$
$\displaystyle=\left(\left.\frac{d}{ds}\right|_{s=t}\theta_{s}\right)(W_{t})+\theta_{t}\left(\left.\frac{d}{ds}\right|_{s=t}W_{s}\right)=\left(\frac{d}{dt}\theta_{t}\right)(Z)+\theta_{t}\left(\left.\frac{d}{ds}\right|_{s=t}W_{s}\right).$

The proof is finished if we show that

$\left.\frac{d}{ds}\right|_{s=t}W_{s}=-[V_{t},Z].$ (33)

Denoting by $U_{t}$ the time-dependent vector field corresponding to the
inverse isotopy $\underline{\smash{\varphi}}_{t}^{-1}$, we have

$\left.\frac{d}{ds}\right|_{s=t}W_{s}=\left.\frac{d}{ds}\right|_{s=t}(\underline{\smash{\varphi}}_{s}^{-1})^{*}(X)=(\underline{\smash{\varphi}}_{t}^{-1})^{*}\big{(}[U_{t},X]\big{)}=\big{[}(\underline{\smash{\varphi}}_{t}^{-1})^{*}(U_{t}),Z\big{]}.$
(34)

It is well-known that the time-dependent vector fields $U_{t}$ and $V_{t}$ are
related by

$U_{t}=-(\underline{\smash{\varphi}}_{t})^{*}V_{t},$

see [1, §1.1]. Inserting this equality into (34), we see that the equation
(33) holds. ∎

We can now state and prove the anticipated generalization of the Gray
stability theorem. In what follows, given a line bundle automorphism
$\Phi\colon\ell\to\ell$, we use the pushforward of derivations $D\Phi\colon
D\ell\rightarrow D\ell$ and the pullback of Atiyah forms
$\Phi_{t}^{\ast}\colon\Omega^{\bullet}_{D}(\ell)\rightarrow\Omega^{\bullet}_{D}(\ell)$,
which are recalled in the Appendix.

######  Proposition 4.3.

Given a line bundle $\ell\to C$, consider a smooth path of pre-contact forms
$\theta_{t}\in\Omega^{1}(C,\ell)$ and let $\varpi_{t}\in\Omega^{2}_{D}(\ell)$
be the corresponding smooth path of pre-symplectic Atiyah forms. Denote by
$\mathcal{F}_{t}$ the characteristic foliation of $(C,\theta_{t})$ and by
$K_{t}$ the kernel of $\varpi_{t}$, so that $T\mathcal{F}_{t}=\sigma(K_{t})$.
Then the following are equivalent:

  1. (1)

there exists an isotopy of foliated diffeomorphisms
$\underline{\smash{\varphi}}_{t}\colon(C,\mathcal{F}_{0})\longrightarrow(C,\mathcal{F}_{t})$,

  2. (2)

there exists an isotopy of LB-automorphisms $\Phi_{t}\colon\ell\to\ell$ such
that $\Phi_{t}^{\ast}\varpi_{t}=\varpi_{0}$,

  3. (3)

there exists an isotopy of LB-automorphisms $\Phi_{t}\colon\ell\to\ell$ such
that $\Phi_{t}^{\ast}\theta_{t}=\theta_{0}$.

###### Proof.

(2)$\Longleftrightarrow$(3) This is easily proved using the relation
$\varpi_{t}=d_{D}(\sigma^{\ast}\theta_{t})$ or equivalently, the relation
$\iota_{\mathbbm{1}}\varpi_{t}=\sigma^{\ast}\theta_{t}$, together with the
natural identities
$\Phi_{t}^{\ast}\iota_{\mathbbm{1}}=\iota_{\mathbbm{1}}\Phi_{t}^{\ast}$ and
$\Phi_{t}^{\ast}d_{D}=d_{D}\Phi_{t}^{\ast}$.

(2)$\Longrightarrow$(1) Denote by $\underline{\smash{\varphi}}_{t}\colon C\to
C$ the isotopy of diffeomorphisms underlying $\Phi_{t}\colon\ell\to\ell$. Then
the hypothesis $\varpi_{0}=\Phi_{t}^{\ast}\varpi_{t}$ can be written
explicitly as follows:

$(\varpi_{0})_{x}(\delta_{x},\delta_{x}^{\prime})=(\Phi_{t})_{x}^{-1}\left((\varpi_{t})_{\underline{\smash{\varphi}}_{t}(x)}\left((D_{x}\Phi_{t})\delta_{x},(D_{x}\Phi_{t})\delta_{x}^{\prime}\right)\right),$

for all $x\in C$ and $\delta_{x},\delta_{x}^{\prime}\in D_{x}\ell$. This
implies that the isomorphism $D_{x}\Phi_{t}\colon
D_{x}\ell\overset{\sim}{\longrightarrow}D_{\underline{\smash{\varphi}}_{t}(x)}\ell$
induces an isomorphism

$D_{x}\Phi_{t}:(K_{0})_{x}\overset{\sim}{\longrightarrow}(K_{t})_{\underline{\smash{\varphi}}_{t}(x)}.$

Since $\sigma\colon D\ell\to TC$ induces a Lie algebroid isomorphism
$K_{t}\overset{\sim}{\rightarrow}T\mathcal{F}_{t}$ and
$\sigma\circ(D\Phi_{t})=(\underline{\smash{\varphi}}_{t})_{*}\circ\sigma$, it
follows that $(\underline{\smash{\varphi}}_{t})_{*}$ descends to an
isomorphism

$(\underline{\smash{\varphi}}_{t})_{*}:T_{x}\mathcal{F}_{0}\overset{\sim}{\longrightarrow}T_{\underline{\smash{\varphi}}_{t}(x)}\mathcal{F}_{t}.$

(1)$\Longrightarrow$(2) Following the usual Moser argument, we will obtain the
desidered isotopy of LB automorphisms as the flow of a suitable smooth path in
$\Gamma(D\ell)$. For any isotopy of LB automorphisms
$\Phi_{t}\colon\ell\to\ell$ generated by a smooth path
$\square_{t}\in\Gamma(D\ell)$, we can compute

$\frac{d}{dt}\Phi_{t}^{\ast}\varpi_{t}=\Phi_{t}^{\ast}\left(\mathcal{L}_{\square_{t}}\varpi_{t}+\accentset{\mbox{\large.}}{\varpi}_{t}\right)=\Phi_{t}^{\ast}d_{D}\left(\iota_{\square_{t}}\varpi_{t}+\sigma^{\ast}\accentset{\mbox{\large.}}{\theta}_{t}\right),$

where we also used Prop. A.5. Since the der-complex is acyclic (see Rem. A.3),
this shows that there is an isotopy $\Phi_{t}$ satisfying
$\Phi_{t}^{\ast}\varpi_{t}=\varpi_{0}$ if and only if we can find smooth paths
$\square_{t}\in\Gamma(D\ell)$ and $\lambda_{t}\in\Gamma(\ell)$ such that

$\iota_{\square_{t}}\varpi_{t}=j^{1}\lambda_{t}-\sigma^{\ast}\accentset{\mbox{\large.}}{\theta}_{t}.$
(35)

Here we used that $d_{D}\lambda=j^{1}\lambda$ for any $\lambda\in\Gamma(\ell)$
under the identification $(D\ell)^{\ast}\otimes\ell\cong J^{1}\ell$.

Denote by $V_{t}\in\mathfrak{X}(C)$ the time-dependent vector field generating
the isotopy $\underline{\smash{\varphi}}_{t}$. Setting
$\lambda_{t}=-\theta_{t}(V_{t})$, we claim that the equation (35) can be
solved for $\square_{t}$. To prove this, we have to check that
$j^{1}\lambda_{t}-\sigma^{\ast}\accentset{\mbox{\large.}}{\theta}_{t}\in\Gamma(J^{1}\ell)$
belongs to the image of $\varpi_{t}^{\flat}\colon D\ell\rightarrow J^{1}\ell$,
which coincides with the annihilator of $K_{t}:=\ker\varpi_{t}$. For any
$\Delta\in\Gamma(K_{t})$, we can compute

$\displaystyle\left\langle\sigma^{\ast}\accentset{\mbox{\large.}}{\theta}_{t}+j^{1}\big{(}\theta_{t}(V_{t})\big{)},\Delta\right\rangle$
$\displaystyle=\accentset{\mbox{\large.}}{\theta}_{t}(\sigma(\Delta))+\Delta\big{(}\theta_{t}(V_{t})\big{)}$
$\displaystyle=\accentset{\mbox{\large.}}{\theta}_{t}(\sigma(\Delta))+\nabla^{\ell}_{\sigma(\triangle)}\theta_{t}(V_{t})$
$\displaystyle=\accentset{\mbox{\large.}}{\theta}_{t}(\sigma(\Delta))+\theta_{t}\big{(}[\sigma(\triangle),V_{t}]\big{)}$
$\displaystyle=\theta_{t}\big{(}[V_{t},\sigma(\triangle)]\big{)}+\theta_{t}\big{(}[\sigma(\triangle),V_{t}]\big{)}$
$\displaystyle=0,$

where we used the diagram (15) in the second equality and Lemma 4.2 in the
fourth equality.

This shows that setting $\lambda_{t}:=-\theta_{t}(V_{t})$, the equation (35)
can be solved for $\square_{t}$. Moreover, the kernels of $\varpi_{t}$ for
$t\in[0,1]$ form a vector bundle over $C\times[0,1]$, and one can choose a
complementary subbundle inside the pullback bundle
$\operatorname{pr}_{1}^{\ast}D\ell$, where $\operatorname{pr}_{1}\colon
C\times[0,1]\rightarrow C$ denotes the projection. Requiring that
$\square_{t}$ takes values in this subbundle uniquely determines
$\square_{t}$, and shows in particular that $\square_{t}$ can be chosen in a
smooth manner. Since $C$ is compact, we can integrate $\square_{t}$ globally
to an isotopy of LB automorphisms $\Phi_{t}\colon\ell\to\ell$ satisfying
$\Phi_{t}^{\ast}\varpi_{t}=\varpi_{0}$ for all $t$. This finishes the proof. ∎

######  Remark 4.4.

A weaker version of the implication (1)$\Longrightarrow$(2) in the above
proposition has appeared in [12, Section 4.2]. There one claims (without
proof) that, if the foliation $\mathcal{F}_{0}$ is simple, one can find an
isotopy of LB automorphisms $\Phi_{t}\colon\ell\to\ell$ satisfying
$D\Phi_{t}(K_{0})=K_{t}$ for any $t$. We proved a stronger statement, namely
that the isotopy $\Phi_{t}$ can be chosen so that
$\Phi_{t}^{\ast}\varpi_{t}=\varpi_{0}$ for any $t$, without imposing any
restrictions on the foliation $\mathcal{F}_{0}$.

As anticipated, the second ingredient for the proof of our main result is the
following uniqueness statement concerning coisotropic embeddings in contact
manifolds.

######  Proposition 4.5.

Let $L\to M$ be a line bundle, $\Omega_{t}\in\Omega_{D}^{2}(L)$ a smooth path
of symplectic Atiyah forms for $0\leq t\leq 1$, and $C\subseteq M$ a
submanifold. Set $\ell:=L|_{C}$ and let $I\colon\ell\to L$ be the inclusion.
Assume that $C$ is a compact regular coisotropic submanifold of
$(M,\Omega_{t})$ for all $0\leq t\leq 1$, such that additionally

$I^{\ast}\Omega_{t}=I^{\ast}\Omega_{0}.$ (36)

Then there exist a neighborhood $\mathcal{U}\subset M$ of $C$ and a path of LB
morphisms $\Psi_{t}\colon L|_{\mathcal{U}}\to L$ covering a path of embeddings
$\psi_{t}\colon\mathcal{U}\to M$, such that

$\Psi_{0}=\operatorname{id}_{\mathcal{U}},\quad\text{and}\quad\Psi_{t}^{\ast}\Omega_{t}=\Omega_{0}|_{\mathcal{U}}.$
(37)

Moreover, $\Psi_{t}$ induces an isotopy of LB automorphisms
$\Psi_{t}|_{C}\colon\ell\to\ell$ covering the isotopy of diffeomorphisms
$\psi_{t}|_{C}\colon C\to C$, such that
$\Psi_{t}|_{C}^{\ast}(I^{\ast}\Omega_{0})=I^{\ast}\Omega_{0}.$

###### Proof.

Set
$\square_{t}=-\Omega_{t}^{\sharp}(\iota_{\mathbbm{1}}\accentset{\mbox{\large.}}{\Omega}_{t})\in\Gamma(DL)$.
We claim that $\square_{t}|_{C}\in\Gamma(D\ell)$, or in other words that
$\sigma(\square_{t})$ is tangent to $C$. To see this, note that by (36), we
have $I^{\ast}\accentset{\mbox{\large.}}{\Omega}_{t}=0$. Since
$\mathbbm{1}|_{C}\in\Gamma(D\ell)$, this implies that for all
$\Delta\in\Gamma(D\ell)$,

$\big{.}\big{(}\iota_{\mathbbm{1}}\accentset{\mbox{\large.}}{\Omega}_{t}\big{)}\big{|}_{C}(\Delta)=(I^{\ast}\accentset{\mbox{\large.}}{\Omega}_{t})(\mathbbm{1}|_{C},\Delta)=0.$

This shows that

$\big{.}\big{(}\iota_{\mathbbm{1}}\accentset{\mbox{\large.}}{\Omega}_{t}\big{)}\big{|}_{C}\in\Gamma((D\ell)^{\circ}).$
(38)

Since $C\subseteq(M,\Omega_{t})$ is a regular coisotropic submanifold for all
$0\leq t\leq 1$, we know by Prop. 1.5 that
$(D\ell)^{\perp_{\Omega_{t}}}\subset D\ell$. Here
$(D\ell)^{\perp_{\Omega_{t}}}$ denotes the $\Omega_{t}$-orthogonal of $D\ell$
inside $DL$. Along with (38), this implies that

$\square_{t}|_{C}=\big{.}\Omega_{t}^{\sharp}(-\iota_{\mathbbm{1}}\accentset{\mbox{\large.}}{\Omega}_{t})\big{|}_{C}\in\Gamma\big{(}\Omega_{t}^{\sharp}((D\ell)^{\circ})\big{)}=\Gamma\big{(}(D\ell)^{\perp_{\Omega_{t}}}\big{)}\subset\Gamma(D\ell).$

This confirms that $\sigma(\square_{t})$ is tangent to $C$. Since $C$ is
compact, there exists a neighborhood $\mathcal{U}\subset M$ of $C$ on which
the flow $\psi_{t}$ of $\sigma(\square_{t})$ is defined for all times $0\leq
t\leq 1$. Hence, the isotopy of line bundle automorphisms $\Psi_{t}\colon L\to
L$ integrating $\square_{t}\in\Gamma(DL)$ is defined on $L|_{\mathcal{U}}$.

The fact that $\Psi_{t}$ satisfies the equation (37) follows from the
computation

$\frac{d}{dt}\Psi_{t}^{\ast}\Omega_{t}=\Psi_{t}^{\ast}\left(\mathcal{L}_{\square_{t}}\Omega_{t}+\accentset{\mbox{\large.}}{\Omega}_{t}\right)=\Psi_{t}^{\ast}d_{D}\left(\iota_{\square_{t}}\Omega_{t}+\iota_{\mathbbm{1}}\accentset{\mbox{\large.}}{\Omega}_{t}\right)=0.$

Here we used that $\iota_{\mathbbm{1}}$ is a contracting homotopy for the der-
complex, see Rem. A.3. At last, since $\sigma(\square_{t})$ is tangent to $C$,
it is clear that $\Psi_{t}$ induces an isotopy of LB automorphisms
$\Psi_{t}|_{C}\colon\ell\to\ell$ covering $\psi_{t}|_{C}\colon C\to C$. This
isotopy automatically satisfies
$\Psi_{t}|_{C}^{\ast}(I^{\ast}\Omega_{0})=I^{\ast}\Omega_{0}$, since
$\Psi_{t}^{\ast}\Omega_{t}=\Omega_{0}|_{\mathcal{U}}$ and
$I^{\ast}\Omega_{t}=I^{\ast}\Omega_{0}$. This finishes the proof. ∎

We can now prove Thm. 4.1. In the following, $C\subset(M,\xi)$ is a compact
regular coisotropic submanifold with pre-contact form
$\theta_{C}\in\Omega^{1}(C,\ell)$, pre-symplectic Atiyah form
$\varpi\in\Omega^{2}_{D}(\ell)$ and characteristic foliation $\mathcal{F}$. To
prove Thm. 4.1, we will work in the local model $(U,\ker\gamma_{G})$ for
$(M,\xi)$ around $C$. Recall from §1.1 that the construction of this model
involves a choice of complement $G$ to $T\mathcal{F}$. The contact structure
$\ker\gamma_{G}$ lives on a neighborhood $U$ of the zero section in the bundle
$p:T^{*}\mathcal{F}\otimes\ell\rightarrow C$, and $\gamma_{G}$ is a
$(p^{*}\ell)$-valued contact form. Working in $(U,\ker\gamma_{G})$ instead of
$(M,\xi)$, the setup is as follows.

  * •

The smooth path $C_{t}$ in $\text{Def}_{\mathcal{F}}(C)$ is identified with a
smooth path of coisotropic sections
$\eta_{t}\in\text{Def}^{U}_{\mathcal{F}}(C)$ with $\eta_{0}=0$.

  * •

By Lemma 1.7, the path $\eta_{t}$ gives rise to a path of pre-contact forms

$\theta_{t}:=\theta_{C}+\text{pr}_{T\mathcal{F}}^{*}\eta_{t}\in\Omega^{1}(C,\ell).$

Denote by $\mathcal{F}_{t}$ the characteristic foliation of the pre-contact
structure $\ker\theta_{t}$.

  * •

By Def. 2.2, we have an isotopy $\underline{\varphi}_{t}$ of foliated
diffeomorphisms

$\underline{\varphi}_{t}:(C,\mathcal{F})\rightarrow(C,\mathcal{F}_{t}).$

  * •

Denote by $\varpi_{t}\in\Omega^{2}_{D}(\ell)$ the pre-symplectic Atiyah form
corresponding to $\theta_{t}$.

Every $\eta\in\Gamma(T^{\ast}\mathcal{F}\otimes\ell)$ determines a fiber-
preserving diffeomorphism

$\tau_{\eta}\colon
T^{\ast}\mathcal{F}\otimes\ell\overset{\sim}{\longrightarrow}T^{\ast}\mathcal{F}\otimes\ell,\
\alpha_{x}\longmapsto\alpha_{x}+\eta_{x},$