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Poisson submanifolds

Recall that for Poisson manifolds $(M_1, \Pi_1)$ , $(M_2, \Pi_2)$ $\vf\: M_1\to M_2$ is a Poisson map if and only if $\vf^{\land 2}_{*,x}(\Pi_1(x))=\Pi_2(\vf(x))$ for all $x\in M_1$ .

Recall that a submanifold of is a pair where is a manifold and $i\: N\hookrightarrow M$ is an injective immersion.
$\begin{defn} Let $(M, \Pi_M)$\ be a Poisson manifold. Then $(N, i)$\ is is a \te... ...N$\ has a Poisson structure $\Pi_N$\ such that $i$\ is a Poisson map. \end{defn}$

$\begin{remark} If $i$\ is an immersion, then $i^{\land 2}_{*,x}$\ is injective a... ... a Poisson manifold are a natural example of Poisson submanifolds. \end{remark}$

$\begin{prop} Every open subset $U$\ of $(M, \Pi_M)$\ is an open Poisson submanif... ...Pi_M)$\ is Poisson if and only if it is a union of symplectic leaves. \end{prop}$

$\begin{proof} From $\Pi$\ being Poisson we have $\Pi\vert _U$\ is Poisson for al... ...N$\ is contained in $N$. Now apply the usual open-closed argument. \end{proof}$

$\begin{example} When $M$\ is a symplectic manifold the only Poisson submanifolds... ... a Poisson manifold was recently introduced (see \cite{crf,xup2}). \end{example}$
An interesting way to construct a Poisson manifold with prescribed Poisson submanifolds is that of gluing symplectic structures on given symplectic leaves. The following theorem ([71], page 26, gives a characterization for such construction. Let us remark that in general using topological constructions in the differential geometrical setting of Poisson manifold is, at the same time, an interesting and difficult procedure, related to what is called flexibility of the geometrical structure. A construction of suspension of Poisson structures on spheres was realized in [7]. For other constructions and some general consideration see [47,23].
$\begin{prop} Let $M$\ be a differentiable manifold and let $\sF$\ be a generaliz... ...ture on $M$\ having symplectic $(F,\omega_F)$\ as symplectic leaves. \end{prop}$

$\begin{example} Let $D=D(0,1)$\ be the disc and let $\Pi_1$, $\Pi_2$\ be Poisson... ...artition of unity argument you can prove smooth gluing. \end{proof}\end{example}$

Pawel Witkowski 2006-06-26