$(A_{S},\mu_{S})$ where the relevant bundle of ideals is
$\mathfrak{k}=\ker\mu_{S}$. In the sequel, we will use the equivalence between
(i)-(iv) without further notice. So, for example, $(A_{S},\mu_{S})$ is
partially split if it admits a partial splitting
$\theta:A_{S}\ltimes\mathfrak{k}^{*}\to T^{*}A_{S}$ as in (ii). This is
equivalent to our original Definition 7.1, which was stated in terms of the
existence of an IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$, as in (iii). The
closed IM 2-form in the local model (7.5) can be written using the partial
splitting $\theta$ from (ii) as

$\mu_{0}=\operatorname{pr}^{*}_{A_{S}}\mu_{S}+\theta^{*}\mu_{\mathrm{can}},$

where $\mu_{\mathrm{can}}\in\Omega^{2}_{\mathrm{IM}}(T^{*}A_{S})$ is the
canonical IM 2-form (see Example A.4), or using the linear IM form
$\mu^{\mathrm{lin}}$ from (iv) as

$\mu_{0}=\operatorname{pr}^{*}_{A_{S}}\mu_{S}+\mu^{\mathrm{lin}}.$

In this expression, the first term is a constant IM 2-form and the second term
is a linear IM 2-form.

######  Remark 7.4.

Let us stress that the partial split condition is not implied by the existence
of a Lie algebroid splitting $r:T^{*}S\to A_{S}$ of $\mu_{S}$ – see Example
9.14.

As in the global case – cf. Proposition 5.9 – the partially split condition
implies certain properties of the isotropy Lie algebras of a first order jet

######  Proposition 7.5 ([27]).

If $(A_{S},\mu_{S})$ is a partially split first order jet of Poisson
structure, a choice of IM connection 1-form $(L,l)$, gives

  1. (i)

a linear connection on $\mathfrak{k}=\ker\mu_{S}$

$\nabla^{L}_{X}\xi:=i_{X}L(\xi),$ (7.6)

which preserves the Lie bracket

$\nabla^{L}_{X}[\xi,\eta]_{\mathfrak{k}}=[\nabla^{L}_{X}\xi,\eta]_{\mathfrak{k}}+[\xi,\nabla^{L}_{X}\eta]_{\mathfrak{k}},\quad
X\in\mathfrak{X}(S),\xi,\eta\in\Gamma(\mathfrak{k});$

  2. (ii)

for each $x\in S$, a decomposition of the isotropy Lie algebra
$\mathfrak{g}_{x}:=\ker\rho_{A_{S}}|_{x}$ into a direct sum of ideals

$\mathfrak{g}_{x}\simeq\mathfrak{k}_{x}\oplus(\mathfrak{g}_{x}\cap\ker l).$

We quote from [27] another interesting characterization of partially split
jets which will be useful later. Given a (usual) connection $\nabla$ on a Lie
algebroid $A_{S}\Rightarrow S$ one has the following associated
$A_{S}$-connections on $A_{S}$ and $TS$

$\displaystyle\overline{\nabla}_{\alpha}\beta$
$\displaystyle:=\nabla_{\rho(\beta)}\alpha+[\alpha,\beta],$
$\displaystyle\overline{\nabla}_{\alpha}X$
$\displaystyle:=\rho(\nabla_{X}\alpha)+[\rho(\alpha),X]$

They satisfy

$\rho(\overline{\nabla}_{\alpha}\beta)=\overline{\nabla}_{\alpha}\rho(\beta).$

The basic curvature of $\nabla$ is defined as the tensor

$R^{\mathrm{bas}}_{\nabla}(\alpha,\beta)(X):=\nabla_{X}([\alpha,\beta])-[\nabla_{X}\alpha,\beta]-[\alpha,\nabla_{X}\beta]-\nabla_{\overline{\nabla}_{\beta}X}\alpha+\nabla_{\overline{\nabla}_{\alpha}X}\beta,$
(7.7)

where $X\in\mathfrak{X}(S)$ and $\alpha,\beta\in\Gamma(A_{S})$. This curvature
arises naturally in the study of _Cartan connections_ (see [1, 2, 3]).
Proposition 5.9 from [27] implies the following.

######  Proposition 7.6.

A first order jet $(A_{S},\mu_{S})$ with $\mathfrak{k}=\ker\mu_{S}$ is
partially split if and only if there is a vector bundle splitting
$l:A_{S}\to\mathfrak{k}$ of the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{S}}$$\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and a (usual) connection $\nabla$ on $A_{S}$ such that

$\overline{\nabla}l=0,\qquad l(R^{\mathrm{bas}}_{\nabla})=0.$ (7.8)

In this case, the pair $(l,\nabla)$ determines a IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ by setting

$L:\Gamma(A_{S})\to\Omega^{1}(S;\mathfrak{k}),\quad
i_{X}L(\alpha):=l(\nabla_{X}\alpha).$

Moreover, every IM connection 1-form $(L,l)$ takes this form for some pair
$(l,\nabla)$ satisfying (7.8).

######  Remark 7.7.

The connection $\nabla$ on $A_{S}$ in the previous proposition is not uniquely
determined by the IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$. In particular, this
connection should not be confused with the connection $\nabla^{L}$ on
$\mathfrak{k}$ from Proposition 7.6. They are related by

$\nabla_{X}^{L}\xi=l(\nabla_{X}\xi),$

for $X\in\mathfrak{X}(S)$ and $\xi\in\Gamma(\mathfrak{k})$.

###  7.3. Existence and uniqueness of the local model

The following proposition gives some geometric meaning to the partially split
condition: it shows that the partially split condition is equivalent to the
algebroid local model $A_{S}\ltimes\mathfrak{k}^{*}$ being the cotangent
algebroid of a Poisson structure extending the given first jet. This is the
infinitesimal version of Proposition 5.7.

######  Proposition 7.8.

A first order jet $(A_{S},\mu_{S})$ with kernel $\mathfrak{k}:=\ker\mu_{S}$ is
partially split if and only if there exists an open neighborhood $S\subset
M_{0}\subset\mathfrak{k}^{*}$ and a closed, non-degenerate IM 2-form
$\mu\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes M_{0})$ whose pullback along the
Lie algebroid map $i:A_{S}\hookrightarrow A_{S}\ltimes\mathfrak{k}^{*}$ is
$\mu_{S}$

$i^{*}\mu=\mu_{S}.$

###### Proof.

If $(A_{S},\mu_{S})$ is partially split, then the local model $\mu_{0}$ from
Proposition 7.2 is an IM 2-form as in the statement. Conversely, let
$\mu\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes M_{0})$ be as in the statement.
We use that canonical decomposition given by the vector bundle structure

$T^{*}_{S}\mathfrak{k}^{*}\simeq\mathfrak{k}\oplus T^{*}S.$

By assumption, $\mu$ takes the following form along $S$

$\mu|_{S}:A_{S}\to\mathfrak{k}\oplus T^{*}S,\quad\mu|_{S}=(l,\mu_{S}).$

Let $m_{t}:A_{S}\ltimes\mathfrak{k}^{*}\to A_{S}\ltimes\mathfrak{k}^{*}$
denote the fiberwise multiplication by $t>0$, which is a Lie algebroid
morphism. We have that

$\lim_{t\to 0}m^{*}_{t}\mu=\operatorname{pr}^{*}\mu_{S}.$

Moreover, $\mu_{t}:=\frac{1}{t}m_{t}^{*}(\mu-\operatorname{pr}^{*}\mu_{S})$ is
a path of closed, IM 2-forms, defined on $m_{t^{-1}}(M_{0})$. Then one can
easily see that the following limit exists

$\mu^{\mathrm{lin}}:=\lim_{t\to 0}\mu_{t},$

(e.g., in coordinates) and it is a linear, closed, IM 2-form
$\mu^{\mathrm{lin}}\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes\mathfrak{k}^{*})$.
If we assume that $l|_{\mathfrak{k}}=\mathrm{Id}$, then, for each $t>0$,
$(\mu_{t}|_{S})|_{\mathfrak{k}}=(\mathrm{Id},0)$, and so this holds also for
the limit $\mu^{\mathrm{lin}}$. Thus the characterization from item (iv) of
the previous subsection of the partially split condition holds. The general
case can be reduced to this using Lemma A.13, which shows that the map
$l|_{\mathfrak{k}}:\mathfrak{k}\to\mathfrak{k}$ is an $A_{S}$-equivariant
linear isomorphism. Let
$\varphi:=(l|_{\mathfrak{k}})^{*}:\mathfrak{k}^{*}\to\mathfrak{k}^{*}$ denote
the transpose map. Then
$(\mathrm{Id},\varphi):A_{S}\ltimes\mathfrak{k}^{*}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{S}\ltimes\mathfrak{k}^{*}$
is a Lie algebroid isomorphism covering $\varphi$. So
$\tilde{M}_{0}:=\varphi(M_{0})$ is an open neighborhood of $S$ over which
$A_{S}\ltimes\tilde{M}_{0}$ carries the non-degenerate IM 2-form
$\tilde{\mu}:=(\mathrm{Id},\varphi^{-1})^{*}(\mu)\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes\tilde{M}_{0})$.
Note that $\tilde{\mu}$ also extends $\mu_{S}$ and by construction, for every
$\beta\in\Gamma(\mathfrak{k})$, we have

$l(\beta)=\mu|_{S}(\beta)=\varphi^{*}\circ\tilde{\mu}|_{S}(\beta)=l(\tilde{\mu}(\beta)),$

which shows that $\tilde{\mu}$ restricted to
$\mathfrak{k}\subset(A_{S}\ltimes\mathfrak{k}^{*})|_{S}$ is the identity map.
Therefore, the above argument applied to $\tilde{\mu}$ shows that
$(A_{S},\mu_{S})$ is partially split. ∎

The infinitesimal multiplicative version of Moser’s argument, proved in
Proposition A.11 (i)), gives the following result.

######  Proposition 7.9.

Let $(A_{S},\mu_{S})$ be a first order jet of a Poisson structure with kernel
$\mathfrak{k}:=\ker\mu_{S}$. Consider two closed, non-degenerate IM 2-forms
$\mu_{0},\mu_{1}\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes M)$, defined on some
neighborhood $S\subset M\subset\mathfrak{k}^{*}$, which extend $\mu_{S}$

$i^{*}\mu_{k}=\mu_{S},\quad k=0,1.$

Then there exists a isomorphism of Lie algebroids

$\Phi:A_{S}\ltimes
M_{0}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{S}\ltimes
M_{1},\quad\Phi^{*}(\mu_{1})=\mu_{0},$

defined between open neighborhoods $S\subset M_{0},M_{1}\subset M$ and whose
base map fixes $S$. In particular, this base map gives a Poisson isomorphism
between the induced Poisson structure

$\phi:(M_{0},\pi_{0})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(M_{1},\pi_{1}).$

We conclude that the local model is well-defined up to isomorphism.

######  Corollary 7.10 (Uniqueness of local models).

Given a first order jet of a Poisson structure $(A_{S},\mu_{S})$, any two
local models associated with distinct IM connection 1-forms are isomorphic
around $S$.

###  7.4. The local model as a Lie-Dirac submanifold

We discuss now the Lie algebroid version of the symplectic groupoid

$(T^{*}\mathcal{G}_{S},\omega_{\mathrm{can}}+\operatorname{pr}^{*}_{\mathcal{G}_{S}}\omega_{S})\rightrightarrows
A_{S}^{*},$ (7.9)

and the infinitesimal version of Corollary 5.6. This gives another perspective
on the partially split condition and the geometry of the local model. The main
result in this section can be stated as follows.

######  Theorem 7.11.

Let $(A_{S},\mu_{S})$ be a first order jet of a Poisson structure. Then

  1. (i)

The Lie algebroid $T^{*}A_{S}\Rightarrow A_{S}^{*}$ carries the closed non-
degenerate IM 2-form:

$\mu:=\mu_{\mathrm{can}}+\operatorname{pr}_{A_{S}}^{*}\mu_{S}:T^{*}A_{S}\to
T^{*}A_{S}^{*},$

where $\mu_{\mathrm{can}}$ is the canonical IM 2-form on $T^{*}A_{S}$ (see
Example A.4).

  2. (ii)

The Poisson structure $\pi_{A_{S},\mu_{S}}$ on $A_{S}^{*}$ corresponding to
$\mu$ is given by

$\pi_{A_{S},\mu_{S}}^{\sharp}=\pi_{A_{S}}^{\sharp}\circ(\mathrm{Id}-\operatorname{pr}_{A_{S}}^{*}\mu_{S}\circ\mu_{\mathrm{can}}^{-1}),$

where $\pi_{A_{S}}$ is the fiberwise linear Poisson structure on $A_{S}^{*}$.

  3. (iii)

If $(A_{S},\mu_{S})$ is partially split, then for any IM connection 1-form
$(L,l)$ the dual $l^{*}:\mathfrak{k}^{*}\hookrightarrow A^{*}_{S}$ realizes
the local model $(M_{0},\pi_{0})$ as a Lie-Dirac submanifold of
$(A_{S}^{*},\pi_{A_{S},\mu_{S}})$.

  4. (iv)

Conversely, let $(N,\pi_{N})\subset(A_{S}^{*},\pi_{A_{S},\mu_{S}})$ be a Lie-
Dirac submanifold containing $S$ such that the corresponding Lie subalgebroid
$A_{N}\subset T^{*}A_{S}^{*}$ satisfies $\mu^{-1}(A_{N}|_{S})=A_{S}$. Then
$(A_{S},\mu_{S})$ is partially split, and $(N,\pi_{N})$ is isomorphic around
$S$ to the local model $(M_{0},\pi_{0})$.

Before we prove this result, let us clarify the terminology used in the
statement. First, a submanifold $i:N\hookrightarrow M$ of a Poisson manifold
$(M,\pi)$ is called a _Poisson-Dirac submanifold_ if the Dirac structure
$L_{\pi}=\mathrm{Graph}(\pi^{\sharp})$ pulls back to a Dirac structure on $N$
which is given by a bivector field: $i^{*}L_{\pi}=L_{\pi_{N}}$. Then $\pi_{N}$
is a Poisson structure. We are interested in the following special type of
Poisson-Dirac submanifolds.

######  Definition 7.12.

A Lie-Dirac submanifold of a Poisson manifold $(M,\pi)$ is an immersed
submanifold, $N\hookrightarrow M$, together with a vector bundle decomposition
$T^{*}_{N}M=A_{N}\oplus(TN)^{\circ}$, for which $A_{N}$ is a Lie subalgebroid
of $T^{*}M$.

These submanifolds were introduced and studied by Xu in [61], who called them
“Dirac submanifolds”, and further investigated in [13] where the term “Lie-
Dirac submanifold” was coined. In these references it is proved that if
$N\subset(M,\pi)$ is a Lie-Dirac submanifold as in the definition then the
following hold (see [61, Theorem 2.3 and Remark 2.7 (iii)], and also [13,
Section 8.3]).

  1. (a)

$N$ is a Poisson-Dirac submanifold, hence has a Poisson structure $\pi_{N}$
which does not depend on the splitting of $T^{*}_{N}M$.

  2. (b)

The restriction map
$r:A_{N}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}N$
is an isomorphism of Lie algebroids, where $T^{*}N$ has the algebroid
structure induced by $\pi_{N}$.

At the groupoid level, Lie-Dirac submanifolds correspond to symplectic
subgroupoids. We have seen in Corollary (5.6) that the groupoid local models
arise as symplectic subgroupoids of the symplectic groupoid (7.9), so Theorem
7.11 gives the infinitesimal analogue of this statement. The theorem also has
the following surprising consequence.

######  Corollary 7.13.

The inclusion of the local model:

$i:=l^{*}:(M_{0},\pi_{0})\hookrightarrow(A_{S}^{*},\pi_{A_{S},\mu_{S}})$

is a backward Dirac map. Therefore, the local model depends only on the base
map of the IM connection 1-form.

We now turn to the proof of the theorem.

###### Proof of (i).

The Lie algebroid $T^{*}A_{S}\Rightarrow A_{S}^{*}$ with the IM 2-form
$\mu=\mu_{\mathrm{can}}+\operatorname{pr}_{A_{S}}^{*}\mu_{S}$ represent the
infinitesimal counterpart of the symplectic groupoid (7.9). Clearly, $\mu$ is
a closed IM 2-form, being the sum of the canonical closed IM 2-from
$\mu_{\mathrm{can}}$ (see Example A.4), and the pullback of the closed IM
2-form $\mu_{S}$ via the Lie algebroid map
$\operatorname{pr}_{A_{S}}:T^{*}A_{S}\to A_{S}$. To show that $\mu$ is
invertible, we use a connection on $A_{S}$ and the dual connection on
$A_{S}^{*}$, to identify

$T^{*}A_{S}\simeq T^{*}S\times_{S}A_{S}^{*}\times_{S}A_{S},\quad
T^{*}A_{S}^{*}\simeq T^{*}S\times_{S}A_{S}\times_{S}A_{S}^{*}.$

Under these isomorphisms, $\mu$ becomes

$\mu(\xi,\alpha,a)=(\xi+\mu_{S}(a),a,\alpha),$

which is clearly invertible. ∎

###### Proof of (ii).

The Poisson structure corresponding to the closed, non-degenerate IM 2-form
$\mu_{\mathrm{can}}$ is the linear Poisson structure $\pi_{A_{S}}$ on
$A_{S}^{*}$, with
$\pi_{A_{S}}^{\sharp}=\rho_{T^{*}A_{S}}\circ\mu_{\mathrm{can}}^{-1}$. Using
the explicit description from the proof of (i), we see that the Poisson
structure $\pi_{A_{S},\mu_{S}}$ corresponding to $\mu$ is given by the formula
from item (ii)

$\pi_{A_{S},\mu_{S}}^{\sharp}=\rho_{T^{*}A_{S}}\circ\mu^{-1}=\pi_{A_{S}}^{\sharp}\circ(\mathrm{Id}-\operatorname{pr}_{A_{S}}^{*}\mu_{S}\circ\mu_{\mathrm{can}}^{-1}).\qed$

###### Proof of (iii).

This follows from the infinitesimal analog of Corollary 5.6.

######  Lemma 7.14.

Let $\theta:A_{S}\ltimes\mathfrak{k}^{*}\to T^{*}A_{S}$ be a partial splitting
with base map $i:=l^{*}:\mathfrak{k}^{*}\to A_{S}^{*}$, and let
$(M_{0},\pi_{0})$ be the corresponding local model. Then
$i:M_{0}\hookrightarrow(A_{S}^{*},\pi_{A_{S},\mu_{S}})$ is a Lie-Dirac
submanifold with Lie algebroid

$A_{M_{0}}=\mathrm{Im}(\mu\circ\theta)\subset T^{*}A_{S}^{*}|_{i(M_{0})},$

and the induced Poisson structure on $M_{0}$ is $\pi_{0}$.

To prove this lemma, as seen in Proposition 7.2, we have the closed IM 2-form

$\mu_{0}\in\Omega^{2}_{\mathrm{IM}}(A_{S}\ltimes\mathfrak{k}^{*}),\quad\mu_{0}=\theta^{*}(\mu)$
(7.10)

which is non-degenerate on $M_{0}\subset\mathfrak{k}^{*}$, and
$\pi_{0}^{\sharp}=\rho_{\ltimes}\circ\mu_{0}^{-1}$. Over $M_{0}$, we have the
following diagram of Lie algebroid maps

${\ignorespaces A_{S}\ltimes M_{0}}$ |  |  | ${\ignorespaces T^{*}M_{0}}$ |
---|---|---|---|---
${\ignorespaces T^{*}A_{S}}$ |  |  | ${\ignorespaces T^{*}A_{S}^{*}}$ |
$\scriptstyle{\ignorespaces\mu_{0}}$$\scriptstyle{\ignorespaces\theta}$$\scriptstyle{\ignorespaces\phi}$$\scriptstyle{\ignorespaces\mu}$

where the horizontal arrows are isomorphisms, the algebroids on the right
correspond to the Poisson structures $\pi_{0}$ and $\pi_{A_{S},\mu_{S}}$,
respectively, and the map $\phi$ is defined such that the diagram commutes.
Clearly, $A_{M_{0}}=\mathrm{Im}(\phi)$. So $A_{M_{0}}$ is a Lie subalgebroid
and $\phi:T^{*}M_{0}\to A_{M_{0}}$ is a Lie algebroid isomorphism. Next, note
that the explicit form of (7.10) is

$\mu_{0}(v)=i^{*}\big{(}\mu\circ\theta(v)\big{)},\quad\forall\,v\in
A_{S}\ltimes M_{0},$

or equivalently, using that $\mu_{0}$ is invertible and the definition of
$\phi$

$\alpha=i^{*}(\phi(\alpha)),\quad\forall\,\alpha\in T^{*}M_{0}.$ (7.11)

This shows that $A_{M_{0}}\cap(Ti(M_{0}))^{\circ}=0$, and so $A_{M_{0}}$ makes
$i:M_{0}\hookrightarrow(A_{S}^{*},\pi_{A_{S},\mu_{S}})$ into a Lie-Dirac
submanifold. We claim that the induced Poisson structure $\widetilde{\pi}_{0}$
on $M_{0}$ coincides with $\pi_{0}$. For this, it suffices that they induce
the same Lie algebroid structure on $T^{*}M_{0}$. We know that both are
isomorphic to $A_{M_{0}}$: the one corresponding to $\pi_{0}$ via the map
$\phi:T^{*}M_{0}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{M_{0}}$,
and by item (b) the one corresponding to it $\widetilde{\pi}_{0}$ via the
pullback map
$i^{*}:A_{M_{0}}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}M_{0}$.
However, by (7.11), these maps are inverse to each other. So, indeed
$\pi_{0}=\widetilde{\pi}_{0}$. ∎

###### Proof of (iv).

Let $S\subset N\subset A_{S}^{*}$ be a Lie-Dirac submanifold such that the
corresponding Lie algebroid $A_{N}\subset T^{*}A_{S}^{*}$ is given above $S$
by

$A_{N}|_{S}=\mu(A_{S})\subset T^{*}A_{S}^{*}.$

By composing the Lie algebroid map $\mu^{-1}|_{A_{N}}:A_{N}\hookrightarrow
T^{*}A_{S}$ with the restriction $T^{*}A_{S}\to A_{S}\ltimes\mathfrak{k}^{*}$,
we obtain a Lie algebroid map $\Phi:A_{N}\to A_{S}\ltimes\mathfrak{k}^{*}$
which covers $\phi=p|_{N}$, where $p:A^{*}_{S}\to\mathfrak{k}^{*}$ is the
projection. We claim that by shrinking $N$ we can ensure that $\Phi$ is a Lie
algebroid isomorphism onto $A_{S}\ltimes M_{0}$, where $M_{0}=\phi(N)$ is an
open neighborhood of $S$. The condition that $N$ is a Lie-Dirac submanifold
with Lie algebroid $A_{N}$ gives

$A_{N}\oplus(TN)^{\circ}=T^{*}_{N}A_{S}^{*}.$

Under the decompositions for $T^{*}A_{S}$ and $T^{*}A_{S}^{*}$ from the proof
of item (i), this condition along $S$ becomes

$\\{(\mu_{S}(a),a)\,:\,a\in A_{S}\\}\oplus(T_{S}N)^{\circ}=T^{*}S\oplus
A_{S},$

where we identify $T^{*}S\oplus A_{S}\simeq T^{*}_{S}A_{S}^{*}$. Since
$\mathfrak{k}=\ker\mu_{S}$ and $TS\subset T_{S}N$, this condition is
equivalent to

$(T_{S}N)^{\circ}\oplus\mathfrak{k}=A_{S}.$

Equivalently, the differential of $\phi$ along $S$ gives a linear isomorphism

$\mathrm{d}\phi:T_{S}N\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T_{S}\mathfrak{k}^{*}.$

Therefore, after shrinking $N$, we may assume that $\phi$ is an open
embedding. By assumption, above $S$ we have a Lie algebroid isomorphism
$\Phi|_{S}:A_{N}|_{S}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{S}$.
Therefore, by shrinking $N$ further, we may assume that $\Phi$ is a Lie
algebroid isomorphism
$\Phi:A_{N}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{S}\ltimes
M_{0}$, where $M_{0}=\phi(N)$ is an open neighborhood of $S$ in
$\mathfrak{k}^{*}$, proving our claim. Now observe that the restriction map
$\mu_{N}:A_{N}\to T^{*}N$ is a closed non-degenerate IM 2-form on $A_{N}$,
therefore defining a Poisson structure $\pi_{N}$ on $N$. This follows since
$N$ is Lie-Dirac with Lie algebroid $A_{N}\subset T^{*}A_{S}^{*}$. Using the
isomorphism $\Phi$, we obtain the closed non-degenerate IM 2-form
$\mu_{0}:=(\Phi^{-1})^{*}(\mu_{N})$ on $A_{S}\ltimes M_{0}$. By the proof of
(i), the pullback of $\mu$ along the inclusion $A_{S}\hookrightarrow
T^{*}A_{S}$ is $\mu_{S}$. This implies that the pullback of $\mu_{0}$ along
$A_{S}\hookrightarrow A_{S}\ltimes M_{0}$ is $\mu_{S}$. Proposition 7.8
implies that $(A_{S},\mu_{S})$ is partially split, and Proposition 7.9 implies
that the Poisson structure $\pi_{0}$ on $M_{0}$ corresponding to $\mu_{0}$ is
isomorphic to the local model around $S$. Since
$\phi:(N,\pi_{N})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(M_{0},\pi_{0})$
is a Poisson diffeomorphism the conclusion follows. ∎

###  7.5. IM connection 1-forms as couplings

In the case of symplectic leaves of Poisson structures the classical approach
to the local model, due to Vorobjev, is via so-called couplings, which
originated in the theory of symplectic fibrations. There exists a similar
approach to our local model, as we now discuss. Consider a partially split
first jet $(A_{S},\mu_{S})$ with a fixed IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$. We use the base map to
split the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\scriptstyle{\mu_{S}}$$\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and to identify $A_{S}\simeq T^{*}S\oplus\mathfrak{k}$. Then we obtain the
following data:

  1. (i)

A linear connection $\nabla^{L}$ on the vector bundle $\mathfrak{k}\to S$,
given by

$\nabla^{L}_{X}\xi:=i_{X}L(\xi).$

  2. (ii)

A tensor $U\in\Gamma(TS\otimes T^{*}S\otimes\mathfrak{k})$, given by

$U(\alpha,X):=-i_{X}L(\alpha).$

That this is a connection and a tensor follows from the symbol equation (A.7)
and the fact that $l|_{\mathfrak{k}}=\mathrm{Id}$. Notice that the connection
$\nabla^{L}$ has already appeared in Proposition 7.5. The following result is
proved in [27, Section 5.4]:

######  Proposition 7.15 ([27]).

The data $(\nabla^{L},U)$ associated with a IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ satisfies the structure
equations

  1. (S1)

the connection $\nabla^{L}$ preserves the Lie bracket
$[\cdot,\cdot]_{\mathfrak{k}}$, i.e.,

$\nabla^{L}_{X}[\xi,\eta]_{\mathfrak{k}}=[\nabla^{L}_{X}\xi,\eta]_{\mathfrak{k}}+[\xi,\nabla^{L}_{X}\eta]_{\mathfrak{k}};$

  2. (S2)

the curvature of $\nabla^{L}$ is related to $[U,\cdot]$ as follows

$\nabla^{L}_{\pi^{\sharp}_{S}(\alpha)}\nabla^{L}_{X}-\nabla^{L}_{X}\nabla^{L}_{\pi^{\sharp}_{S}(\alpha)}-\nabla^{L}_{[\pi^{\sharp}_{S}(\alpha),X]}=[U(\alpha,X),\cdot]_{\mathfrak{k}};$

  3. (S3)

$U$ satisfies the skew-symmetry condition

$U(\alpha,\pi^{\sharp}_{S}(\beta))=-U(\beta,\pi^{\sharp}_{S}(\alpha)).$ (7.12)

and the “mixed” cocyle-type equation

$\displaystyle\nabla^{L}_{\pi^{\sharp}_{S}(\alpha)}U(\beta,X)$
$\displaystyle-\nabla^{L}_{\pi^{\sharp}_{S}(\beta)}U(\alpha,X)+\nabla^{L}_{X}U(\alpha,\pi_{S}^{\sharp}(\beta))$
$\displaystyle+U(\alpha,[\pi^{\sharp}_{S}(\beta),X])-U(\beta,[\pi_{S}^{\sharp}(\alpha),X])=U([\alpha,\beta]_{\pi_{S}},X),$

for all $X\in\mathfrak{X}^{1}(S)$, $\alpha,\beta\in\Omega^{1}(S)$,
$\xi,\eta\in\Gamma(\mathfrak{k})$.

We introduce the following notation.

######  Definition 7.16.

Let $(S,\pi_{S})$ be a Poisson manifold and
$(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})$ a Lie algebra bundle over $S$. A
coupling data is a pair $(\nabla^{L},U)$, where $\nabla^{L}$ is a connection
on $\mathfrak{k}$ and $U\in\Gamma(TS\otimes T^{*}S\otimes\mathfrak{k})$ is a
tensor field satisfying the structure equations (S1), (S2) and (S3).

In the rest of this section we show that the coupling data determines
completely the partially split jet, and hence the local form. We start by
stating the following special case of a result proved in [27, Section 5.4].

######  Proposition 7.17.

Let $(S,\pi_{S})$ be a Poisson manifold,
$(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})$ a Lie algebra bundle over $S$
and $(\nabla^{L},U)$ coupling data. The pair
$(A_{S},\mu_{S}=\mathrm{pr}_{T^{*}S})$ is a partially split first order jet of
a Poisson structure, with IM connection 1-form $(L,l)$ given by

$l=\mathrm{pr}_{\mathfrak{k}},\quad
i_{X}L(\alpha,\xi)=\nabla^{L}_{X}\xi-U(\alpha,X).$ (7.13)

Moreover, any first order jet of a Poisson structure which is partially split
is isomorphic to one of this type.

With the description of a partially split first jet in terms of coupling data
given by the previous propositions, we can now turn to the corresponding
description of the local model. This local model is defined in the open set
$M_{0}\subset\mathfrak{k}^{*}$ over which the IM form $\mu_{0}$ is invertible.
This is clarified in the following result where we continue to use the
notation from Proposition 7.17.

######  Proposition 7.18.

Let $(A_{S},\mu_{S})$ be partially split with a fixed IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ and associated coupling
data $(\nabla^{L},U)$. The local model is defined on the open set
$M_{0}\subset\mathfrak{k}^{*}$ consisting of points $z\in\mathfrak{k}^{*}$
where the linear map

$\mathrm{Id}+\langle z,U\rangle:T^{*}_{x}S\to T^{*}_{x}S,\quad
x=\operatorname{pr}_{S}(z),$

is invertible. At $z\in M_{0}$ the Poisson structure $\pi_{0}$ of the local
model decomposes as

$\pi_{0}|_{z}=\pi_{\mathfrak{k}_{x}}+\mathrm{hor}_{z}\gamma_{z}\in\wedge^{2}\mathrm{Vert}\oplus\wedge^{2}\mathrm{Hor}_{\nabla^{L}},$
(7.14)

where the vertical component $\pi_{\mathfrak{k}_{x}}|_{z}$ is the linear
Poisson structure on the dual of the Lie algebra $\mathfrak{k}_{x}$ and the
horizontal component $\mathrm{hor}_{z}\gamma_{z}$ is the
$\nabla^{L}$-horizontal lift of the bivector $\gamma_{z}\in\wedge^{2}T_{x}S$
given by

$\gamma_{z}^{\sharp}=\pi_{S}^{\sharp}\circ\big{(}\mathrm{Id}+\langle
z,U\rangle\big{)}^{-1}.$

###### Proof.

Proposition 7.17 implies that the anchor of the Lie algebroid
$A_{S}\ltimes\mathfrak{k}^{*}$ is

$\rho_{\ltimes}|_{z}:T^{*}_{x}S\oplus\mathfrak{k}_{x}\to
T_{z}\mathfrak{k}^{*},\quad(\alpha,\xi)\mapsto\pi_{\mathfrak{k}_{x}}^{\sharp}\xi|_{z}+\mathrm{hor}_{z}(\pi_{S}^{\sharp}\alpha),$

where $z\in\mathfrak{k}^{*}_{x}$, $\mathrm{hor}_{z}:T_{x}S\to
T_{z}\mathfrak{k}^{*}$ is the $\nabla^{L}$-horizontal lift and
$\xi\in\mathfrak{k}_{x}$ is thought of as an element of
$T^{*}_{z}\mathfrak{k}_{x}^{*}$. Note that this decomposes the anchor into a
vertical and a $\nabla^{L}$-horizontal component. Now using (7.5) and (7.13),
we find that the IM 2-form $\mu_{0}$ can also be decomposed into a vertical
and a $\nabla^{L}$-horizontal component

$\mu_{0}|_{z}:T^{*}_{x}S\oplus\mathfrak{k}_{x}\to
T^{*}_{z}\mathfrak{k}^{*},\quad(\alpha,\xi)\mapsto\big{(}\alpha+\langle
z,U(\alpha)\rangle\big{)}\circ\mathrm{d}_{z}\operatorname{pr}_{S}+\mathrm{vert}^{*}_{z}(\xi),$

where $\operatorname{pr}_{S}:\mathfrak{k}^{*}\to S$ is the bundle projection,
$\langle z,U(\alpha)\rangle\in T^{*}_{x}S$ and
$\mathrm{vert}_{z}:T_{z}\mathfrak{k}^{*}\to\ker(\mathrm{d}_{z}\operatorname{pr}_{S})\simeq\mathfrak{k}^{*}_{x}$
is the vertical projection corresponding to the connection $\nabla^{L}$. This
shows that $\mu_{0}$ is invertible on the open $M_{0}$ consisting of elements
$z\in\mathfrak{k}^{*}$ where the map $\mathrm{Id}+\langle
z,U\rangle:T^{*}_{x}S\to T^{*}_{x}S$ is invertible. On this open set, the
composition $\rho_{\ltimes}\circ\mu_{0}^{-1}$ yields formula (7.14) in the
statement. Note that skew-symmetry of $\gamma_{z}$ follows from (7.12). ∎

As a consequence of the proposition we see that the push-forward of $\pi_{0}$
along the projection $\operatorname{pr}_{S}:\mathfrak{k}^{*}\to S$ is
$\gamma$, and hence it is tangent to the leaves of $\pi_{S}$.

######  Corollary 7.19.

For any Poisson submanifold $P\subset(S,\pi_{S})$, the pre-image
$\operatorname{pr}_{S}^{-1}(P)\cap M_{0}$ is a Poisson submanifold of the
local model $(M_{0},\pi_{0})$.

When $P$ is a symplectic leaf of $(S,\pi_{S})$, we recover a Vorobjev-type
local model.

######  Proposition 7.20.

Let $(M_{0},\pi_{0})$ be the local model of the first order jet
$(A_{S},\mu_{S})$ corresponding to the IM connection 1-form $(L,l)$. For any
symplectic leaf $(S_{0},\omega_{0})$ of $(S,\pi_{S})$ the Poisson manifold
$\operatorname{pr}_{S}^{-1}(S_{0})\cap M_{0}$ is the Vorobjev local model
corresponding to the transitive first order jet $(A_{S_{0}},\mu_{S_{0}})$ over
$(S_{0},\omega_{0})$ given by

$A_{S_{0}}:=\left(A_{S}/(\ker
l\cap\ker\rho_{S})\right)|_{S_{0}},\quad\mu_{S_{0}}([\alpha]):=\mu_{S}(\alpha)|_{TS_{0}}.$

###### Proof.

Since $\pi_{S}|_{S_{0}}$ is non-degenerate, (7.12) implies that, for any
$z\in\operatorname{pr}_{S}^{-1}(S_{0})$, the map $\langle
z,U\rangle:T_{x}^{*}S\to T_{x}^{*}S$ preserves the isotropy bundle
$\ker\pi_{S}^{\sharp}=(TS_{0})^{\circ}$. Therefore, we have an induced map:
$\langle z,U_{0}\rangle:T^{*}_{x}S_{0}\to T^{*}_{x}S_{0}$. Using property
(7.12) again, we see that we can write
$U_{0}(\alpha)=i_{\pi_{S}^{\sharp}(\alpha)}\mathbb{F}$, for a unique
$\mathbb{F}\in\Omega^{2}(S;\mathfrak{k})$. Hence, the Poisson structure on
$\operatorname{pr}_{S}^{-1}(S_{0})\cap M_{0}$ is the Poisson structure
corresponding to the Vorobjev triple
$(\mathbb{F}+\omega_{S},\nabla^{L_{0}},\pi_{\mathfrak{k}_{0}})$, where
$\nabla^{L_{0}}$ it the pullback of $\nabla^{L}$ to the subbundle
$\mathfrak{k}^{*}|_{S_{0}}$, and $\pi_{\mathfrak{k}_{0}}$ is the linear
Poisson structure dual to $\mathfrak{k}|_{S_{0}}$. In our language, this is
the local model corresponding to the first order jet given in the statement. ∎

######  Remark 7.21.

From well-known properties of the Vorobjev local model (see, e.g., [44]), we
conclude that the symplectic structures on the leaves of the local model
$M_{0}$ lying over a leaf $S_{0}$ of $\pi_{S}$ vary in an affine fashion, in
the following sense.

  1. (i)

The Poisson structure on $\operatorname{pr}_{S}^{-1}(S_{0})\cap M_{0}$ extends
to a Dirac structure $\mathbb{L}_{S_{0}}$ on
$\operatorname{pr}_{S}^{-1}(S_{0})$;

  2. (ii)

If $(S_{v},\omega_{v})$ is the presymplectic leaf of $\mathbb{L}_{S_{0}}$
through $v\in\operatorname{pr}_{S}^{-1}(S_{0})$, then the connected component
of $v$ in $S_{v}\cap M_{0}$ is the symplectic leaf of $M_{0}$ through $v$.

  3. (iii)

The fiberwise multiplication by $\lambda\neq 0$ on
$\operatorname{pr}_{S}^{-1}(S_{0})\to S_{0}$ induces an isomorphism
$m_{\lambda}:S_{v}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}S_{\lambda
v}$ and

$m_{\lambda}^{*}(\omega_{\lambda v})=\lambda\omega_{v}+(1-\lambda)\omega_{0}.$

In Example 9.25, we show that $\pi_{0}$ might not extend to a Dirac structure
on $\mathfrak{k}^{*}$.

In [27, Section 5.5] we called an IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$ leafwise flat if the
induced vector bundle splitting $T^{*}S\to A_{S}$ is a Lie algebroid map. In
terms of the coupling data, this is equivalent to (see [27, Definition 5.14])

$U(\alpha,\pi_{S}^{\sharp}(\beta))=0,\quad\alpha,\beta\in T^{*}S.$

On the other hand, it is easy to see that this is equivalent also to
$\gamma=\pi_{S}$, where we used the notation of Proposition 7.18. Thus, we
obtain the following.

######  Corollary 7.22.

Let $(M_{0},\pi_{0})$ be the local model corresponding to the first order jet
$(A_{S},\mu_{S})$ and the IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$. The restriction of the
bundle projection $p:\mathfrak{k}^{*}\to S$ is a Poisson map

$p|_{M_{0}}:(M_{0},\pi_{0})\to(S,\pi_{S})$

if and only if $(L,l)$ is leafwise flat.

######  Remark 7.23 (Local model in coordinates).

Let $(A_{S},\mu_{S})$ be a partially split first jet, and fix a IM connection
1-form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$, which gives
rise to coupling data $(\nabla^{L},U)$. Choose a local chart $(V,x^{i})$ for
$S$ over which the bundle $\mathfrak{k}$ trivializes, and let $\\{e_{a}\\}$ be
a frame for $\mathfrak{k}|_{V}$. Then we can write all our data as

$\displaystyle\pi_{S}=\frac{1}{2}\pi_{S}^{ij}\partial_{x^{i}}\wedge\partial_{x^{j}}$
$\displaystyle[e_{a},e_{b}]_{\mathfrak{k}}=C_{ab}^{c}e_{c},$
$\displaystyle\nabla^{L}_{\partial_{x^{i}}}e_{a}=\Gamma^{b}_{ia}e_{b},$
$\displaystyle U(\mathrm{d}x^{i},\partial_{x^{j}})=U^{ia}_{j}e_{a}.$

Then one checks that the structure equations become

$\displaystyle
C_{ad}^{c}\Gamma^{d}_{ib}+C_{db}^{c}\Gamma^{d}_{ia}-C_{ab}^{d}\Gamma^{c}_{id}=\frac{\partial
C_{ab}^{c}}{\partial x^{i}}$
$\displaystyle\pi^{ik}_{S}\left(\frac{\partial\Gamma^{b}_{ja}}{\partial
x^{k}}-\frac{\partial\Gamma^{b}_{ka}}{\partial
x^{j}}+\Gamma^{d}_{ja}\Gamma^{b}_{kd}-\Gamma^{d}_{ka}\Gamma^{b}_{jd}\right)=U^{id}_{j}C^{b}_{da}$
$\displaystyle\pi^{il}_{S}\left(\frac{\partial U^{ja}_{k}}{\partial
x^{l}}-\frac{1}{2}\frac{\partial U^{ja}_{l}}{\partial
x^{k}}+\Gamma^{a}_{ld}U^{jd}_{k}-U^{jd}_{l}\Gamma^{a}_{kd}\right)+\frac{1}{2}\frac{\partial\pi^{il}_{S}}{\partial
x^{k}}U^{ja}_{l}$ $\displaystyle\qquad-\pi^{jl}_{S}\left(\frac{\partial
U^{ia}_{k}}{\partial x^{l}}-\frac{1}{2}\frac{\partial U^{ia}_{l}}{\partial
x^{k}}+\Gamma^{a}_{ld}U^{id}_{k}-U^{id}_{l}\Gamma^{a}_{kd}\right)-\frac{1}{2}\frac{\partial\pi^{jl}_{S}}{\partial
x^{k}}U^{ia}_{l}=\frac{\partial\pi^{ij}_{S}}{\partial x^{l}}U^{la}_{k}$
$\displaystyle U^{ia}_{k}\pi^{jk}_{S}+U^{ja}_{k}\pi^{ik}_{S}=0$

Note that for the structure equation (S3) we have used its skew-symmetric
version, arising from the relation (7.12) – the extra equation is precisely
the coordinate version of (7.12). The local coordinate expression associated
with the Poisson structure $\pi_{0}$ of the local model, is the following. In
coordinates $(x^{i},z_{a})$, where $z_{a}$ are fiber coordinates on
$\mathfrak{k}^{*}$ defined by the basis dual to $\\{e_{a}\\}$, it has a matrix
of structure functions given by the product of the block-matrices

$\displaystyle\pi_{0}=$
$\displaystyle\left(\begin{array}[]{cc}\mathrm{Id}&0\\\
\Gamma&\mathrm{Id}\end{array}\right)\left(\begin{array}[]{cc}\pi_{S}\cdot(\mathrm{Id}+\langle
z,U\rangle)^{-1}&0\\\
0&\pi_{\mathfrak{k}}\end{array}\right)\left(\begin{array}[]{cc}\mathrm{Id}&\Gamma^{t}\\\
0&\mathrm{Id}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\delta^{i}_{j}&0\\\
\Gamma_{ib}^{c}z_{c}&\delta^{a}_{b}\end{array}\right)\left(\begin{array}[]{cc}\pi^{ij}_{S}&0\\\
0&C^{c}_{ab}z_{c}\end{array}\right)\left(\begin{array}[]{cc}\delta^{i}_{j}+U_{j}^{ic}z_{c}&0\\\
0&\delta^{a}_{b}\end{array}\right)^{-1}\left(\begin{array}[]{cc}\delta^{i}_{j}&\Gamma_{jb}^{c}z_{c}\\\
0&\delta^{a}_{b}\end{array}\right).$

##  8\. Linearization and normal form theorems

In the previous section, we have seen that a partially split first order jet
has a well-defined local model. When this is also a normal form, we use the
following terminology.

######  Definition 8.1.

A Poisson manifold $(M,\pi)$ is said to be linearizable around a Poisson
submanifold $S\subset M$, if the first jet $J^{1}_{S}\pi=(A_{S},\mu_{S})$ is
partially split and $(M,\pi)$ is isomorphic around $S$ to the local model
$(M_{0},\pi_{0})$.

In other words, $(M,\pi)$ is linearizable around a Poisson submanifold $S$,
when there are open neighborhoods $S\subset U\subset M$ and $S\subset
V\subset\mathfrak{k}^{*}$, and a Poisson diffeomorphism
$\varphi:(U,\pi|_{U})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(V,\pi_{0}|_{V})$,
such that $\varphi|_{S}=\mathrm{Id}_{S}$. If $U$ and $V$ can be chosen to be
saturated, then we say that $\pi$ is invariantly linearizable around $S$. In
this section we will discuss conditions which imply that the local model
represents a local normal form, i.e., we discuss normal form theorems.

###  8.1. A linearization result

Our aim in this subsection is to prove the following:

######  Theorem 8.2.

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a Poisson submanifold.
Then the following are equivalent

  1. (i)

$(M,\pi)$ is linearizable around $S$;

  2. (ii)

The Lie algebroid $T^{*}M$ is linearizable around $S$.

Let us start by recalling a few facts about linearization of Lie algebroids.
First, given a Lie algebroid $A\to M$ with an invariant, embedded submanifold
$S\subset M$, the restricted Lie algebroid $A_{S}:=A|_{S}$ has a canonical
representation on the normal bundle $\nu(S)$ to $S$, which generalizes the
Bott connection from foliation theory. It can be defined by

$\nabla^{\nu}:\Gamma(A_{S})\times\Gamma(\nu(S))\longrightarrow\Gamma(\nu(S)),\quad\nabla^{\nu}_{\alpha}(v):=[\rho_{A}(\tilde{\alpha}),\tilde{v}]|_{S}\quad\mathrm{mod}\
TS,$

where $\widetilde{\alpha}\in\Gamma(A)$ and $\widetilde{v}\in\mathfrak{X}(M)$
are any smooth extensions of $\alpha$ and $v$, respectively. The linear
approximation of $A$ around $S$ is the action Lie algebroid associated with
this representation

$(A_{S}\ltimes\nu(S),[\cdot,\cdot]_{\ltimes},\rho_{\ltimes}).$

######  Definition 8.3.

We say that $A$ is linearizable around $S$ if there are open neighborhoods
$S\subset U\subset M$ and $S\subset V\subset\nu(S)$ and an isomorphism of Lie
algebroids

$\phi:A_{S}\ltimes
V\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A|_{U},$

which is the identity on $A_{S}$.

In order to obtain more geometric insight into the linear approximation, fix a
tubular neighborhood whose differential along $S$ induces the identity of
$\nu(S)$

$\varphi:\nu(S)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}U\subset
M.$

Fix also a vector bundle isomorphism covering $\varphi$

$\phi:A_{S}\times_{S}\nu(S)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A|_{U},$

such that $\phi|_{A_{S}\times 0_{S}}=\mathrm{Id}_{A_{S}}$. We use $\phi$ to
transport the Lie algebroid structure of $A|_{U}$ to obtain a Lie algebroid
structure on $A_{S}\times_{S}\nu(S)\to\nu(S)$, which we denote

$(A_{1},[\cdot,\cdot]_{1},\rho_{1}).$

Let $m_{t}$ be the multiplication by $t>0$ on the second component

$\vbox{\hbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.10852pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A_{S}\times_{S}\nu(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
24.10854pt\vrule
width=0.0pt,depth=0.0pt,height=0.0pt}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.20474pt\raise 5.12202pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.89186pt\hbox{$\scriptstyle{m_{t}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern-0.2474pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\xyatipfont\char 111\relax\xybtipfont\char
111\relax}}}}}{\hbox{\kern 0.03038pt\vrule
width=-0.1389pt,depth=0.0pt,height=0.0pt}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
0.0pt\vrule width=0.0pt,depth=5.5pt,height=-5.5pt}}\ignorespaces{\hbox{\kern
0.0pt\raise-13.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\xyatipfont\char
15\relax\xybtipfont\char 15\relax}}}}}{\hbox{\kern 0.0pt\vrule
width=0.0pt,depth=13.5pt,height=-5.5pt}}{\hbox{\kern 0.0pt\vrule
width=0.0pt,depth=13.5pt,height=-13.5pt}}{\hbox{\kern-0.10852pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A_{S}\times_{S}\nu(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
24.0pt\vrule width=0.0pt,depth=5.5pt,height=-5.5pt}}\ignorespaces{\hbox{\kern
24.0pt\raise-13.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\xyatipfont\char
15\relax\xybtipfont\char 15\relax}}}}}{\hbox{\kern 24.0pt\vrule
width=0.0pt,depth=13.5pt,height=-5.5pt}}{\hbox{\kern 24.0pt\vrule
width=0.0pt,depth=13.5pt,height=-13.5pt}}{\hbox{\kern-12.71297pt\raise-24.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\nu(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
12.71297pt\vrule
width=0.0pt,depth=24.0pt,height=-24.0pt}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.20474pt\raise-18.87798pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.89186pt\hbox{$\scriptstyle{m_{t}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 11.28703pt\raise-24.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\xyatipfont\char 47\relax\xybtipfont\char
47\relax}}}}}{\hbox{\kern 11.28703pt\vrule
width=0.0pt,depth=24.0pt,height=-24.0pt}}{\hbox{\kern
11.28703pt\raise-24.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\nu(S)}$}}}}}}}\ignorespaces}}}}\ignorespaces}}\qquad
m_{t}(a,v)=(a,m_{t}(v))=(a,tv).$

By pulling back the Lie algebroid structure of $A_{1}$ along $m_{t}$, we
obtain Lie algebroids

$(A_{t},[\cdot,\cdot]_{t},\rho_{t}),\quad t>0.$

These Lie algebroids are on the same vector bundle
$A_{t}:=A_{S}\times_{S}\nu(S)\to\nu(S)$, and the structure maps are uniquely
determined by the condition that for $\alpha,\beta\in\Gamma(A_{S})$

$[\overline{\alpha},\overline{\beta}]_{t}:=m_{t}^{*}([\overline{\alpha},\overline{\beta}]_{1}),\quad\rho_{t}(\overline{\alpha}):=(m^{0}_{t})^{*}(\rho_{1}(\overline{\alpha})),$

where the bar indicates the corresponding constant section. In other words, we
have constructed a family of Lie algebroid isomorphisms

$m_{t}:A_{t}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{1}\cong
A,\quad t>0.$

Moreover, it is easy to see that this path of Lie algebroids extends smoothly
at $t=0$, and in the limit one obtains precisely the action Lie algebroid

$A_{0}=A_{S}\ltimes\nu(S),\quad\textrm{with}\quad\lim_{t\to
0}[\cdot,\cdot]_{t}=[\cdot,\cdot]_{\ltimes},\quad\ \lim_{t\to
0}\rho_{t}=\rho_{\ltimes}.$

This follows because bundle maps $m_{t}:A_{0}\to A_{0}$ are Lie algebroid
automorphisms of the linear approximation and this actually characterizes it.
Let us go back to our case, where $A=T^{*}M$ is the cotangent bundle of a
Poisson manifold $(M,\pi)$ and $S\subset M$ is a Poisson submanifold. As we
saw before, we have the adjoint representation $\nabla^{\mathfrak{k}}$ on the
conormal bundle

$\nu^{*}(S)=(TS)^{0}=\ker\mu_{S}=\mathfrak{k},$

while the dual bundle $\nu(S)=\mathfrak{k}^{*}$ carries the dual
representation $\nabla^{\mathfrak{k}^{*}}$. Next, we show that this
representation is nothing else but the Bott connection.

######  Lemma 8.4.

The two representations of $A_{S}$ on $\nu(S)=\mathfrak{k}^{*}$ coincide:
$\nabla^{\nu}=\nabla^{\mathfrak{k}^{*}}$.

###### Proof.

Given sections $\alpha\in\Gamma(A_{S})$ and $\beta\in\Gamma(\mathfrak{k})$,
denote by $\widetilde{\alpha},\widetilde{\beta}\in\Omega^{1}(M)$ extensions of
$\alpha$ and $\beta$ to $M$. Also, given a section $v\in\nu(S)$, let
$\widetilde{v}\in\mathfrak{X}(M)$ be a vector field whose restriction
$\widetilde{v}|_{S}$ represents $v$. Using the expression of the bracket on
the Lie algebroid $(T^{*}M,[\cdot,\cdot]_{\pi},\pi^{\sharp})$, we have that

$\mathscr{L}_{\pi^{\sharp}(\widetilde{\alpha})}(\widetilde{\beta}(\widetilde{v}))=[\widetilde{\alpha},\widetilde{\beta}]_{\pi}(\widetilde{v})+\widetilde{\beta}([\pi^{\sharp}(\widetilde{\alpha}),\widetilde{v}])+\mathrm{d}\widetilde{\alpha}(\pi^{\sharp}(\widetilde{\beta}),\widetilde{v}).$

Restricting to $S$, we have that
$\pi^{\sharp}(\widetilde{\beta})|_{S}=\rho_{A_{S}}(\beta)=0$. Therefore, the
above formula restricts to the duality relation

$\mathscr{L}_{\rho_{A_{S}}(\alpha)}\langle\beta,v\rangle=\langle\nabla^{\mathfrak{k}}_{\alpha}\beta,v\rangle+\langle\beta,\nabla_{\alpha}^{\nu}v\rangle.\qed$

We conclude that the linear approximation to the cotangent Lie algebroid of
$(M,\pi)$ along the Poisson submanifold $S$ coincides with the action
algebroid $A_{S}\ltimes\mathfrak{k}^{*}$ we have used to construct the local
model. Moreover, it is completely determined by the first jet $J^{1}_{S}\pi$
or, equivalently, by the pair $(A_{S},\mu_{S})$. We can now give a proof of
the main result in this subsection.

###### Proof of Theorem 8.2.

Assume first that we have a Lie algebroid automorphism $\phi:A_{S}\ltimes
V\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}U$
giving a linearization of $T^{*}M$ around $S$. By assumption,
$\phi|_{A_{S}}=\mathrm{Id}_{A_{S}}$. The identity map of $T^{*}U$ is a non-
degenerate, closed IM 2-form, and therefore, so is its pullback
$\mu_{0}:=\phi^{*}(\mathrm{Id}):A_{S}\ltimes V\to T^{*}V$. We have that
$i^{*}(\mu_{0})=\mu_{S}$, where $i:A_{S}\to A_{S}\ltimes V$ is the inclusion.
Proposition 7.8 implies that $(A_{S},\mu_{S})$ is partially split, and
Proposition 7.9 says that the Poisson local model is isomorphic around $S$ to
$(V,\pi_{V}:=\rho_{\ltimes}\circ\mu_{0}^{-1})$. We conclude that the base map
of $\phi$ is a Poisson isomorphism

$\varphi:(V,\pi_{V})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(U,\pi|_{U}),$

and so $\pi$ is linearizable around $S$. For the converse, assume that
$(M,\pi)$ is linearizable around $S$ and consider a Poisson isomorphism
$\varphi:(U,\pi|_{U})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(V,\pi_{0}|_{V})$
between open neighborhoods $S\subset U\subset M$ and $S\subset
V\subset\mathfrak{k}^{*}$, with $\varphi|_{S}=\mathrm{Id}_{S}$. The
pushforward along $\varphi$ is an isomorphism between the cotangent Lie
algebroids

$\varphi_{*}:T^{*}U\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}V,\quad\varphi_{*}(\xi):=(\mathrm{d}\varphi^{-1})^{*}(\xi).$

On the other hand, $\pi_{0}$ is constructed from the Lie algebroid local model
$(A_{S}\ltimes V,\mu_{0})$, where the IM 2-form is nondegenerate. Hence, we
have a Lie algebroid isomorphism

$\mu_{0}:A_{S}\ltimes
V\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}V,$

which extends $\mu_{S}$, i.e., $i^{*}(\mu_{0})=\mu_{S}$, where
$i:A_{S}\hookrightarrow A_{S}\ltimes V$ is the inclusion. Combining the two
maps, we obtain a Lie algebroid isomorphism

$\phi:=(\varphi_{*})^{-1}\circ\mu_{0}:A_{S}\ltimes
V\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}U,$

which covers $\varphi^{-1}$. We still need to fix $\phi$ so that its
restriction
$\phi_{A_{S}}:=\phi|_{A_{S}}:A_{S}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{S}$
is the identity. This is done in the following lemma.

######  Lemma 8.5.

The map $\phi_{A_{S}}$ restricts to a bundle isomorphism
$\phi_{\mathfrak{k}}:=\phi|_{\mathfrak{k}}:\mathfrak{k}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}$,
and we have a Lie algebroid isomorphism

$(\phi_{A_{S}},\phi_{\mathfrak{k}^{*}}):A_{S}\ltimes\mathfrak{k}^{*}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{S}\ltimes\mathfrak{k}^{*},$

where
$\phi_{\mathfrak{k}^{*}}:=(\phi_{\mathfrak{k}}^{*})^{-1}:\mathfrak{k}^{*}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}^{*}$.

By precomposing $\phi$ with the inverse of the isomorphism in this lemma, we
obtain an isomorphism of Lie algebroids,

$\phi^{\prime}:A_{S}\ltimes
V^{\prime}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}U,\quad
V^{\prime}=\phi_{\mathfrak{k}^{*}}(V)\subset\mathfrak{k}^{*}$

which satisfies $\phi^{\prime}|_{A_{S}}=\mathrm{Id}_{A_{S}}$. Hence, $T^{*}M$
is linearizable around $S$. To prove the lemma, we note that $\phi_{A_{S}}$
covers the identity of $S$, so we have that
$\rho_{A_{S}}\circ\phi_{A_{S}}=\rho_{A_{S}}$ and
$\mu_{S}\circ\phi_{A_{S}}=\mu_{S}$ (since this holds for $\mu_{0}$ and
$\phi_{*}$). Therefore, the restriction
$\phi_{\mathfrak{k}}:=\phi|_{\mathfrak{k}}:\mathfrak{k}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}$
is a vector bundle automorphism of $\mathfrak{k}=\ker\mu_{S}$,
$\phi_{\mathfrak{k}}:=\phi|_{\mathfrak{k}}:\mathfrak{k}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}$.
Since $\phi_{A_{S}}$ is a Lie algebroid map, we have that

$\phi_{\mathfrak{k}}(\nabla^{\mathfrak{k}}_{\alpha}\beta)=\phi_{A_{S}}([\alpha,\beta]_{A_{S}})=[\phi_{A_{S}}(\alpha),\phi_{A_{S}}(\beta)]_{A_{S}}=\nabla^{\mathfrak{k}}_{\phi_{A_{S}}(\alpha)}\phi_{\mathfrak{k}}(\beta).$

This implies the analog equation for the dual representation on
$\mathfrak{k}^{*}$ and the automorphism
$\phi_{\mathfrak{k}^{*}}:=(\phi_{\mathfrak{k}}^{*})^{-1}:\mathfrak{k}^{*}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}^{*}$.
This completes the proof of the lemma. ∎

######  Remark 8.6.

As discussed before, the linearization procedure for Lie algebroids yields a
smooth family of Lie algebroid structures $A_{t}$ connecting $A_{1}\simeq
T^{*}M$ to the linear model $A_{0}=A_{S}\ltimes\mathfrak{k}^{*}$. Since
$m_{t}:A_{t}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}A_{1}$
is a Lie algebroid isomorphism for $t>0$, we have the family of Poisson
structures

$\pi_{t}=\rho_{t}\circ(m_{t}^{*}(\mu_{1}))^{-1},$ (8.1)

where
$\mu_{1}:A_{1}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}T^{*}M$
is the identification we started with. However, in general, $\lim_{t\to
0}\pi_{t}$ may not exist (even if local models exists!), and if this limit
exists, it may not be isomorphic to the local model (see Section 9.1). It
would be interesting to understand better the following.

######  Question.

If the first order jet of $(M,\pi)$ at a Poisson submanifold $S\subset M$ is
partially split, can $\pi$ be connected to a local model by a smooth path of
Poisson structure $\pi_{t}$, $t\in[0,1]$, defined around $S$, with constant
first order jet along $S$?

A possible candidate can be constructed as follows. Consider the path $A_{t}$
of Lie algebroid structures which connects $T^{*}M$ to the linearization
$A_{0}$. Then the path of bivectors
$\tilde{\pi}_{t}=\rho_{t}\circ\mu_{t}^{-1}$, where

$\mu_{t}:=m_{t}^{*}\big{(}\mathrm{pr}^{*}\mu_{S}+t^{-1}\,(\mu-\mathrm{pr}^{*}\mu_{S})\big{)},$
(8.2)

has constant first order jet along $S$ and the limit at $t=0$ exists. However,
these bivectors may not be Poisson since $\mathrm{pr}^{*}\mu_{S}$ may fail to
be a closed IM 2-form. The problem is that the projection
$\mathrm{pr}:A_{1}\to A_{S}$ is usually not a Lie algebroid morphism. The
question has positive answers in the following situations:

  * -

$M=S\times N$ is a product of a Poisson manifold $(S,\pi_{S})$ and a Poisson
manifold $(N,\pi_{N})$ the a zero $\pi_{N}(x_{0})=0$;

  * -

$S$ is a symplectic leaf (see [53, 54, 44]) – instead of
$\mathrm{pr}^{*}\mu_{S}$, one uses in formula (8.2) the IM 2-form
corresponding to the closed de Rham 2-form $r^{*}(\pi_{S}^{-1})$, where
$r:U\to S$ is the retraction of a tubular neighborhood;

  * -

there exists a Lie algebroid map $p:T^{*}U\to A_{S}$, defined on some open set
$S\subset U$, such that $p|_{A_{S}}=\mathrm{Id}_{A_{S}}$ – this case includes
the previous two;

  * -

$\pi$ is linearizable around $S$.

###  8.2. The normal form theorem

The following result implies Theorem 4 in the Introduction.

######  Theorem 8.7 (Normal form).

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a Poisson submanifold. If
$T^{*}_{S}M$ is integrable by a compact, Hausdorff, Lie groupoid whose target
fibers have trivial 2nd de Rham cohomology, then $\pi$ is invariantly
linearizable around $S$.

This result is a generalization to Poisson submanifolds of the normal form
theorem around leaves from [17, 46], which in turn generalizes Conn’s
linearization theorem around points [12]. Conn’s original proof uses the Nash-
Moser fast convergence method to build a linearization map. Later, a geometric
proof was obtained by Crainic and Fernandes [15]. Both methods have been
extended to symplectic leaves in [46] and [17], respectively, and the results
in these papers can be adapted to the general setting of Poisson submanifolds,
to give two proofs of Theorem 8.7. We will explain these two approaches in the
following subsection.

###  8.3. The geometric method

With the geometric method, developed in [15] and extended in [17], we obtain a
version of Theorem 8.7 under a stronger assumption.

###### Proof of Theorem 8.7, with an extra assumption.

We assume that the groupoid integrating $T^{*}_{S}M$ from the statement has,
in addition, 1-connected target fibers. Under these assumptions, we claim
that:

  * •

there is a saturated neighborhood $U\subset M$ of $S$ such that $T^{*}U\subset
T^{*}M$ is integrable by a Hausdorff, target-proper, Lie groupoid
$\mathcal{G}\rightrightarrows U$.

Since Hausdorff, target-proper, groupoids are invariantly linearizable around
invariant submanifolds, it then follows that the Lie algebroid $T^{*}U$ (and
hence $T^{*}M$) is invariantly linearizable around $S$, so one can apply
Theorem 8.2. It remains to prove the claim. When $S$ is symplectic, i.e., when
$T^{*}_{S}M$ is transitive, this claim is proved in Section 5 of [17].
However, one observes that the proof of the claim given in [17] does not use
that $S$ is a symplectic leaf, and remains valid in the non-transitive case. ∎

###  8.4. The analytic method

For the proof of Theorem 8.7, we will use the following result, obtained using
the Nash-Moser method.

######  Theorem 8.8 (Rigidity [46]).

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a compact Poisson
submanifold. If $T^{*}M$ is integrable by a Hausdorff Lie groupoid whose
target fibers are compact and have trivial 2nd de Rham cohomology, then any
Poisson structure $\pi^{\prime}$ with $J^{1}_{S}\pi=J^{1}_{S}\pi^{\prime}$ is
locally isomorphic to $\pi$ around $S$.

###### Proof of Theorem 8.7.

Let $\mathcal{G}_{S}\rightrightarrows S$ be a Lie groupoid integrating
$A_{S}:=T^{*}_{S}M$ which is compact, Hausdorff and whose target fibers have
trivial second de Rham cohomology. Since $\mathcal{G}_{S}$ is proper and
Hausdorff, by Theorem 5.10, the bundle of ideals
$\mathfrak{k}:=(TS)^{\circ}\subset A_{S}$ is partially split for
$\mathcal{G}_{S}$, and so also for $A_{S}$. After choosing a partial splitting
and a tubular neighborhood
$\phi:\nu(S)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}M_{0}\subset
M$ of $S$, we obtain the Poisson structure of the local model
$\pi_{0}\in\mathfrak{X}^{2}(M_{0})$, which satisfies
$J^{1}_{S}\pi_{0}=J^{1}_{S}\pi$. The algebroid $T^{*}M_{0}$ is isomorphic to
the action algebroid $A_{S}\ltimes\mathfrak{k}^{*}$. Since $A_{S}$ integrates
to $\mathcal{G}_{S}\rightrightarrows S$, and the action of $A_{S}$ on the
bundle of ideals $\mathfrak{k}$ integrates to the action by conjugation of the
target-connected Lie groupoid $\mathcal{G}_{S}$ (see, e.g., [46, Lemma B.1]),
it follows that the action algebroid $A_{S}\ltimes\mathfrak{k}^{*}$ also
integrates to the action groupoid
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}$. We
conclude that $T^{*}M_{0}$ integrates to a Hausdorff Lie groupoid whose target
fibers are compact and have trivial 2nd de Rham cohomology. Hence, by Theorem
8.8, there exists a Poisson diffeomorphism
$\phi:(U,\pi_{0})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(V,\pi)$,
between neighborhoods $U$ and $V$ of $S$. Finally, since the action groupoid
$\mathcal{G}_{S}\ltimes\mathfrak{k}^{*}\rightrightarrows\mathfrak{k}^{*}$ is
target-proper, it follows that $S$ has a basis of open neighborhoods in
$\mathfrak{k}^{*}$ that are invariant (see, e.g.,[46, Lemma A.1]). So we may
assume that $U$ is invariant. Since all the leaves of $\pi_{0}|_{U}$ are
compact, so are also those of $\pi|_{V}$. This implies that $V$ is invariant
as well. Hence, $\pi$ is invariantly linearizable around $S$. ∎

From the proof, we extract the following result.

######  Corollary 8.9.

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a Poisson submanifold. If
the hypotheses of Theorem 8.7 are satisfied, then a neighborhood of $S$ in $M$
is integrable by a target-proper, Hausdorff Lie groupoid, whose target-fibers
have trivial second de Rham cohomology.

So we may apply [46, Theorem 2] and obtain the following, which implies
Corollary 5 in the Introduction (for a complete version of (ii), see [46]).

######  Corollary 8.10 (Rigidity).

Let $(M,\pi)$ be a Poisson manifold and $S\subset M$ a Poisson submanifold. If
the hypotheses of Theorem 8.7 are satisfied, then

  1. (i)

For any Poisson structure $\pi^{\prime}\in\Pi(M,S)$, with
$J^{1}_{S}\pi=J^{1}_{S}\pi^{\prime}$, there is a Poisson diffeomorphism
$\phi:(U,\pi)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(U^{\prime},\pi^{\prime})$
between open neighborhoods $U$ and $U^{\prime}$ of $S$, with
$\phi|_{S}=\mathrm{Id}_{S}$;

  2. (ii)

There is a neighborhood of $\pi$ in the space of all Poisson structures on
$M$, $\mathcal{V}\subset\Pi(M)$, such that for every
$\pi^{\prime}\in\mathcal{V}$ there is a Poisson diffeomorphism
$\phi:(U,\pi)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(U^{\prime},\pi^{\prime})$
between open neighborhoods $U$ and $U^{\prime}$ of $S$.

##  9\. Examples of first order jets and applications

In this section we illustrate the constructions and results of the previous
sections with several examples. We follow the notation introduced there.

###  9.1. Products

Consider a product Poisson structure

$(M,\pi):=(S\times\mathbb{R}^{n},\pi_{S}+\gamma),$ (9.1)

where $\pi_{S}$ is a Poisson structure on $S$ and $\gamma$ is a Poisson
structure on $\mathbb{R}^{n}$ vanishing at $0$. We consider the first order
jet $(A_{S},\mu_{S})$ of $\pi$ along Poisson submanifold $S=S\times\\{0\\}$,
which can be identified as follows. Denote by
$\mathfrak{g}=T^{*}_{0}\mathbb{R}^{n}$ the isotropy Lie algebra of $\gamma$ at
$0$. We have the trivial Lie algebra bundle
$\mathfrak{k}=S\times\mathfrak{g}\to S$ and

$A_{S}=T^{*}_{S}M=T^{*}S\times\mathfrak{g}$

is the product Lie algebroid, with closed IM form the projection
$\mu_{S}=\operatorname{pr}_{T^{*}S}$. We call the pair $(A_{S},\mu_{S})$ a
trivial product jet. The trivial product jet $(A_{S},\mu_{S})$ is partially
split with a canonical choice of an IM connection 1-form. This follows by
observing that we have canonical isomorphisms

$T^{*}A_{S}=T^{*}(T^{*}S)\times(\mathfrak{g}\ltimes\mathfrak{g}^{*}),\quad
A_{S}\ltimes\mathfrak{k}^{*}=T^{*}S\times(\mathfrak{g}\ltimes\mathfrak{g}^{*}),$

in terms of which the projection

$p:T^{*}(T^{*}S)\times(\mathfrak{g}\ltimes\mathfrak{g}^{*})\to
T^{*}S\times(\mathfrak{g}\ltimes\mathfrak{g}^{*}).$

is the bundle projection in the first factor. Hence, we have the partial
splitting

$\theta:T^{*}S\times(\mathfrak{g}\ltimes\mathfrak{g}^{*})\to
T^{*}(T^{*}S)\times(\mathfrak{g}\ltimes\mathfrak{g}^{*})\oplus\mathfrak{k}\oplus\mathfrak{k}^{*},\quad(\alpha,v,\xi)\mapsto(0_{\alpha},v,\xi).$

The corresponding linear IM 2-form
$\mu^{\mathrm{lin}}=\theta^{*}\mu_{\mathrm{can}}\in\Omega^{2}_{\mathrm{IM}}(T^{*}S\times(\mathfrak{g}\ltimes\mathfrak{g}^{*}))$
is

$\mu^{\mathrm{lin}}:T^{*}S\times(\mathfrak{g}\ltimes\mathfrak{g}^{*})\to
T^{*}(S\times\mathfrak{g}^{*})=T^{*}S\times(\mathfrak{g}\ltimes\mathfrak{g}^{*}),\quad(\alpha,v,\xi)\mapsto(0,v,\xi),$

so the closed IM 2-form (7.5) of the local model is just the identity map

$\mu_{0}=\operatorname{pr}^{*}\mu_{S}+\theta^{*}\mu_{\mathrm{can}}=\mathrm{Id}.$

The resulting local model Poisson manifold is the product

$(M_{0},\pi_{0}):=(S\times\mathbb{R}^{n},\pi_{S}+\gamma_{\mathrm{lin}}).$

where $\gamma_{\mathrm{lin}}$ is the linear Poisson structure on
$\mathbb{R}^{n}=\mathfrak{g}^{*}$, i.e. the linear approximation to $\gamma$
at $0$. Following Remark 8.6, let us look at the path of bivectors (8.1).
Using the obvious identification $A_{S}\times\mathbb{R}^{n}\simeq T^{*}M$, we
find

$\pi_{t}=\pi_{S}+m_{t}^{*}(\gamma)=\pi_{S}+\frac{1}{t}\gamma_{\mathrm{lin}}+\gamma_{2}+t\,\gamma_{3}+\ldots,$

where each $\gamma_{k}$ a bivector whose coefficients are homogeneous
polynomials of degree $k$. Thus $\lim_{t\to 0}\pi_{t}$ exists if and only if
$\gamma_{\mathrm{lin}}=0$, i.e., if and only if the isotropy Lie algebra of
$\gamma$ at 0 is abelian, in which case we obtain

$\lim_{t\to 0}\pi_{t}=\pi_{S}+\gamma_{2}\not=\pi_{0}.$

On the other hand, in this example the projection $\mathrm{pr}:A_{1}\to A_{S}$
is a Lie algebroid morphism so $\mathrm{pr}^{*}\mu_{S}$ is a closed IM 2-form.
Hence, applying (8.2) we obtain a smooth path of Poisson structures connecting
$\pi$ to the local model $\pi_{0}$

$\tilde{\pi}_{t}:=\pi_{S}+t\,m_{t}^{*}(\gamma).$

###  9.2. Jets over zero Poisson structures

As we saw in Example 6.1, first order jets $(A_{S},\mu_{S})$ with $\pi_{S}=0$
may fail be partially split. The following result, which is a direct
consequence of [27, Proposition 6.1] and of Proposition 7.18, shows that the
obstructions are precisely those found in Proposition 7.5.

######  Proposition 9.1.

A first order jet of a Poisson structure $(A_{S},\mu_{S})$ with $\pi_{S}=0$ is
partially split if and only if the following two conditions hold:

  1. (i)

$\mathfrak{k}=\ker\mu_{S}$ is a Lie algebra bundle;

  2. (ii)

there exists a splitting $A_{S}\simeq T^{*}S\oplus\mathfrak{k}$, for which the
fiberwise Lie bracket is a direct product with abelian first component.

Moreover, if these conditions hold, then the associated linear model is
isomorphic to the bundle of linear Poisson structure
$(\mathfrak{k}^{*},\pi_{\mathfrak{k}})$.

######  Remark 9.2.

Let $(\mathfrak{k},[\cdot,\cdot]_{\mathfrak{k}})\to S$ be a bundle of Lie
algebras. The fiberwise linear Poisson structure $\pi_{\mathfrak{k}}$ on
$\mathfrak{k}^{*}$ has the zero section as a Poisson submanifold with
$\pi_{S}=0$. The corresponding first order jet is
$(A_{S}=T^{*}S\oplus\mathfrak{k},\mu_{S}=\operatorname{pr}_{T^{*}S})$, where
sections of $T^{*}S$ commute with all other sections, and on $\mathfrak{k}$ we
have the bracket $[\cdot,\cdot]_{\mathfrak{k}}$. By Proposition 9.1, this
first order jet is partially split if and only if $\mathfrak{k}$ is a Lie
algebra bundle. In this case, the local model is $\pi_{0}=\pi_{\mathfrak{k}}$.
However, even if no partial splitting exits, we have a Poisson embedding

$i:(\mathfrak{k}^{*},\pi_{\mathfrak{k}})\hookrightarrow(A_{S}^{*}=TS\oplus\mathfrak{k}^{*},\pi_{A_{S}}),\quad\xi\mapsto(0,\xi),$

where $\pi_{A_{S}}=\pi_{A_{S},\mu_{S}}$. When $\mathfrak{k}$ is locally
trivial this inclusion realizes $\mathfrak{k}^{*}$ as a Lie-Dirac submanifold
(see Theorem 7.11).

######  Example 9.3.

Consider the first order jet of the linear Poisson structure

$(\mathbb{R}^{2},\pi=x\,\partial_{x}\wedge\partial_{y})$

along the Poisson submanifold $S=\\{x=0\\}$. In Example 6.1 we gave a groupoid
argument to see that it is not partially split. Proposition 9.1 gives now an
infinitesimal argument and one can also see this by applying Corollary 7.13:
it is easy to see that the pullback of the Dirac structure corresponding to
$\pi_{A_{S},\mu_{S}}$ via any linear splitting
$i:\mathfrak{k}^{*}\hookrightarrow A_{S}^{*}$ of the projection is not a
smooth Dirac structure. Similarly, we can use Proposition 9.1 to show that the
first order jet of the linear Poisson structure

$(\mathbb{R}^{3},\pi=z\,\partial_{x}\wedge\partial_{y})$

along the Poisson submanifold $S=\\{z=0\\}$ is not partially split. However,
in this example, the pullback of the Dirac structure corresponding to
$\pi_{A_{S}}=\pi_{A_{S},\mu_{S}}$ via any linear splitting
$i:\mathfrak{k}^{*}\hookrightarrow A_{S}^{*}$ of the projection is smooth, and
corresponds to the zero Poisson structure $\pi_{0}=0$ on $\mathfrak{k}^{*}$.
But note that $\pi_{0}$ is not a first order local model around $S$ because
$J^{1}_{S}\pi_{0}=0\neq J^{1}_{S}\pi$.

###  9.3. Transitive jets

Let $(A_{S},\mu_{S})$ be a first order jet with $A_{S}\Rightarrow S$
transitive. Such jets arise at symplectic leaves. That $A_{S}$ is transitive
is equivalent to the induced Poisson structure $\pi_{S}$ being non-degenerate,
and so its inverse is a symplectic structure $\omega$ on $S$. The IM 2-form is
determined by $\omega$ via the relation
$\mu_{S}(\alpha)=-i_{\rho(\alpha)}\omega$. Also, we have that
$\mathfrak{k}=\ker\rho$. If we choose a splitting of the anchor

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{l}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{0}$

we can define a linear operator $L:\Gamma(A_{S})\to\Omega^{1}(S,\mathfrak{k})$
by setting

$i_{X}L(\alpha):=l([\tau(X),\alpha]).$

One checks easily that the pair $(L,l)$ satisfies (A.2), so it is a
$\mathfrak{k}$-valued, IM 1-form, with $l|_{\mathfrak{k}}=\mathrm{Id}$. Hence,
$L$ is a IM connection 1-form so that $(A_{S},\mu_{S})$ is partially split.
Conversely, given any IM connection 1-form
$(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S},\mathfrak{k})$, the bundle map
$l:A_{S}\to\mathfrak{k}$ determines a splitting of the anchor. These two
constructions are inverse to each other (see [27, Subsection 6.3]). We
conclude that transitive jets are partially split and a choice of IM
connection 1-form is equivalent to a choice of splitting of its anchor. The
resulting local model coincides with Vorobjev’s local model for Poisson
structures around symplectic leaves [54].

###  9.4. Cartan connections

A Cartan connection on a Lie algebroid $A_{S}$ is a connection $\nabla$ whose
basic curvature vanishes identically

$R^{\mathrm{bas}}_{\nabla}\equiv 0.$

Hence, an immediate consequence of Proposition 7.6 is the following.

######  Corollary 9.4.

Let $(A_{S},\mu_{S})$ be a first order jet equipped with a Cartan connection
$\nabla$ and a splitting $l:A_{S}\to\mathfrak{k}$ which is
$\overline{\nabla}$-invariant. Then $(A_{S},\mu_{S})$ is partially split.

######  Example 9.5 (Action Lie algebroids).

Let $A_{S}=\mathfrak{g}\ltimes S\Rightarrow S$ be the action Lie algebroid
associated with a Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}(S)$.
The canonical flat connection $\nabla$ on $A_{S}$ has vanishing basic
curvature, hence it is a Cartan connection. Given a bundle of ideals
$\mathfrak{k}\subset A_{S}$ a splitting $l:A_{S}\to\mathfrak{k}$ is
$\overline{\nabla}$-invariant if and only if it is $\mathfrak{g}$-equivariant

$l([v,w]_{\mathfrak{g}})=[v,l(w)]_{\mathfrak{g}\ltimes S},$

for all $v,w\in\mathfrak{g}$ (here we identify elements of $\mathfrak{g}$ with
constant sections of $A_{S}$). Hence, we recover the infinitesimal version of
the example is Subsection 6.3: a first order jet on an action algebroid
admitting a $\mathfrak{g}$-equivariant splitting is partially split. One class
of examples is obtained when $\mathfrak{g}$ is a Lie algebra of compact type.
Then $\mathfrak{g}$ admits an $\operatorname{ad}$-invariant scalar product
$\langle\cdot,\cdot\rangle$, which yields a $\overline{\nabla}$-invariant
fiberwise metric $\eta$ on $A_{S}$. Hence any bundle of ideals on
$A_{S}=\mathfrak{g}\ltimes S$ admits a $\mathfrak{g}$-equivariant splitting,
so it is partially split.

######  Example 9.6 (Poisson submanifolds of $\mathfrak{g}^{*}$).

Let us consider the case where $S\subset\mathfrak{g}^{*}$ is a closed
$\operatorname{ad}^{*}$-invariant submanifold. Then $S$ is a Poisson
submanifold of the linear Poisson manifold
$(\mathfrak{g}^{*},\pi_{\mathrm{lin}})$ and its first order jet
$(A_{S},\mu_{S})$ is the action algebroid $A_{S}=\mathfrak{g}\ltimes
S\Rightarrow S$ obtained by restriction of the coadjoint action. Hence, we can
use the results of the previous example. Let us assume then that
$\mathfrak{g}$ is a Lie algebra of compact type and choose an
$\operatorname{ad}$-invariant inner product. Then
$\mathfrak{k}^{*}=(\ker\mu_{S})^{*}$ is the normal bundle of $S$ in
$\mathfrak{g}^{*}$

$\mathfrak{k}^{*}=\\{(w,x):x\in
S,w\in(T_{x}S)^{\perp}\\}\subset\mathfrak{g}^{*}\times\mathfrak{g}^{*}.$

The Riemannian exponential is the $\mathfrak{g}$-equivariant map

$\exp:\mathfrak{k}^{*}\to\mathfrak{g}^{*},\quad(w,x)\mapsto w+x.$

and so it induces a map of Lie algebroids

$A_{0}=\mathfrak{g}\ltimes\mathfrak{k}^{*}\to
T^{*}\mathfrak{g}^{*}=\mathfrak{g}\ltimes\mathfrak{g}^{*}.$

This map is an isomorphism around $S$. On the other hand, the inner product
determines a $\mathfrak{g}$-equivariant splitting giving a closed IM form
$\mu_{0}$ on $A_{0}$

$\mu_{0}=\operatorname{pr}^{*}\mu_{S}+\mu^{\mathrm{lin}}.$

Under the identification $A_{0}\simeq T^{*}\mathfrak{g}^{*}$ the two terms of
$\mu_{0}$ correspond to the orthogonal projections on $T^{*}S$ and
$(T^{*}S)^{\perp}$, respectively,

$\operatorname{pr}^{*}\mu_{S}|_{\xi}=\operatorname{pr}_{T^{*}_{\xi}S}:T_{\xi}\mathfrak{g}^{*}\to
T^{*}_{\xi}S,\qquad\mu^{\mathrm{lin}}|_{\xi}=\operatorname{pr}_{(T^{*}_{\xi}S)^{\perp}}:T_{\xi}\mathfrak{g}^{*}\to(T^{*}_{\xi}S)^{\perp}.$

Therefore, $\mu_{0}=\mathrm{Id}$. We conclude that for a compact Lie algebra
$\mathfrak{g}$, the Poisson manifold $\mathfrak{g}^{*}$ is linearizable around
any closed Poisson submanifold $S\subset\mathfrak{g}^{*}$. This holds already
at the groupoid level, as was shown in Subsection 6.3. The result agrees with
Theorem 8.2 since the Lie algebroid $T^{*}\mathfrak{g}^{*}$ is linearizable
around $S$ if $\mathfrak{g}$ is compact. On the other hand, Theorem 8.7 can
only be applied when $\mathfrak{g}$ is compact, semi-simple, since only in
this case $T^{*}_{S}\mathfrak{g}^{*}$ is integrable by a compact groupoid with
target fibers having vanishing 2nd de Rham cohomology.

###  9.5. Jets of principal type

We discuss the infinitesimal counterpart of the over-symplectic groupoids of
principal type from Subsections 4.5 and 6.2. Let $B\Rightarrow S$ be a
transitive Lie algebroid over a Poisson manifold $(S,\pi_{S})$. Consider the
Lie algebroid fibre product of $B$ with the cotangent Lie algebroid of
$\pi_{S}$, i.e.,

$A_{S}:=T^{*}S\times_{TS}B:=\\{(\alpha,v)\,:\,\pi_{S}^{\sharp}(\alpha)=\rho_{B}(v)\\},$

where the structure is such that the inclusion in the product
$A_{S}\hookrightarrow T^{*}S\times B$ is a Lie algebroid morphism. The
projection $\mu_{S}:=\operatorname{pr}_{T^{*}S}:A_{S}\to T^{*}S$ is a closed
IM 2-form, which is surjective, because $B$ is transitive. The resulting pair
$(A_{S},\mu_{S})$ will be called the first order jet of principal type
associated to $(S,\pi_{S})$ and $B$. Note that $\mathfrak{k}:=\ker\mu_{S}$ can
be identified with $\ker\rho_{B}$ via $\operatorname{pr}_{B}$. As discussed in
Section 9.3, a splitting $l_{B}:B\to\mathfrak{k}$ of $\rho_{B}$ determines a
IM connection 1-form form
$(L_{B},l_{B})\in\Omega^{1}_{\mathrm{IM}}(B,\mathfrak{k})$. Pulling this back
to $A_{S}$, we obtain an IM connection 1-form

$(L:=L_{B}\circ\operatorname{pr}_{B},l:=l_{B}\circ\mathrm{pr}_{B})\in\Omega^{1}_{\mathrm{IM}}(A_{S},\mathfrak{k}),$

which also satisfies $l|_{\mathfrak{k}}=\mathrm{Id}_{\mathfrak{k}}$. So first
order jets of principal type are partially split. The associated coupling data
is described as follows. The splitting $l_{B}$ gives an identification
$B\simeq TS\oplus\mathfrak{k}$, where the anchor becomes
$\operatorname{pr}_{TS}$ and the bracket

$[(X,\xi),(Y,\eta)]_{B}=([X,Y],\Omega(X,Y)+\nabla^{B}_{X}\eta-\nabla^{B}_{Y}\xi+[\xi,\eta]_{\mathfrak{k}}),$
(9.2)

for all $X,Y\in\mathfrak{X}^{1}(S)$, $\xi,\eta\in\Gamma(\mathfrak{k})$. Here,

  1. -

$\Omega$ is $C^{\infty}(S)$-bilinear, so that
$\Omega\in\Omega^{2}(S;\mathfrak{k})$;

  2. -

$\nabla^{B}$ is a connection on $\mathfrak{k}$ preserving
$[\cdot,\cdot]_{\mathfrak{k}}$ with curvature
$R^{\nabla^{B}}=\operatorname{ad}(\Omega)$.

Then, one finds that

$i_{X}L(\alpha,\xi)=\nabla^{B}_{X}(\xi)+\Omega(X,\pi^{\sharp}_{S}(\alpha)),$

and the coupling data (see Definition 7.16) is given by

$\nabla^{L}=\nabla^{B},\qquad U(\alpha,X)=\Omega(\pi_{S}^{\sharp}(\alpha),X).$

The local model of a first order jet of principal type $(A_{S},\mu_{S})$ for
which the transitive Lie algebroid $B$ is integrable has a nice description:
it is a Poisson geometric version of the “symplectic induction” construction
(see e.g. [39]). For this, identify $B$ with the Atiyah algebroid of a
principal $G$-bundle $P$

$B\simeq TP/G.$

Fix a principal connection $\theta\in\Omega^{1}(P)\otimes\mathfrak{g}$. Using
the associated 1-form on $P\times\mathfrak{g}^{*}$

$\widetilde{\theta}\in\Omega^{1}(P\times\mathfrak{g}^{*}),\ \
\widetilde{\theta}_{(x,\xi)}:=\operatorname{pr}_{P}^{*}\langle\theta_{x}|\xi\rangle,$

we build a Dirac structure on $P\times\mathfrak{g}^{*}$ by setting

$\mathbb{L}_{\theta}:=e^{-\mathrm{d}\tilde{\theta}}\left(\operatorname{pr}_{S}^{!}\mathbb{L}_{\pi_{S}}\right)=\\{v+\operatorname{pr}_{S}^{*}(\xi)-i_{v}\mathrm{d}\widetilde{\theta}\,:\,\mathrm{d}\operatorname{pr}_{S}(v)=\pi_{S}^{\sharp}(\xi)\\}.$
(9.3)

This Dirac structure is invariant under the diagonal action of $G$ and
corresponds to a Poisson structure at points in $P\times\\{0\\}$. Hence, there
is a $G$-invariant open set

$P\times\\{0\\}\subset U\subset P\times\mathfrak{g}^{*}$

on which $\mathbb{L}_{\theta}$ is the graph of a Poisson structure
$\pi_{\theta}\in\mathfrak{X}^{2}(U)$. The $G$-action on $(U,\pi_{U})$ is
proper and free and the local model is the quotient Poisson manifold

$(M_{0},\pi_{0}):=(U,\pi_{\theta})/G,\quad
M_{0}\subset\mathfrak{k}^{*}:=P\times_{G}\mathfrak{g}^{*}.$

Note that the action of $G$ on $(U,\pi_{\theta})$ is Hamiltonian with
$G$-equivariant moment map
$\operatorname{pr}_{\mathfrak{g}^{*}}:U\to\mathfrak{g}^{*}$.

######  Example 9.7 (Local model around the Marsden-Weinstein reduction).

Given a proper and free $G$-Hamiltonian action on a Poisson manifold
$(X,\pi_{X})$ with equivariant moment map $\mu:X\to\mathfrak{g}^{*}$, we have
the Marsden-Weinstein reduction

$S:=X/\\!\\!/G=\mu^{-1}(0)/G.$

This is a Poisson submanifold $(S,\pi_{S})$ of the Poisson quotient

$(M,\pi_{M}):=(X,\pi_{X})/G.$

If we equip the principal $G$-bundle $P:=\mu^{-1}(0)\to S$ with a connection
$\theta$, we have the local model above

$(U,\pi_{\theta})\subset(P\times\mathfrak{g}^{*},L_{\theta}).$

The following is proven in [32].

######  Theorem 9.8.

Around $P=\mu^{-1}(0)$, the $G$-Hamiltonian spaces

$(X,\pi_{X},\mu)\quad\textrm{and}\quad(U,\pi_{\theta},\operatorname{pr}_{\mathfrak{g}^{*}})$

are isomorphic. Hence, the quotient Poisson manifold
$(M,\pi_{M}):=(X,\pi_{X})/G$ is linearizable around the Marsden-Weinstein
quotient $(S=\mu^{-1}(0)/G,\pi_{S})$.

######  Remark 9.9.

In the symplectic case $(X,\pi_{X}=(\omega_{X})^{-1})$, the Marsden-Weinstein
reduction is the usual symplectic quotient $(S,\pi_{S}=(\omega_{S})^{-1})$. In
this case, the Dirac structure $\mathbb{L}_{\theta}$ on
$P\times\mathfrak{g}^{*}$ is the graph of the closed 2-form
$\omega_{\theta}:=\operatorname{pr}_{S}^{*}\omega_{S}-\mathrm{d}\tilde{\theta}$,
which is the classical “coupling construction” due to Guillemin, Sternberg and
Weinstein (see, e.g., [39]). It is a standard result in symplectic geometry
that $(U,\omega_{\theta})$ provides a local $G$-equivariant model for
$(X,\omega_{X})$ around $\mu^{-1}(0)$ (see, e.g., [40, Proposition 5.2] or
[48, Theorem 6.1]).

Finally, we recall the following result from [27, Subsection 6.6], which gives
a large class of first jets that are of principal type, hence partially split.

######  Proposition 9.10 ([27]).

Let $(A_{S},\mu_{S})$ be a first jet of a Poisson structure such that
$\mathfrak{k}=\ker\mu_{S}$ is a Lie algebra bundle, whose typical fiber
$(\mathfrak{g},[\cdot,\cdot])$ satisfies

$H^{0}(\mathfrak{g},\mathfrak{g})=H^{1}(\mathfrak{g},\mathfrak{g})=0.$ (9.4)

Then $(A_{S},\mu_{S})$ is of principal type, with transitive Lie algebroid the
Atiyah Lie algebroid of the principal
$\operatorname{Aut}(\mathfrak{g},[\cdot,\cdot])$-bundle of
$\mathfrak{g}$-frames

$P=\\{\varphi:\mathfrak{g}\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{k}_{x}\,:\,x\in
S,\,\varphi\,\textrm{is a Lie algebra isomorphism}\\}.$

In particular, the assumptions hold if $\mathfrak{k}$ is a bundle of semi-
simple Lie algebras.

###  9.6. Codimension one

We now consider in detail the case of first order jets $(A_{S},\mu_{S})$ in
codimension one, i.e., such that $\ker\mu_{S}$ is a line bundle. We will use
the results of [27, Subsection 6.8] concerning IM connections for bundles of
ideals of rank one. If we choose a splitting $l_{0}:A_{S}\to\mathfrak{k}$ of
the short exact sequence of $\mu_{S}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{S}}$$\scriptstyle{l_{0}}$$\textstyle{T^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{0}}$$\textstyle{0}$

we obtain a flat $T^{*}S$-connection $\nabla^{\mathfrak{k}}$ on the line
bundle $\mathfrak{k}$

$\nabla^{\mathfrak{k}}_{\alpha}\xi:=[\tau_{0}(\alpha),\xi],$

and an isomorphism of vector bundles

$A_{S}\simeq T^{*}S\oplus\mathfrak{k}.$

Under this identification the IM-form becomes the projection
$\mu_{S}=\operatorname{pr}_{T^{*}S}$, while the anchor and the Lie bracket are
given by

$\displaystyle\rho_{A_{S}}(\alpha,\xi)=\pi_{S}^{\sharp}(\alpha),$
$\displaystyle[(\alpha,\xi),(\beta,\eta)]_{A_{S}}=([\alpha,\beta]_{\pi_{S}},\lambda_{0}(\alpha,\beta)+\nabla^{\mathfrak{k}}_{\alpha}\eta-\nabla^{\mathfrak{k}}_{\beta}\xi).$

Notice that:

  * •

The flat $T^{*}S$-connection $\nabla^{\mathfrak{k}}$ is independent of the
choice of splitting of $A_{S}$;

  * •

The line bundle $\mathfrak{k}$ is canonically a representation of $T^{*}S$, so
it has a characteristic class (see, e.g., Section 11.1 in [16])

$c_{1}(\mathfrak{k})\in H^{1}_{\pi_{S}}(S).$

In the sequel we will assume, for simplicity, that $\mathfrak{k}$ is
orientable. If we fix a trivialization $\mathfrak{k}\simeq S\times\mathbb{R}$,
then

$\nabla^{\mathfrak{k}}_{\alpha}(f)=\mathscr{L}_{\pi_{S}^{\sharp}(\alpha)}f+i_{V}(\alpha)f,$

for a unique Poisson vector field $V\in\mathfrak{X}^{1}(S)$ representing the
class $c_{1}(\mathfrak{k})$. This vector field is independent of the choice of
splitting of $A_{S}$. If we change the trivialization $\mathfrak{k}\simeq
S\times\mathbb{R}$ by multiplying with a non-zero function $h$, $V$ changes to
$V+\mathrm{d}_{\pi_{S}}\log(h)$.

  * •

The $\mathfrak{k}$-valued 2-vector field $\lambda_{0}$ is a Poisson 2-cocycle
which depends on the choice of splitting $l_{0}$, but its cohomology class
does not

$c_{2}(A_{S}):=[\lambda_{0}]\in H^{2}_{\pi_{S}}(S,\mathfrak{k}).$

If $l:A_{S}\to\mathfrak{k}\simeq S\times\mathbb{R}$ is a second splitting,
then $l=l_{0}+i_{Z}\circ\mu_{S}$, for a unique vector field $Z$ on $S$. Under
this change, $\lambda_{0}$ becomes
$\lambda=\lambda_{0}+\mathrm{d}_{\pi_{S}}Z$.

We can describe all possible IM connection 1-forms of a codimension one jet.

######  Proposition 9.11.

Let $(T^{*}S\oplus\mathbb{R},\mu_{S}=\operatorname{pr}_{T^{*}S})$ be a
codimension one first jet with a choice of trivialization $\mathfrak{k}\simeq
S\times\mathbb{R}$. Then IM connection 1-forms are in 1-to-1 correspondence
with triples $\theta\in\Omega^{1}(S)$, $Z\in\mathfrak{X}(S)$ and $U:T^{*}S\to
T^{*}S$, that satisfy

$\pi_{S}^{\sharp}(\theta)=V,\qquad
i_{\pi_{S}^{\sharp}(\beta)}U(\alpha)=\lambda_{0}(\alpha,\beta)+\mathrm{d}_{\pi_{S}}Z(\alpha,\beta),$

and the structure equations

$\displaystyle i_{\pi^{\sharp}_{S}(\alpha)}\mathrm{d}\theta=0,$ (S2”)
$\displaystyle
U([\alpha,\beta]_{\pi_{S}})=\mathscr{L}_{\pi_{S}^{\sharp}(\alpha)}U(\beta)-i_{\pi^{\sharp}_{S}(\beta)}\mathrm{d}U(\alpha)+$
(S3”)
$\displaystyle\qquad\qquad\qquad\qquad+\pi_{S}(U(\alpha),\beta)\theta+\pi_{S}(\theta,\alpha)U(\beta)-\pi_{S}(\theta,\beta)U(\alpha),$

for all $\alpha,\beta\in\Omega^{1}(S)$.

###### Proof.

As we saw in Section 7.5, IM connection 1-forms $(L,l)$ correspond to coupling
data $(\nabla^{L},U)$ satisfying the structure equations (S1)-(S3) in
Proposition 7.15. As we observed above, the splitting $l:A_{S}\to\mathfrak{k}$
is related to our fixed splitting $l_{0}$ by $l=l_{0}+i_{Z}\circ\mu_{S}$, for
a unique vector field $Z$ on $S$. On the other hand, the connection
$\nabla^{L}$ is given by

$\nabla^{L}_{X}=\mathscr{L}_{X}+\theta(X),$

for some $\theta\in\Omega^{1}(S)$. By Proposition 7.17 and the discussion
preceding the proposition, we see that we can codify the coupling data in
terms of a triple $(\theta,U,Z)$, related to $V$ and $\lambda_{0}$ as in the
statement and we only have to take care of the structure equations. In
codimension one the first structure equation in Proposition 7.15 is always
satisfied. One the other hand, one easily checks that the second and third
structure equations now take the form (S2”) and (S3”). ∎

######  Remark 9.12.

Equation (S2”) has the following interpretation. For the (possibly singular)
symplectic foliation $\mathcal{F}$ of $(S,\pi_{S})$, we denote by
$H^{\bullet}_{\mathcal{F}-\mathrm{bas}}(S)$ the $\mathcal{F}$-basic
cohomology, i.e., the cohomology of differential forms $\eta$ satisfying

$i_{\pi_{S}^{\sharp}(\alpha)}\eta=0\quad\textrm{and}\quad
i_{\pi_{S}^{\sharp}(\alpha)}\mathrm{d}\eta=0,\qquad\forall\,\alpha\in T^{*}S,$

endowed with the de Rham differential. Then (S2”) means that
$\mathrm{d}\theta$ is an $\mathcal{F}$-basic form. If we change the
trivialization, $\theta$ changes to $\theta+\mathrm{d}f$, so
$\mathrm{d}\theta$ stays the same. If we change the IM connection 1-form, then
$\theta$ changes to $\theta+\eta$, where $\eta$ is $\mathcal{F}$-basic. This
shows that, the class

$[\mathrm{d}\theta]\in H^{2}_{\mathcal{F}\textrm{-bas}}(S)$

is independent of the choice of IM connection 1-form and trivialization. It is
easy to see that the vanishing of this class is equivalent to
$c_{1}(\mathfrak{k})$ being in the image of the canonical map

$\pi_{S}^{\sharp}:H^{1}(S)\to H^{1}_{\pi_{S}}(S),$

and also equivalent to the existence of a (usual) flat connection $\nabla$ on
$\mathfrak{k}$ inducing $\nabla^{\mathfrak{k}}$, i.e., such that
$\nabla^{\mathfrak{k}}_{\alpha}=\nabla_{\pi_{S}^{\sharp}(\alpha)}$.

From the proposition, we obtain a simple class of codimension one jets for
which it is easy to decide whether they are partially split:

######  Corollary 9.13.

A Poisson manifold $(S,\pi_{S})$ together with a Poisson vector field
$V\in\mathfrak{X}_{\pi_{S}}(S)$ yield a first jet of a Poisson structure on
$A_{S}=T^{*}S\oplus\mathbb{R}$, with

$\nabla^{\mathfrak{k}}_{\alpha}f=\mathscr{L}_{\pi_{S}^{\sharp}(\alpha)}f+(i_{V}\alpha)\,f\quad\textrm{and}\quad\lambda_{0}\equiv
0.$ (9.5)

This is partially split if and only if $V=\pi_{S}^{\sharp}(\theta)$ for some
1-form $\theta$ with $\mathrm{d}\theta$ is $\mathcal{F}$-basic.

###### Proof.

If $V=\pi_{S}^{\sharp}(\theta)$, with $\mathrm{d}\theta$ $\mathcal{F}$-basic,
then (S2”) holds. Letting $U=0$, (S3”) also holds, so we obtain an IM
connection 1-form. Conversely, if the first order jet (9.5) is partially
split, by (S2”) we have
$c_{1}(\mathfrak{k})=[V]=[\pi_{S}^{\sharp}(\theta_{0})]$, for some 1-from with
$\mathrm{d}\theta_{0}$ $\mathcal{F}$-basic. So there is a function $f$ such
that $V=\pi_{S}^{\sharp}(\theta_{0}+\mathrm{d}f)$, and we can set
$\theta:=\theta_{0}+\mathrm{d}f$. ∎

The next examples fit into the setting of this corollary.

######  Example 9.14.

Consider an orientable log-symplectic manifold $(M^{2n},\pi)$, with singular
locus $S:=(\wedge^{n}\pi)^{-1}(0)$. Then, as shown in [38], $S$ has a tubular
neighborhood $M_{0}\subset S\times\mathbb{R}$ in which

$\pi=\pi_{S}+V\wedge t\partial_{t},$

where $V\in\mathfrak{X}^{1}(S)$ is the restriction to $S$ of the modular
vector field of $\pi$. Under the induced isomorphism $\mathfrak{k}\simeq
S\times\mathbb{R}$, the first order jet along $S$ has classes
$c_{1}(\mathfrak{k})=[V]$ and $c_{2}(A_{S})=[\lambda]=0$. Since $V$ is
everywhere transverse to the symplectic leaves of $\pi_{S}$, this first order
jet is not partially split. Note also that, since $\lambda=0$, the inclusion
$T^{*}S\hookrightarrow A_{S}$ corresponding to the splitting is a Lie
algebroid homomorphism. This supports the claim made in Remark 7.4.

######  Example 9.15.

On $S:=\mathbb{R}^{3}$, consider

$\pi_{S}:=(x\partial_{y}-y\partial_{x})\wedge\partial_{z},\quad
V:=(x^{2}+y^{2})\partial_{z}=\pi_{S}^{\sharp}(\theta_{0}),\quad\textrm{where}\quad\theta_{0}:=x\mathrm{d}y-y\mathrm{d}x.$

The corresponding first order jet is not partially split. Indeed, assume that
$V=\pi_{S}^{\sharp}(\theta)$, for some 1-form $\theta$ with $\mathrm{d}\theta$
$\mathcal{F}$-basic. Since the symplectic foliation $\mathcal{F}$ has
codimenion 1, it follows that $\mathrm{d}\theta=0$, and therefore
$\theta=\mathrm{d}f$, for some $f\in C^{\infty}(\mathbb{R}^{3})$. But the
restriction of $\theta_{0}$ to a leaf $L_{r}:=\\{x^{2}+y^{2}=r^{2}\\}$ is not
exact.

######  Example 9.16.

On the 3-torus $S:=\\{(\varphi_{1},\varphi_{2},\varphi_{3})\,:\,\varphi_{i}\in
S^{1}\\}$, consider

$\pi_{S}:=\partial_{\varphi_{1}}\wedge\partial_{\varphi_{2}},\quad
V:=\pi_{S}^{\sharp}(\theta_{0}),\quad\textrm{where}\quad\theta_{0}=\cos(\varphi_{3})\mathrm{d}\varphi_{1}.$

The corresponding first order jet is not partially split. Again, if we assume
that $V=\pi_{S}^{\sharp}(\theta)$, with $\mathrm{d}\theta$
$\mathcal{F}$-basic, it follows that $\mathrm{d}\theta=0$. Hence, the integral
$\int_{\gamma}\theta$ would depend only on the homology class of
$\gamma:S^{1}\to S$. However, this is not true, since

$\int_{S^{1}\times\\{\varphi_{2}\\}\times\\{\varphi_{3}\\}}\theta=\int_{S^{1}\times\\{\varphi_{2}\\}\times\\{\varphi_{3}\\}}\theta_{0}=2\pi\cos(\varphi_{3}),$

where we used that $\theta-\theta_{0}$ vanishes on the leaves of $\pi_{S}$.

######  Example 9.17.

Let $V\in\mathfrak{X}^{1}(S^{1})$ be the generator of the rotation and
$W\in\mathfrak{X}^{1}(S^{3})$ be the generator of the $S^{1}$-action of the
Hopf fibration $S^{3}\to S^{2}$. The first order jet on $S=S^{1}\times S^{3}$
corresponding to $\pi_{S}=V\wedge W$ and $V$ is partially split. Namely,
$V=-\pi_{S}^{\sharp}(\theta)$, where $\theta$ is a principal connection for
the Hopf fibration. Note that the class

$[\mathrm{d}\theta]\in H^{2}_{\mathcal{F}-\mathrm{bas}}(S)\simeq H^{2}(S^{2})$

is the first Chern class of the Hopf fibration, and so it is non-trivial.
Therefore, by Remark 9.12, $\nabla^{\mathfrak{k}}$ is not induced by a flat
connection and it follows from Proposition 9.18 (iii) below that this first
order jet is not of principal type.

An IM connection 1-form $(L,l)\in\Omega^{1}_{\mathrm{IM}}(A_{S};\mathfrak{k})$
is called kernel flat if the associated connection $\nabla^{L}$ is flat. We
refer to [27] for a detailed discussion of this condition. If $\nabla^{L}$ is
flat, then the third structure equation (S3”) also has a nice geometric
interpretation: it means that $(\mathrm{d}^{\nabla}U,U)$ is a closed IM 2-form
with coefficients in $\mathfrak{k}$. We have the following consequence of [27,
Proposition 6.9].

######  Proposition 9.18.

Let $(A_{S},\mu_{S})$ be a codimension one first order jet inducing a Poisson
structure $\pi_{S}$. Then

  1. (i)

$(A_{S},\mu_{S})$ is isomorphic to the product jet
$(S,\pi_{S})\times\mathbb{R}$ if and only if

$c_{1}(\mathfrak{k})=0\quad\text{and}\quad c_{2}(A_{S})=0;$

  2. (ii)

$(A_{S},\mu_{S})$ admits a kernel flat IM connection 1-form if and only
$\mathfrak{k}$ admits a flat connection $\nabla$ inducing
$\nabla^{\mathfrak{k}}$ and $c_{2}(A_{S})$ is in the image of the canonical
map

$H^{2}_{\mathrm{IM}}(A_{S};\mathfrak{k})\to H^{2}_{\pi_{S}}(S;\mathfrak{k});$

  3. (iii)

$(A_{S},\mu_{S})$ is of principal type if and only if $\mathfrak{k}$ admits a
flat connection $\nabla$ inducing $\nabla^{\mathfrak{k}}$ and the class
$c_{2}(A_{S})$ is in the image of the canonical map

$\pi_{S}^{\sharp}:H^{2}(S;\mathfrak{k})\to H^{2}_{\pi_{S}}(S;\mathfrak{k}).$

######  Example 9.19.

Let $\mathfrak{h}$ be a compact semi-simple Lie algebra, and consider the unit
sphere $S\subset\mathfrak{h}^{*}$, as in the example in Subsection 4.6. Under
the diffeomorphism

$S\times(0,\infty)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathfrak{h}^{*}\backslash\\{0\\},\quad(x,t)\mapsto\frac{1}{t}x,$

the linear Poisson structure on $\mathfrak{h}^{*}$ corresponds to the Poisson
structure $t\cdot\pi_{S}$ on $S\times(0,\infty)$. This yields a splitting of
$A_{S}\simeq T^{*}S\oplus\mathbb{R}$, under which one has

$\lambda(\alpha,\beta)=\pi_{S}(\alpha,\beta),\quad\nabla^{\mathfrak{k}}_{\alpha}=\mathscr{L}_{\pi^{\sharp}_{S}(\alpha)}.$

This first order jet is partially split with coupling data

$\nabla^{L}_{X}=\mathscr{L}_{X},\quad U=-\mathrm{Id}:T^{*}S\to T^{*}S.$

Let $H$ be the 1-connected Lie group integrating $\mathfrak{h}$, In the
example in Subsection 4.6, we have seen that if
$\mathfrak{h}\not\simeq\mathfrak{so}(3,\mathbb{R})$ the over-symplectic
groupoid $\mathcal{G}_{S}=H\ltimes S$ is not of principal type. By applying
Proposition 9.18 (iii), we can see that this is also true at the Lie algebroid
level. Namely, since $H^{1}(S)=0$, any flat connection on $\mathfrak{k}\simeq
S\times\mathbb{R}$ is isomorphic to the trivial one. Since
$\mathfrak{h}\not\simeq\mathfrak{so}(3,\mathbb{R})$, we have that
$H^{2}(S)=H^{2}(S,\mathfrak{k})=0$. If this first jet was of principal type,
then the corollary would imply that $\lambda=\pi_{S}$ is exact. To see that
this is not the case, assume that $X$ is a primitive of $\pi_{S}$. Then
$\mathscr{L}_{X}(\pi_{S})=-\mathrm{d}_{\pi_{S}}X=-\pi_{S}$, and so
$(\phi_{X}^{t})^{*}\pi_{S}=e^{-t}\pi_{S}$. In particular, for each symplectic
leaf $(L,\omega_{L})$, we have that $\phi_{X}^{t}(L)$ is a symplectic leaf
symplectomorphic to $(L,e^{t}\omega_{L})$. Then the symplectic volume of
leaves in the sphere $S$ would be unbounded. It is well-known that this volume
is bounded (see, e.g., [46, Lemma 2.2]).

The previous example, although not globally, is of principal type when
restricted to small open sets. The following example violates even this
condition

######  Example 9.20.

Consider the first order jet $A_{S}\simeq T^{*}S\oplus\mathbb{R}$ on
$S=\mathbb{R}^{2}$ with data

$\lambda=\pi_{S}=(x^{2}+y^{2})\partial_{x}\wedge\partial_{y},\quad V\equiv 0.$

As in the previous example, this first order jet is partially split, with
coupling data

$\nabla^{L}_{X}=\mathscr{L}_{X},\quad U=-\mathrm{Id}.$

We claim that there is no open neighborhood $S_{0}\subset\mathbb{R}^{2}$ of 0
on which the restriction of the jet is of principal type. It is proved in [35]
that the class $[\pi_{S}]$ is non-trivial in the formal Poisson cohomology at
0, so it is non-trivial on any open neighborhood $S_{0}$ of $0$. Since $S_{0}$
can be assumed contractible, the claim follows from Proposition 9.18 (iii).

######  Remark 9.21 (Locally trivial jets).

In the previous example the jet is not locally a trivial product jet. We call
a general jet $(A_{S},\mu_{S})$ locally trivial, if $S$ can be covered by open
subsets on which the restriction of the jet is isomorphic to a trivial product
jet: $A_{S}|_{U}\simeq T^{*}U\times\mathfrak{g}$ (see Example 9.1). We note
the following properties of this class of jets

  1. (i)

First order jets of principal type are locally trivial: any transitive Lie
algebroid $B\Rightarrow S$ is locally isomorphic to a product $B|_{U}\simeq
TU\times\mathfrak{g}$ [25].

  2. (ii)

A partially split first order jet is locally trivial if and only if the local
model is locally isomorphic to a product
$(U,\pi_{S}|_{U})\times(\mathfrak{g}^{*},\pi_{\mathfrak{g}})$ (see Corollary
7.10).

  3. (iii)

In codimension one, a first order jet is locally trivial if and only if the
classes $c_{1}(\mathfrak{k})$ and $c_{2}(A_{S})$ vanish locally (see
Proposition 9.18 (i)).

  4. (iv)

A locally trivial first order jet is not necessarily partially split: Example
9.16 is locally trivial because $c_{2}(A_{S})=0$ and $c_{1}(\mathfrak{k})$
vanishes locally.

  5. (v)

A partially split first order jet is not necessarily locally trivial: the
first order jet from Example 9.20 is partially split, however it is not
locally trivial because $c_{2}(A_{S})$ does not vanish around 0.

In codimension one, Proposition 7.18 gives an explicit form of the local
model.

######  Proposition 9.22.

Let $(A_{S},\mu_{S})$ be a partially split first order jet of a Poisson
structure with $\mathfrak{k}\simeq S\times\mathbb{R}$. For an IM connection
1-form $(L,l)$ with coupling data $(\nabla^{L}=\mathrm{d}+\theta,U)$, the
corresponding local model $(M_{0},\pi_{0})$ is an open neighborhood
$M_{0}\subset S\times\mathbb{R}$ of $S\times\\{0\\}$ with Poisson structure

$\pi_{0}=\gamma_{t}+\gamma_{t}^{\sharp}(\theta)\wedge t\partial_{t},$

where $\gamma_{t}\in\mathfrak{X}^{2}(S_{t})$ is the bivector on the plaque
$S_{t}:=M_{0}\cap(S\times\\{t\\})$ given by

$\gamma^{\sharp}_{t}=\pi_{S}^{\sharp}\circ(\mathrm{Id}+tU)^{-1}.$ (9.6)

The following examples discuss the special cases where either $\theta$ or $U$
vanish.

######  Example 9.23.

If in Proposition 9.22 we assume that $U=0$, then

$\pi_{0}=\pi_{S}+\pi_{S}^{\sharp}(\theta)\wedge
t\partial_{t}\in\mathfrak{X}^{2}(S\times\mathbb{R}).$

Note that these Poisson structures fit into a more general class: for any
Poisson vector field $V$ on $(S,\pi_{S})$, one has a Poisson structure
$\pi_{S}+V\wedge t\partial_{t}$, which has $S$ as a Poisson submanifold, but
its first order jet will in general not be partially split (see Corollary 9.13
or Example 9.14).

######  Example 9.24.

If in Proposition 9.22 we assume that $\theta=0$, then the terms in the second
line of (S3”) vanish $U:T^{*}S\to T^{*}S$ is a closed IM 2-form. In this case,
the local model $(M_{0},\pi_{0})$ can be thought of as a deformation
$\\{\gamma_{t}\\}_{t\in\mathbb{R}}$ of the Poisson structure $\pi_{S}$

$\pi_{0}^{\sharp}|_{(x,t)}=\gamma_{t}^{\sharp}|_{x}=\pi_{S}^{\sharp}\circ(\mathrm{Id}+tU)^{-1}|_{x}.$

If we assume that $\pi_{S}$ is integrable, this deformation has a nice
geometric interpretation. Namely, let $(\Sigma,\omega)\rightrightarrows S$ be
a target 1-connected symplectic groupoid integrating $\pi_{S}$ and let
$\eta\in\Omega^{2}_{\mathrm{M}}(\Sigma)$ be the multiplicative closed 2-form
corresponding to $U$. Then, for small $t$, $(\Sigma,\omega+t\eta)$ is a
symplectic groupoid (at least locally), and the corresponding Poisson
structures is precisely $\gamma_{t}$ (see e.g. [9, Section 3.2.2]).

######  Example 9.25.

Let us consider the particular case of the previous example where $\theta=0$
and $U=-\mathrm{Id}$. The local model becomes simply

$M_{0}=S\times(\mathbb{R}\backslash\\{1\\}),\quad\pi_{0}=\frac{1}{1-t}\pi_{S}.$

We claim that $\pi_{0}$ extends to a smooth Dirac structure on
$\mathfrak{k}^{*}=S\times\mathbb{R}$ if and only if $\pi_{S}$ is a regular
Poisson structure. This is in contrast with what happens for Vorobjev’s local
model around symplectic leaves, which always comes from a global Dirac
structure on $\mathfrak{k}^{*}$ (see, e.g., [44]). One direction is clear: if
$\pi_{S}$ corresponds to a regular foliation $\mathcal{F}$ with leafwise
symplectic form $\omega$, then $\pi_{0}$ can be extended by the regular Dirac
structure with foliation $T\mathcal{F}_{0}:=T\mathcal{F}\times\mathbb{R}$ and
leafwise presymplectic form:
$\omega_{0}=(1-t)\operatorname{pr}_{S}^{*}\omega|_{T\mathcal{F}_{0}}$. For the
converse, we show that the graph of a non-regular $\pi_{0}$ does not extend to
a Dirac structure on $S\times\mathbb{R}$. By invoking Weinstein’s splitting
theorem, we can reduce to the case when $\pi_{S}|_{x_{0}}=0$ and $\pi_{S}$
does not vanish identically in a neighborhood of $x_{0}$. Assume that the
graph of $\pi_{0}^{\sharp}$ extends to a Dirac structure $\mathbb{L}$ on
$S\times\mathbb{R}$. Since
$\mathbb{L}_{(x_{0},t)}=T^{*}_{(x_{0},t)}(S\times\mathbb{R})$ for $t\neq 1$,
by continuity, we must have also that
$\mathbb{L}_{(x_{0},1)}=T^{*}_{(x_{0},1)}(S\times\mathbb{R})$. This implies
that $\mathbb{L}$ corresponds to a Poisson structure $\pi_{\mathbb{L}}$ around
$(x_{0},1)$, which of course must be $\pi_{0}$ outside of $S\times\mathbb{R}$.
This is a contradiction, because $\pi_{0}$ does not extend continuously at
$(x_{0},1)$.

###  9.7. Other local models

The local model for a Poisson manifold $(M,\pi)$ around a Poisson submanifold
$S$ that we have introduced is characterized by the fact that the underlying
Lie algebroid is the linear approximation to $T^{*}M$ around $S$. In some
cases, it is possible that our local model does not exist, but there may exist
some other local model. Many such local models have been considered in the
literature, and they often turn out to be homogeneous of a certain degree. For
a detailed version of the following discussion, see for example the recent PhD
thesis of Aldo Witte [60]. However, these local models have little overlap
with the ones we are considering. Let $(M,\pi)$ be a Poisson manifold and let
$S\subset M$ be a Poisson submanifold. By passing to a tubular neighborhood,
we can replace $M$ by a vector bundle $E\to S$. Denote by $m_{t}(e)=te$ the
fiberwise multiplication by $t$. The bivector field $m_{t}^{*}(\pi)$ has a
Laurent expansion of the following form

$m_{t}^{*}(\pi)\simeq\frac{1}{t}\pi_{-1}+\pi_{0}+t\pi_{1}+\ldots,$

where each term $\pi_{i}\in\mathfrak{X}^{2}(E)$ is homogeneous of degree $i$,
in the sense that

$m_{t}^{*}(\pi_{i})=t^{i}\pi_{i}.$

Let $\pi_{k}$ be the first non-zero term of the sequence
$\\{\pi_{i}\\}_{i\geq-1}$. Then $\pi_{k}$ is itself a Poisson structure and is
independent of the chosen tubular neighborhood (see [60]). The pair
$(E,\pi_{k})$ is the homogeneous local model of $\pi$ around $S$. In most
situations, the homogeneous local model is very different from ours. First of
all, in general, $\pi_{k}$ depends on the $(k+2)$-th jet of the Poisson
structure at $S$. Let us discuss the geometric structure encoded by the first
non-zero term $\pi_{k}$. $k=-1$. Poisson structures on the vector bundle $E$
such that $m_{t}^{*}{\pi_{-1}}=t^{-1}\pi_{-1}$ are in 1-to-1 correspondence
with Lie algebroid structures on $E^{*}$. The condition that $S$ is a Poisson
submanifold is equivalent to $\mathfrak{k}:=E^{*}$ being a bundle of Lie
algebras. Then $\pi_{S}=0$ and $\pi_{-1}$ coincides with the linear Poisson
structure on $\mathfrak{k}^{*}=E$. As we saw in Proposition 9.1, the partial
split condition is satisfied precisely when $\mathfrak{k}$ is a Lie algebra
bundle. In that case $\pi_{-1}$ coincides with our linear model. $k=0$. The
condition $\pi_{-1}=0$ imply that $S$ is a Poisson submanifold for which the
bundle of Lie algebras $\mathfrak{k}:=\ker\mu_{S}$ is abelian. Therefore,
homogeneous Poisson structures exclude many examples where our local model
exists (see Example 9.23 for the case of codimension one). Poisson structures
on the vector bundle $E$ such that $m_{t}^{*}{\pi_{0}}=\pi_{0}$, called
_quadratic Poisson structures_ , are discussed in the recent PhD thesis [47]
(see also [60, Proposition 5.1.55]). Several “weakly degenerate” structures
admit quadratic normal forms around their singularities. As mentioned in
Example 9.14, log-symplectic manifolds admit a quadratic normal form around
the singular locus, and these are not partially split. Elliptic Poisson
structures [11, 60] also admit a quadratic normal form around the singular
locus. $k=1$. In this case, the first order jet is very simple: $A_{S}\simeq
T^{*}S\oplus\mathfrak{k}$ with bracket
$[(\alpha,\xi),(\beta,\eta)]=(0,\lambda(\alpha,\beta))$, for some
$\lambda\in\mathfrak{X}^{2}(S,\mathfrak{k})$. This jet is partially split if
and only if $\lambda\equiv 0$. However, also in this case interesting
geometric structures can appear. For example, if the codimension of $S$ in $M$
is one, there is a 1-to-1 correspondence between Poisson structures $\pi_{1}$
on the line bundle $E$ satisfying $m_{t}^{*}(\pi_{1})=t\pi_{1}$ and Jacobi
structures $J$ on $E$ (see [52]). $k=2$. The local model for scattering
Poisson structures [41] is 2-homogeneous. For $k\geq 2$, the first order jet
of $\pi$ at $S$ vanishes identically, so our local model is the zero Poisson
structure.

##  Appendix A The infinitesimal multiplicative Moser method

Multiplicative forms and infinitesimal multiplicative (IM) forms play an
important role in Poisson Geometry and, in particular, in this paper. In this
appendix we give the main definitions and some basic results concerning such
forms that are needed throughout the paper. We will use the notations and
conventions from [16] and the Appendix of [27], and we refer to [5, 19, 24]
for proofs and details. In the last section of this appendix we give an IM-
version of the well-known Moser method from symplectic geometry, which does
not seem to be known, and which is used in the paper to prove uniqueness of
the local model.

###  A.1. Multiplicative forms

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with source/target
$\mathbf{s},\mathbf{t}:\mathcal{G}\to M$ and multiplication
$m:\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{G}\to\mathcal{G}$.
A differential form $\omega\in\Omega^{k}(\mathcal{G})$ is called
multiplicative if

$m^{*}\omega=\operatorname{pr}_{1}^{*}\omega+\operatorname{pr}_{2}^{*}\omega\in\Omega^{k}(\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{G}),$

where
$\operatorname{pr}_{i}:\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{G}\to\mathcal{G}$
are the projections on the factors. The differential of a multiplicative form
is again a multiplicative form, so we have a complex of multiplicative
differential forms $(\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G}),\mathrm{d})$.
A basic fact is that two multiplicative forms
$\omega,\omega^{\prime}\in\Omega^{k}_{\mathrm{M}}(\mathcal{G})$ such that
$\omega|_{M}=\omega^{\prime}|_{M}$ and
$(\mathrm{d}\omega)|_{M}=(\mathrm{d}\omega^{\prime})|_{M}$ actually coincide.
Denote by $(A,[\cdot,\cdot],\rho)$ the Lie algebroid of
$\mathcal{G}\rightrightarrows M$. Given a multiplicative form
$\omega\in\Omega^{k}_{\mathrm{M}}(\mathcal{G})$ one defines two vector bundle
maps $\mu:A\to\wedge^{k-1}T^{*}M$ and $\zeta:A\to\wedge^{k}T^{*}M$ by setting

$\mu(a)=i_{a}\omega|_{TM},\quad\zeta(a)=i_{a}\mathrm{d}\omega|_{TM}.$ (A.1)

The pair $(\mu,\zeta)$ satisfies the following set of equations for any
$\alpha,\beta\in\Gamma(A)$

$\displaystyle i_{\rho(\beta)}\mu(\alpha)$
$\displaystyle=-i_{\rho(\alpha)}\mu(\beta),$
$\displaystyle\mu([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\rho(\alpha)}\mu(\beta)-i_{\rho(\beta)}\mathrm{d}\mu(\alpha)-i_{\rho(\beta)}\zeta(\alpha),$
(A.2) $\displaystyle\zeta([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\rho(\alpha)}\zeta(\beta)-i_{\rho(\beta)}\mathrm{d}\zeta(\alpha).$

This leads to the notion of infinitesimal multiplicative $k$-form on an
arbitrary Lie algebroid $A\Rightarrow M$ integrable or not: a pair
$(\mu,\zeta)$, where $\mu:A\to\wedge^{k-1}T^{*}M$ and
$\zeta:A\to\wedge^{k}T^{*}M$ are bundle maps satisfying (A.1). The space of IM
forms is denoted by $\Omega_{\mathrm{IM}}^{k}(A)$ and it becomes a cochain
complex with differential given by

$\mathrm{d}_{\mathrm{IM}}:\Omega_{\mathrm{IM}}^{k}(A)\to\Omega_{\mathrm{IM}}^{k+1}(A),\quad\mathrm{d}_{\mathrm{IM}}(\mu,\zeta):=(\zeta,0).$
(A.3)

For a _target 1-connected_ Lie groupoid $\mathcal{G}\rightrightarrows M$ with
Lie algebroid $A\Rightarrow M$ the assignment $\omega\mapsto(\mu,\zeta)$ given
by (A.1) is an isomorphism of complexes

$(\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G}),\mathrm{d})\simeq(\Omega^{\bullet}_{\mathrm{IM}}(A),\mathrm{d}_{\mathrm{IM}}).$

We denote closed IM forms $(\mu,0)$ simply by $\mu$. For us the most relevant
IM-forms will be closed IM 2-forms $\mu:A\to T^{*}M$ which are non-degenerate,
meaning that they are bundle isomorphisms.

######  Example A.1 (IM exact forms).

Any differential form $\omega\in\Omega^{k}(M)$ induces a IM-form

$\rho^{*}(\omega)\in\Omega_{\mathrm{IM}}^{k}(A),$

with components $\rho^{*}(\omega)=(\mu,\zeta)$ given by

$\mu(\alpha)=i_{\rho(\alpha)}\omega\quad\zeta(\alpha)=i_{\rho(\alpha)}\mathrm{d}\omega.$

Forms of this type will be called IM-exact.

######  Example A.2 (IM forms and Poisson structures).

If $\mu:A\to T^{*}M$ is a non-degenerate closed IM 2-form, we obtain a Poisson
structure $\pi\in\mathfrak{X}^{2}(M)$ by letting

$\pi^{\sharp}:=\rho\circ\mu^{-1}:T^{*}M\to TM.$ (A.4)

This in turn induces a cotangent Lie algebroid structure
$(T^{*}M,[\cdot,\cdot]_{\pi},\pi^{\sharp})$, where

$[\alpha_{1},\alpha_{2}]_{\pi}:=\mathscr{L}_{\pi^{\sharp}(\alpha_{1})}\alpha_{2}-\mathscr{L}_{\pi^{\sharp}(\alpha_{2})}\alpha_{1}-\mathrm{d}\pi(\alpha_{1},\alpha_{2}).$

The IM form $\mu:A\to T^{*}M$ then becomes a Lie algebroid isomorphism.
Conversely, a Poisson structure $\pi\in\mathfrak{X}^{2}(M)$ yields a Lie
algebroid structure on the cotangent bundle
$(T^{*}M,[\cdot,\cdot]_{\pi},\pi^{\sharp})$ such that the identity map
$\mathrm{Id}:T^{*}M\to T^{*}M$ is a non-degenerate closed IM 2-form. The
groupoid version of this is as follows: starting with a symplectic groupoid
$(\mathcal{G},\omega)$, i.e., a multiplicative non-degenerate closed 2-form
$\omega\in\Omega^{2}_{\mathrm{M}}(\mathcal{G})$, we obtain

  1. (i)

a non-degenerate closed IM 2-form on its Lie algebroid $\mu:A\to T^{*}M$, and

  2. (ii)

a unique Poisson structure $\pi\in\mathfrak{X}^{2}(M)$ for which the target
map $\mathbf{t}:\mathcal{G}\to M$ is a Poisson map.

Then $\mu$ and $\pi$ are related by (A.4). In this way one recovers the well-
known correspondence between target 1-connected symplectic groupoids and
integrable Poisson manifolds.

######  Example A.3 (IM forms and Dirac structures).

Generalizing the previous example, a closed IM 2-form $\mu:A\to T^{*}M$ is
called robust if the map

$(\rho,\mu):A\to TM\oplus T^{*}M$ (A.5)

has constant rank equal to the dimension of $M$. Then the image of this map

$\mathbb{L}:=(\rho,\mu)(A)\subset TM\oplus T^{*}M$

is a Dirac structure on $M$ [7]. If $\mu$ is robust and $(\rho,\mu)$ is
injective, then we call $\mu$ Dirac non-degenerate. In this case, we have a
Lie algebroid isomorphism
$(\rho,\mu):A\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\mathbb{L}$.
Conversely, for any Dirac structure $\mathbb{L}$, the projection
$\mu:=\operatorname{pr}_{T^{*}M}:\mathbb{L}\to T^{*}M$ is Dirac non-
degenerate.

######  Example A.4 (The canonical IM 2-form).

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with Lie algebroid
$A\Rightarrow M$. The canonical symplectic structure $\omega_{\mathrm{can}}$
on the cotangent bundle $T^{*}\mathcal{G}\rightrightarrows A^{*}$ is
multiplicative. Moreover,
$(T^{*}\mathcal{G},\omega_{\mathrm{can}})\rightrightarrows A^{*}$ is the
symplectic groupoid integrating the linear Poisson structure on $A^{*}$. At
the infinitesimal level, we have the canonical “reverse isomorphism”
$\mu_{\mathrm{can}}:T^{*}A\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}TA^{*}$
(see [43]), which is a closed, non-degenerate IM 2-form
$\mu_{\mathrm{can}}\in\Omega^{2}_{\mathrm{IM}}(T^{*}A)$.

###  A.2. Multiplicative forms with coefficients

We also need to consider multiplicative forms and IM forms with coefficients
in a representation. Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid
and let $p:E\to M$ be a $\mathcal{G}$-representation. We will work with
differential forms on $\mathcal{G}$ with coefficients in $\mathbf{s}^{*}E$
(instead of $\mathbf{t}^{*}E$, as in some of the references), which we denote
by

$\Omega^{\bullet}(\mathcal{G};E):=\Omega^{\bullet}(\mathcal{G};\mathbf{s}^{*}E).$

We refer to these simply as multiplicative forms on $\mathcal{G}$ with
coefficients in $E$. Similarly, we denote by
$\Omega^{\bullet}(\mathcal{G}^{(k)};E)$ the space of differential forms on the
manifold $\mathcal{G}^{(k)}$, of composable $k$-strings of arrows, with values
in the vector bundle
$(\mathbf{s}\circ\operatorname{pr}_{k})^{*}(E)\to\mathcal{G}^{(k)}$, where
$\operatorname{pr}_{i}:\mathcal{G}^{(k)}\to\mathcal{G}$ is the projection onto
the $i$-th component.

######  Definition A.5.

A form $\omega\in\Omega^{\bullet}(\mathcal{G};E)$ is called multiplicative if
it satisfies

$m^{*}\omega|_{g_{1}g_{2}}=\operatorname{pr}_{1}^{*}\omega|_{g_{1}}+\operatorname{pr}_{2}^{*}\omega|_{g_{2}},\quad\forall\,(g_{1},g_{2})\in\mathcal{G}^{(2)}.$

We denote by $\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G};E)$ the space of
$E$-valued multiplicative $k$-forms.

Let us pass to the infinitesimal level, so denote by $A\Rightarrow M$ the Lie
algebroid of $\mathcal{G}\rightrightarrows M$ and $\nabla$ the induced
representation of $A$ on $E$. Now, if
$\omega\in\Omega^{k}_{\mathrm{M}}(\mathcal{G};E)$ is an $E$-valued
multiplicative $k$-form, we define a linear operator
$L:\Gamma(A)\to\Omega^{k}(M;E)$ and a vector bundle map
$l:A\to\wedge^{k-1}T^{*}M\otimes E$ by

$\displaystyle L(\alpha)_{x}(v_{1},\dots,v_{k})$
$\displaystyle:=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Big{|}_{\epsilon=0}\phi_{\alpha^{L}}^{\epsilon}(x)\cdot\omega(\mathrm{d}_{x}\phi_{\alpha^{L}}^{\epsilon}(v_{1}),\dots,\mathrm{d}_{x}\phi_{\alpha^{L}}^{\epsilon}(v_{k})),$
(A.6) $\displaystyle l(a)$ $\displaystyle:=(i_{a}\omega)|_{TM}.$

where $\phi_{\alpha^{L}}^{\epsilon}$ denotes the flow of the left-invariant
vector field $\alpha^{L}$. The resulting operator $L$ is a kind of
differential operator with symbol $l$, in the sense that for any $f\in
C^{\infty}(M)$ and $\alpha\in\Gamma(A)$ it satisfies

$L(f\alpha)=fL(\alpha)+\mathrm{d}f\wedge l(\alpha).$ (A.7)

Furthermore, the pair $(L,l)$ satisfies the following set of equations

$\displaystyle i_{\rho(\alpha)}l(\beta)$
$\displaystyle=-i_{\rho(\beta)}l(\alpha),$ $\displaystyle L([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\alpha}L(\beta)-\mathscr{L}_{\beta}L(\alpha),$
(A.8) $\displaystyle l([\alpha,\beta])$
$\displaystyle=\mathscr{L}_{\alpha}l(\beta)-i_{\rho(\beta)}L(\alpha),$

where, for $\alpha\in\Gamma(A)$ and $\gamma\in\Omega^{k}(M;E)$, we denoted

$\mathscr{L}_{\alpha}\gamma(X_{1},\dots,X_{k}):=\nabla_{\alpha}(\gamma(X_{1},\dots,X_{k}))-\sum_{i=1}^{k}\gamma(X_{1},\dots,[\rho(\alpha),X_{i}],\dots,X_{k}).$

These equations make sense for a general, possibly non-integrable Lie
algebroid.

######  Definition A.6.

Let $(E,\nabla)$ be a representation of a Lie algebroid $A\Rightarrow M$. An
$E$-valued IM $k$-form is a pair $(L,l)$, where
$L:\Gamma(A)\to\Omega^{k}(M,E)$ is a linear map and
$l:A\to\wedge^{k-1}T^{*}M\otimes E$ is a vector bundle map satisfying (A.7)
and (A.2).

We denote by $\Omega^{\bullet}_{\mathrm{IM}}(A;E)$ the space of $E$-valued IM
$k$-forms. For historical reasons, these are also sometime called _Spencer
operators_. For a _target 1-connected_ Lie groupoid
$\mathcal{G}\rightrightarrows M$ with Lie algebroid $A\Rightarrow M$ the
assignment $\omega\mapsto(L,l)$ given by (A.6) gives an isomorphism

$\Omega^{\bullet}_{\mathrm{M}}(\mathcal{G};E)\simeq\Omega^{\bullet}_{\mathrm{IM}}(A;E).$

###  A.3. Multiplicative Cartan calculus

Recall that a vector field $X\in\mathfrak{X}(\mathcal{G})$ is called
multiplicative if its flow $\phi^{t}_{X}:\mathcal{G}\to\mathcal{G}$ is by Lie
groupoid automorphisms. Equivalently, $X:\mathcal{G}\to T\mathcal{G}$ is a
groupoid morphism, and this amounts to the relation

$\mathrm{d}m(X_{g},X_{h})=X_{gh},\quad\forall(g,h)\in\mathcal{G}\tensor[_{\mathbf{s}}]{\times}{{}_{\mathbf{t}}}\mathcal{G}.$

We denote by
$\mathfrak{X}_{\mathrm{M}}(\mathcal{G})\subset\mathfrak{X}(\mathcal{G})$ the
subspace of multiplicative vector fields. A multiplicative vector field
$X\in\mathfrak{X}(\mathcal{G})$ covers vector field
$\sigma_{X}\in\mathfrak{X}(M)$

$\mathbf{s}_{*}(X)=\mathbf{t}_{*}(X)=\sigma_{X}.$

The Lie bracket of multiplicative vector fields is multiplicative. If
$X\in\mathfrak{X}_{\mathrm{M}}(\mathcal{G})$ and
$\omega\in\Omega^{k}_{\mathrm{M}}(\mathcal{G})$ then the contraction
$i_{X}\omega$ is still a multiplicative form. In this way, all the usual
formulas from Cartan calculus restrict to multiplicative objects. At the
infinitesimal level, multiplicative vector fields correspond the Lie algebroid
derivations, i.e., to $\mathbb{R}$-linear maps $D:\Gamma(A)\to\Gamma(A)$ for
which there exists a vector field $\sigma_{D}\in\mathfrak{X}(M)$, called
symbol of $D$, such that

$D(f\alpha)=fD(\alpha)+\mathscr{L}_{\sigma_{D}}(f)\alpha,\quad\forall f\in
C^{\infty}(M),\alpha\in\Gamma(A),$

and which act as derivations of the Lie bracket

$D([\alpha,\beta])=[D(\alpha),\beta]+[\alpha,D(\beta)],\quad\forall\alpha,\beta\in\Gamma(A).$

For example, a section $\beta\in\Gamma(A)$ defines the inner derivation

$D_{\beta}:=[\beta,\cdot],$

with symbol $\sigma_{D_{\beta}}=\rho(\beta)$. Given a multiplicative vector
field $X\in\mathfrak{X}_{\mathrm{M}}(\mathcal{G})$ the corresponding
derivation $D_{X}\in\operatorname{Der}(A)$ has symbol $\sigma_{X}$, and can be
defined by

$(D_{X}(\alpha))^{L}:=[X,\alpha^{L}],$

where $\beta^{L}\in\mathfrak{X}(\mathcal{G})$ denotes the left-invariant
vector field on $\mathcal{G}$ determined by $\beta\in\Gamma(A)$. For a _target
1-connected_ Lie groupoid $\mathcal{G}\rightrightarrows M$ with Lie algebroid
$A\Rightarrow M$ the assignment $X\mapsto D_{X}$ establishes an isomorphism
between the space of multiplicative vector fields
$\mathfrak{X}_{\mathrm{M}}(\mathcal{G})$ and the space of Lie algebroid
derivations $\operatorname{Der}(A)$. For a Lie algebroid $A$ the flow of a
derivation $D\in\operatorname{Der}(A)$ is the unique (local) 1-parameter group
of Lie algebroid automorphisms $\varphi^{t}_{D}:A\to A$ satisfying

$\frac{\mathrm{d}}{\mathrm{d}t}(\varphi^{t}_{D})^{*}(\alpha)=D((\varphi^{t}_{D})^{*}(\alpha)),\quad\varphi^{t}_{D}=\mathrm{Id}.$

It covers the flow $\phi^{t}_{\sigma_{D}}$ of the symbol of $D$ and it is
defined as long as the flow of the symbol is defined. If
$D_{X}\in\operatorname{Der}(A)$ is a derivation corresponding to a
multiplicative vector field $X\in\mathfrak{X}_{\mathrm{M}}(\mathcal{G})$ then
its flow is the 1-parameter group of Lie algebroid automorphisms induced by
the flow of $X$

$\varphi^{t}_{D_{X}}=(\phi^{t}_{X})_{*}:A\to A.$

The assignments mapping multiplicative forms and vector fields to IM forms and
algebroid derivations can be used to transfer the Cartan Calculus on
multiplicative forms to infinitesimal multiplicative forms. Given a IM form
$(\mu,\xi)\in\Omega^{k}_{\mathrm{IM}}(A)$ and a Lie algebroid derivation
$D\in\operatorname{Der}(A)$ with symbol $\sigma_{D}\in\mathfrak{X}(M)$ one has

  1. (i)

the interior product of $(\mu,\xi)$ by $D$ is the IM form of degree $k-1$

$i_{D}(\mu,\xi):=(-i_{\sigma_{D}}\mu,\mathscr{L}_{D}\mu+i_{\sigma_{D}}\xi).$

  2. (ii)

the Lie derivative of $(\mu,\xi)$ along $D$ is the IM form of degree $k$

$\mathscr{L}_{D}(\mu,\xi):=(\mathscr{L}_{D}\mu,\mathscr{L}_{D}\xi).$

Here, for a bundle map $\nu:A\to\wedge^{k}T^{*}M$, we have denoted by
$\mathscr{L}_{D}\nu:A\to\wedge^{k}T^{*}M$ its Lie derivative, which is the
bundle map defined by

$(\mathscr{L}_{D}\nu)(\alpha):=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}(\phi^{t}_{\sigma_{D}})^{*}\nu((\varphi^{t}_{D})_{*}(\alpha))=\mathscr{L}_{\sigma_{D}}(\nu(\alpha))-\nu(D(\alpha)).$

For example, for an inner derivation $D_{\beta}=[\beta,\cdot]$ and any IM form
$(\mu,\xi)$ one obtains

$\mathscr{L}_{D_{\beta}}(\mu,\xi)=\rho^{*}(\mathrm{d}\mu(\beta)-\xi(\beta)),$

i.e., the form is IM-exact, in the sense of Example A.1.

######  Proposition A.7.

The operators
$\mathrm{d}_{\mathrm{IM}}:\Omega_{\mathrm{IM}}^{\bullet}(A)\to\Omega_{\mathrm{IM}}^{\bullet+1}(A)$,
$\mathscr{L}_{D}:\Omega_{\mathrm{IM}}^{\bullet}(A)\to\Omega_{\mathrm{IM}}^{\bullet}(A)$
and
$i_{D}:\Omega_{\mathrm{IM}}^{\bullet}(A)\to\Omega_{\mathrm{IM}}^{\bullet-1}(A)$
satisfy

  1. (i)

$[\mathrm{d}_{\mathrm{IM}},i_{D}]=\mathscr{L}_{D}$;

  2. (ii)

$[\mathscr{L}_{D},\mathscr{L}_{D^{\prime}}]=\mathscr{L}_{[D.D^{\prime}]}$;

  3. (iii)

$[\mathscr{L}_{D},i_{D^{\prime}}]=i_{[D,D^{\prime}]}$;

  4. (iv)

$[i_{D},i_{D^{\prime}}]=0$.

For a symplectic groupoid $(\mathcal{G},\omega)$, the relation
$i_{X}\omega=\eta$ gives a 1-to-1 correspondence between multiplicative vector
fields $X$ and multiplicative 1-forms $\eta$, which sends multiplicative exact
vector fields to multiplicative exact forms. In other words, we have the
commutative diagram

$\textstyle{\mathfrak{X}_{\mathrm{M}}(\mathcal{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{\flat}}$$\textstyle{\Omega^{1}_{\mathrm{M}}(\mathcal{G})}$$\textstyle{\Gamma(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\mu}$$\textstyle{\Omega^{1}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$
(A.9)

where the horizontal rows are isomorphisms and the vertical arrows are the
simplicial differentials

$\displaystyle\delta:\Gamma(A)\to\mathfrak{X}_{\mathrm{M}}(\mathcal{G}),\qquad$
$\displaystyle\delta(\beta)=\beta^{R}-\beta^{L},$
$\displaystyle\delta:\Omega^{1}(M)\to\Omega^{1}_{\mathrm{M}}(\mathcal{G}),\qquad$
$\displaystyle\delta(\theta)=\mathbf{t}^{*}\theta-\mathbf{s}^{*}\theta.$

We give now the infinitesimal version of this result.

######  Lemma A.8.

A closed, non-degenerate IM 2-form $\mu:A\to T^{*}M$ gives a bijection

$\operatorname{Der}(A)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\Omega^{1}_{\mathrm{IM}}(A),\quad
D\mapsto i_{D}(\mu,0),$

which maps inner derivations to IM-exact 1-forms.

###### Proof.

For injectivity, $i_{D}(\mu,0)=0$ is equivalent to the relations

$-i_{\sigma_{D}}\mu(\alpha)=0,\quad\mathscr{L}_{\sigma_{D}}(\mu(\alpha))-\mu(D(\alpha))=0,\quad\forall\alpha\in\Gamma(A).$

Since $\mu$ is a bijection, the first relation implies that $\sigma_{D}=0$.
From the second relation, we obtain then that also $D=0$. For surjectivity,
let $(\nu,\xi)\in\Omega^{1}_{\mathrm{IM}}(A)$. Then we need to find a
derivation $D$ such that, for all $\alpha\in\Gamma(A)$, we have

$-i_{\sigma_{D}}\mu(\alpha)=\nu(\alpha),\quad\mathscr{L}_{\sigma_{D}}(\mu(\alpha))-\mu(D(\alpha))=\xi(\alpha).$

Since $\mu$ is a bijection, these relations have the unique solution

$\sigma_{D}:=-\nu\circ\mu^{-1},\quad
D(\alpha):=\mu^{-1}\big{(}\mathscr{L}_{\sigma_{D}}(\mu(\alpha))-\xi(\alpha)\big{)}.$

The Leibniz rule for $D$ with respect to $\sigma_{D}$ follows because $\mu$
and $\xi$ are linear over $C^{\infty}(M)$. It remains to show that $D$ is a
derivation. First, we compute

$\displaystyle-i_{\rho(D(\alpha))}\mu(\beta)$
$\displaystyle=i_{\rho(\beta)}\mu(D(\alpha))$
$\displaystyle=i_{\rho(\beta)}\big{(}\mathscr{L}_{\sigma_{D}}(\mu(\alpha))-\xi(\alpha)\big{)}$
$\displaystyle=-i_{\rho(\beta)}\mathrm{d}\nu(\alpha)-i_{\rho(\beta)}\xi(\alpha)+i_{\rho(\beta)}i_{\sigma_{D}}\mathrm{d}\mu(\alpha)$
$\displaystyle=\nu([\alpha,\beta])-\mathscr{L}_{\rho(\alpha)}\nu(\beta)-i_{\sigma_{D}}i_{\rho(\beta)}\mathrm{d}\mu(\alpha)$
$\displaystyle=-i_{\sigma_{D}}\mu([\alpha,\beta])+\mathscr{L}_{\rho(\alpha)}i_{\sigma_{D}}\mu(\beta)-i_{\sigma_{D}}i_{\rho(\beta)}\mathrm{d}\mu(\alpha)$
$\displaystyle=-i_{\sigma_{D}}\mathscr{L}_{\rho(\alpha)}\mu(\beta)+\mathscr{L}_{\rho(\alpha)}i_{\sigma_{D}}\mu(\beta)$
$\displaystyle=i_{[\rho(\alpha),\sigma_{D}]}\mu(\beta),$

where we used the IM conditions for $(\mu,0)$ and $(\nu,\xi)$. Since $\mu$ is
a bijection, we obtain the relation

$\rho(D(\alpha))=[\sigma_{D},\rho(\alpha)].$ (A.10)

Using this, we compute

$\displaystyle\mu([D(\alpha),\beta])$ $\displaystyle=-\mu([\beta,D(\alpha)])$
$\displaystyle=-\mathscr{L}_{\rho(\beta)}\mu(D(\alpha))+i_{\rho(D(\alpha))}\mathrm{d}\mu(\beta)$
$\displaystyle=\mathscr{L}_{\rho(\beta)}\xi(\alpha)-\mathscr{L}_{\rho(\beta)}\mathscr{L}_{\sigma_{D}}\mu(\alpha)+i_{[\sigma_{D},\rho(\alpha)]}\mathrm{d}\mu(\beta)$
$\displaystyle\mu([\alpha,D(\beta)])$
$\displaystyle=\mathscr{L}_{\rho(\alpha)}\mu(D(\beta))-i_{\rho(D(\beta))}\mathrm{d}\mu(\alpha)$
$\displaystyle=-\mathscr{L}_{\rho(\alpha)}\xi(\beta)+\mathscr{L}_{\rho(\alpha)}\mathscr{L}_{\sigma_{D}}\mu(\beta)-i_{[\sigma_{D},\rho(\beta)]}\mathrm{d}\mu(\alpha)$
$\displaystyle\mu(D([\alpha,\beta]))$
$\displaystyle=\mathscr{L}_{\sigma_{D}}(\mathscr{L}_{\rho(\alpha)}\mu(\beta)-i_{\rho(\beta)}\mathrm{d}\mu(\alpha))-\xi([\alpha,\beta]).$

For the terms containing $\xi$, we have

$\displaystyle\xi([\alpha,\beta])-\mathscr{L}_{\rho(\alpha)}\xi(\beta)+\mathscr{L}_{\rho(\beta)}\xi(\alpha)$
$\displaystyle=\mathrm{d}i_{\rho(\beta)}\xi(\alpha)$
$\displaystyle=\mathrm{d}\big{(}i_{\rho(\beta)}\mathscr{L}_{\sigma_{D}}\mu(\alpha)-i_{\rho(\beta)}\mu(D(\alpha))\big{)}$
$\displaystyle=\mathrm{d}\big{(}i_{\rho(\beta)}\mathscr{L}_{\sigma_{D}}\mu(\alpha)+i_{[\sigma_{D},\rho(\alpha)]}\mu(\beta)\big{)}$

So, we obtain

$\displaystyle\mu\big{(}$ $\displaystyle
D([\alpha,\beta])-[D(\alpha),\beta]-[\alpha,D(\beta)]\big{)}=$
$\displaystyle=\big{(}-\mathscr{L}_{\sigma_{D}}i_{\rho(\beta)}\mathrm{d}-\mathrm{d}i_{\rho(\beta)}\mathscr{L}_{\sigma_{D}}+\mathscr{L}_{\rho(\beta)}\mathscr{L}_{\sigma_{D}}+i_{[\sigma_{D},\rho(\beta)]}\mathrm{d}\big{)}\mu(\alpha)$
$\displaystyle+\big{(}\mathscr{L}_{\sigma_{D}}\mathscr{L}_{\rho(\alpha)}-\mathrm{d}i_{[\sigma_{D},\rho(\alpha)]}-i_{[\sigma_{D},\rho(\alpha)]}\mathrm{d}-\mathscr{L}_{\rho(\alpha)}\mathscr{L}_{\sigma_{D}}\big{)}\mu(\beta)=0,$

which, by injectivity of $\mu$, implies that $D$ is indeed a derivation. Let
$D_{\beta}=[\beta,\cdot]$ be an inner derivation associated with
$\beta\in\Gamma(A)$, and denote $(\nu,\xi):=i_{D_{\beta}}(\mu,0)$. Using the
IM-conditions for $\mu$, we obtain

$\displaystyle\nu(\alpha)$
$\displaystyle=-i_{\rho(\beta)}\mu(\alpha)=i_{\rho(\alpha)}\mu(\beta),$
$\displaystyle\xi(\alpha)$
$\displaystyle=\mathscr{L}_{\rho(\beta)}(\mu(\alpha))-\mu([\beta,\alpha])=i_{\rho(\alpha)}\mathrm{d}\mu(\beta),$

for all $\alpha\in\Gamma(A)$. This is equivalent to

$i_{D_{\beta}}(\mu,0)=\rho^{*}\mu(\beta),$ (A.11)

which proves that $i_{D_{\beta}}(\mu,0)$ is an IM-exact 1-form. ∎

For Dirac structures the situation is similar, but slightly more involved. At
the groupoid level, for a presymplectic groupoid $(\mathcal{G},\omega)$, in
the commutative diagram (A.9) the horizontal rows are no more isomorphisms.
Instead they define a quasi-isomorphism, i.e., the induced map in cohomology
is an isomorphism

$\mathfrak{X}_{\mathrm{M}}(\mathcal{G})/\delta(\Gamma(A))\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\Omega^{1}_{\mathrm{M}}(\mathcal{G})/\delta(\Omega^{1}(M)).$
(A.12)

This amounts to the fact that we have a short exact sequence of vector spaces

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{X}_{\mathrm{M}}(\mathcal{G})\oplus\Omega^{1}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{\mathrm{M}}(\mathcal{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
(A.13)

where

$\displaystyle\Gamma(A)\to\mathfrak{X}_{\mathrm{M}}(\mathcal{G})\oplus\Omega^{1}(M),\qquad$
$\displaystyle\beta\mapsto(\delta(\beta),\mu(\beta)),$
$\displaystyle\mathfrak{X}_{\mathrm{M}}(\mathcal{G})\oplus\Omega^{1}(M)\to\Omega^{1}_{\mathrm{M}}(\mathcal{G}),\qquad$
$\displaystyle(X,\theta)\mapsto i_{X}\omega-\delta(\theta).$

######  Remark A.9.

Given a closed multiplicative 2-form $\omega$ on a Lie groupoid $\mathcal{G}$
it is proved in [23, Section 5.2] that the non-degeneracy condition
$\ker\omega\cap\ker\mathrm{d}\mathbf{t}\cap\ker\mathrm{d}\mathbf{s}=0$ is
equivalent to the contraction $\omega^{\flat}:T\mathcal{G}\to
T^{*}\mathcal{G}$ being a VB Morita map. The authors show that such a map
induces an isomorphism between the fiberwise linear VB cohomologies. The two
quotients in (A.12) can be identified with the fiberwise linear VB
cohomologies in degree one of $T^{*}\mathcal{G}$ (left side) and
$T\mathcal{G}$ (right side). Hence, the isomorphism (A.12) follows from the
results in [23].

At the algebroid level, we have the corresponding statement.

######  Lemma A.10.

Let $\mu:A\to T^{*}M$ be a Dirac non-degenerate, closed IM 2-form. We have a
short exact sequence of vector spaces

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Der}(A)\oplus\Omega^{1}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{1}_{\mathrm{IM}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
(A.14)

where

$\displaystyle\Gamma(A)\to\operatorname{Der}(A)\oplus\Omega^{1}(M),\qquad$
$\displaystyle\beta\mapsto([\beta,\cdot],\mu(\beta)),$
$\displaystyle\operatorname{Der}(A)\oplus\Omega^{1}(M)\to\Omega^{1}_{\mathrm{IM}}(A),\qquad$
$\displaystyle(D,\theta)\mapsto i_{D}(\mu,0)-\rho^{*}(\theta).$

Moreover, the choice of a complement in $TM\oplus T^{*}M$ of the corresponding
Dirac structure $\mathbb{L}$ induces a splitting of (A.14).

###### Proof.

That the sequence is indeed a cochain complex follows from (A.11)

$\beta\mapsto(D_{\beta},\mu(\beta))\mapsto
i_{D_{\beta}}(\mu,0)-\rho^{*}(\mu(\beta))=0.$

We start by checking exactness at $\Gamma(A)$. Let $\beta\in\Gamma(A)$ be in
the kernel if the first map. Then $[\beta,\cdot]=0$, which implies that
$\rho(\beta)=0$. Since also $\mu(\beta)=0$ and the map $(\rho,\mu)$ is
injective, we have that $\beta=0$. We show now surjectivity of the last map.
For this, consider $(\nu,\xi)\in\Omega^{1}_{\mathrm{IM}}(A)$, and we try to
solve

$(\nu,\xi)=i_{D}(\mu,0)-\rho^{*}(\theta),$ (A.15)

with $D\in\operatorname{Der}(A)$ and $\theta\in\Omega^{1}(M)$. The first
component can be written as

$(\rho,\mu)^{*}(\theta,\sigma_{D})=-\nu.$ (A.16)

By assumption, $(\rho,\mu)$ is injective, therefore its dual map is
surjective. So a solution to the above equation exists, and we fix such a
solution $(\theta,\sigma_{D})$. The derivation $D$ will be determined by the
following equations

$\left\\{\begin{array}[]{l}\rho(D(\alpha))=[\sigma_{D},\rho(\alpha)]\\\ \\\
\mu(D(\alpha))=\mathscr{L}_{\sigma_{D}}(\mu(\alpha))-i_{\rho(\alpha)}\mathrm{d}\theta-\xi(\alpha)\end{array}\right.$
(A.17)

where the first equation holds for any derivation and the second is the second
from (A.15). By assumption, the map $(\rho,\mu)$ is an isomorphism from $A$ to
the Dirac structure $\mathbb{L}$. Therefore, if we show that the right-hand
side of the above set of equations defines an element in $\mathbb{L}$, then it
follows that the system determines a unique $D(\alpha)\in\Gamma(A)$, for any
$\alpha\in\Gamma(A)$. We need to check that

$[\sigma_{D},\rho(\alpha)]+\mathscr{L}_{\sigma_{D}}(\mu(\alpha))-i_{\rho(\alpha)}\mathrm{d}\theta-\xi(\alpha)\in\mathbb{L}$

for all $\alpha\in\Gamma(A)$. Since $\mathbb{L}$ is Lagrangian, this is
equivalent to

$\langle[\sigma_{D},\rho(\alpha)],\mu(\beta)\rangle+\langle\rho(\beta),\mathscr{L}_{\sigma_{D}}(\mu(\alpha))\rangle=\langle\rho(\beta),i_{\rho(\alpha)}\mathrm{d}\theta+\xi(\alpha)\rangle,$

for all $\alpha,\beta\in\Gamma(A)$. This is proven as follows

$\displaystyle\langle[\sigma_{D},\rho(\alpha)],\mu(\beta)\rangle+\langle\rho(\beta),\mathscr{L}_{\sigma_{D}}(\mu(\alpha))\rangle=-\mathscr{L}_{\rho(\alpha)}\langle\sigma_{D},\mu(\beta)\rangle$
$\displaystyle+\langle\sigma_{D},\mathscr{L}_{\rho(\alpha)}\mu(\beta)\rangle+i_{\rho(\beta)}\mathrm{d}\langle\sigma_{D},\mu(\alpha)\rangle-\langle\sigma_{D},i_{\rho(\beta)}\mathrm{d}\mu(\alpha)\rangle$
$\displaystyle=(i_{\sigma_{D}}\mu)([\alpha,\beta])-\mathscr{L}_{\rho(\alpha)}(i_{\sigma_{D}}\mu)(\beta)+i_{\rho(\beta)}\mathrm{d}(i_{\sigma_{D}}\mu)(\alpha)$
$\displaystyle=i_{\rho(\beta)}\big{(}\xi(\alpha)+i_{\rho(\alpha)}\mathrm{d}\theta\big{)}$