\documentclass[11pt,leqno]{book} \textwidth 4.7in \textheight 7.5in \usepackage{amsmath,amscd,theorem} \usepackage{epsfig} \usepackage{amssymb} \usepackage[all]{xy} \input{jdg.sty} \renewcommand{\thepage}{\footnotesize\arabic{page}} \renewcommand{\thefootnote}{}{} \newcommand{\qed}{\quad q.e.d.} \vfuzz2pt % Don't report over-full v-boxes if over-edge is small \hfuzz2pt % Don't report over-full h-boxes if over-edge is small % THEOREMS --------------------------------------------------------------- \newtheorem{THM}{{\!}}[section] \newtheorem{THMX}{{\!}} \renewcommand{\theTHMX}{} % not numbered % \newtheorem{thm}{Theorem}[subsection] \newtheorem{thmx}{Theorem}[subsection] \renewcommand{\thethmx}{} % not numbered \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \newtheorem{ex}[thm]{Example} \numberwithin{equation}{section} % MATH ------------------------------------------------------------------- \newcommand{\Rr}{\mathbb R} \newcommand{\Zz}{\mathbb Z} \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\seq}[1]{\left\langle#1\right\rangle} \newcommand{\eps}{\varepsilon} \newcommand{\al}{\alpha} \newcommand{\e}{\mathbf{e}} \newcommand{\To}{\longrightarrow} \newcommand{\BX}{\mathbf{B}(X)} \newcommand{\A}{\mathcal{A}} \newcommand{\D}{\mathcal{D}} \newcommand{\F}{\mathcal{F}} \newcommand{\Ho}{\mathcal{H}} \newcommand{\M}{\mathcal{M}} \newcommand{\N}{\mathcal{N}} \newcommand{\X}{\mathcal{X}} \newcommand{\Lie}{\mathcal{L}} \newcommand{\Comp}{\mathcal{K}} \newcommand{\Basis}{\mathcal{B}} \newcommand{\Agerm}{\mathfrak{Aut}} \newcommand{\Ogerm}{\mathfrak{Out}} \newcommand{\Dgerm}{\mathfrak{Diff}} \renewcommand{\gg}{\mathfrak{g}} \newcommand{\hh}{\mathfrak{h}} \newcommand{\dd}{\mathfrak{d}} \newcommand{\kk}{\mathfrak{k}} \newcommand{\mm}{\mathfrak{m}} \newcommand{\gl}{\mathfrak{gl}} \newcommand{\Exp}{\textrm{Exp}\,} \newcommand{\Ker}{\textrm{Ker}~} \newcommand{\Ad}{\textrm{Ad}\,} \newcommand{\ad}{\textrm{ad}\,} \newcommand{\Aut}{\textrm{Aut}\,} \newcommand{\Out}{\textrm{Out}\,} \newcommand{\Inn}{\textrm{Inn}\,} \newcommand{\rank}{\textrm{rank}\,} \newcommand{\tr}{\textrm{tr}\,} % COMMENTS % -------------------------------------------------------------- % -- This will show our comments. %\newcommand {\comment}[1]{{\marginpar{*}\scriptsize{\ #1 \ }}} % -- This will hide our comments. \newcommand{\comment}[1]{} %%% --------------------------------------------------- \setcounter{page}{303} \begin{document} \title{Connections in Poisson Geometry I:\\ Holonomy and Invariants} \author{Rui Loja Fernandes} \pagestyle{myheadings} \markboth{\centerline{\normalsize\sc rui loja fernandes}} {\centerline{\sc connections in poisson geometry}} \footnote{Received February 10, 2000, and, in revised form, February 22, 2000. Supported in part by FCT grant PCEX/C/MAT/44/96 and PRAXIS XXI through the Research Units Pluriannual Funding Program.} %%% ---------------------------------------------------------------------- \abstract{We discuss contravariant connections on Poisson manifolds. For vector bundles, the corresponding operational notion of a contravariant derivative had been introduced by I.~Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds and we use them to define Poisson holonomy and new invariants of Poisson manifolds. } \section*{Introduction} \renewcommand{\thefootnote}{\footnotesize\arabic{footnote}} \setcounter{footnote}{0} Let $M$ be a Poisson manifold and suppose that we require the existence of a linear connection on $M$, compatible with the Poisson tensor $\Pi$. Since parallel transport preserves the rank of the Poisson tensor, the Poisson manifold must be regular in order for such a connection to exist. Therefore, the usual notion of a covariant connection is not appropriate for the study of Poisson manifolds, as some of the most interesting examples of Poisson manifolds are non-regular. For non-regular Poisson manifolds the symplectic foliation is singular and the dimension of the leaves varies, so one can only hope to compare tangent spaces at different points of the same symplectic leaf. One possible way around this difficulty is to use families of connections para\-me\-terized by the leaves. However, there are examples showing that the symplectic foliation can be wild, so the space of leaves will not be easy to parameterize. A much more efficient and direct approach, to be introduced in this paper, is through the notion of a contravariant connection, a concept that mimics for the case of Poisson manifolds the usual notion of a covariant connection. Assume we are given a principal bundle over a manifold $M$: \[ \xymatrix{P \ar@(ur,dr)^G \ar[d]_p\\ M} \] then a covariant connection $\Gamma$ on this principal bundle is defined by a $G$-invariant horizontal distribution $u\mapsto H_u$ in $P$. Given a connection $\Gamma$, we have a notion of \emph{horizontal lift}: $h(u,v)\in T_u P$ is the unique tangent vector to $H_u$ which projects to the vector $v\in T_{p(u)}M$. Conversely, the horizontal lift $h$ defines the horizontal distribution $H_u=\set{h(u,v): v\in T_{p(u)}M}$, so $h$ completely determines the connection. We shall define a contravariant connection on a principal bundle over a Poisson manifold by defining analogously the horizontal lift of cotangent vectors. To formulate this notion, observe that $h$ is defined precisely for pairs $(u,v)$ in $p^*TM$, the pullback bundle by $p$ of the tangent bundle over $M$. Denote by $\widehat{p}:p^*TM\to TM$ the induced bundle map so we have the commutative diagram %$$\begin{CD} %\end{CD} %$$ \[ \xymatrix{ p^*TM\ar[r]^{\widehat{p}}\ar[d]_{\widehat{\pi}}& TM \ar[d]^{\pi} \\ P\ar[r]_{p} &M } \] Then we can define a covariant connection to be a bundle map $h:p^*TM\to TP$, such that: \begin{enumerate} \item[(CI)] $h$ is horizontal, i.~e., the following diagram commutes: \[ \xymatrix{ p^*TM\ar[r]^{h}\ar[d]_{\widehat{p}}& TP \ar[d]^{p_*} \\ TM\ar[r]_{\text{id}} &TM } \] \item[(CII)] $h$ is $G$-invariant: $h(ua,v)=(R_a)_*h(u,v)$, for all $a\in G$; \end{enumerate} Assume now that $M$ is a Poisson manifold. According to a general philosophical principle, in Poisson geometry sometimes the cotangent bundle plays the role of the tangent bundle. Hence, we replace $TM$ by $T^*M$ in the diagrams above, whenever it makes sense. Thus we are lead to the notion of a \emph{contravariant connection} on a Poisson manifold: this is a bundle map $h:p^*T^*M\to TP$, such that: \begin{enumerate} \item[(CI)$^*$] The following diagram commutes: \[ \xymatrix{ p^*T^*M\ar[r]^{h}\ar[d]_{\widehat{p}}& TP \ar[d]^{p_*} \\ T^* M\ar[r]_{\#} &TM } \] where $\#:T^*M\to TM$ is the bundle map induced by the Poisson tensor; \item[(CII)$^*$] $h$ is $G$-invariant: $h(u a,\al)=(R_a)_*h(u,\al)$, for all $a\in G$; \end{enumerate} Given a point $x$ in $M$ and a covector $\al\in T^*_x M$, the vector $h(u,\al)\in T_u P$ will be called the \emph{horizontal lift} of $\al$ to the point $u$ in the fiber over $x$. On any fibration one can also consider \emph{generalized contravariant connections} which satisfy only (CI)$^*$. With such a definition at hand one can then develop the usual concepts of pa\-rallelism, curvature, holonomy, geodesic, etc. In particular, for a contravariant connection on a vector bundle $p:E\to M$, one obtains in a way entirely analogous to the covariant case, the notion of a \emph{contravariant derivative} operator $D$: for each 1-form $\al$ on $M$, $D_\al$ maps sections of $E$ to sections of $E$ and satisfies \begin{enumerate} \item[i)] $D_{\al+\beta}\phi=D_\al\phi+D_\beta\phi$; \item[ii)] $D_\al(\phi+\psi)=D_\al\phi+D_\al\psi$; \item[iii)] $D_{f\al}=fD_\al\phi$; \item[iv)] $D_\al(f\phi)=fD_\al\phi+\#\al(f)\phi$; \end{enumerate} where $\al,\beta\in \Omega^1(M)$, $\phi$, $\psi$ are sections of $E$, and $f\in C^{\infty}(M)$. Conversely, every such operator is induced by a contravariant connection. Moreover, one can show that there always exists a linear connection preserving the Poisson tensor. In \cite{Vaisman:art:1} Vaisman introduces the notion of contravariant derivative using i)-iv) as axioms. In spite of its formal similarities with covariant connections, there are striking differences in contravariant Poisson geometry. For example, the holonomy of a connection may be non-discrete when the connection is flat, contravariant connections cannot be pushed back or forward, etc. However, just like in ordinary geometry, contravariant connections are useful to study global properties of Poisson manifolds. Recall that the local structure of a Poisson manifold is given by the Weinstein splitting theorem, also known as the generalized Darboux theorem (see \cite{Weinstein:article:1}, Thm.~2.1). In a neighborhood of a point, the Poisson structure splits as a direct product of a symplectic structure and a Poisson structure which vanishes at the point. So on the normal space to each symplectic leaf we have a notion of \emph{transverse Poisson structure}. In global Poisson geometry one would like to understand the geometry and topology of the symplectic foliation. Using generalized contravariant connections we show that we have a notion of \emph{Poisson holonomy} of the symplectic foliation, analogous to the holonomy in the theory of regular foliations. The corresponding linear holonomy coincides with the \emph{linear Poisson holonomy} introduced by Ginzburg and Golubev in \cite{Ginzburg:article:1}. The Poisson holonomy map is by Poisson automorphisms of the transverse Poisson structure. Poisson holonomy is not homotopy invariant, but factoring out the inner Poisson automorphisms one obtains a notion of \emph{reduced Poisson holonomy} invariant by homotopy, and we can prove the following analogue of the Reeb stability theorem: \begin{thmx}\hspace*{-2.8mm}{\bf .} Let $S$ be a compact, transversely stable leaf, with finite reduced Poisson holonomy. Then $S$ is stable, i.~e., $S$ has arbitrarily small neighborhoods which are invariant under all hamiltonian automorphisms. Moreover, each symplectic leaf of $M$ near $S$ is a bundle over $S$ whose fiber is a finite union of symplectic leaves of the transverse Poisson structure. \end{thmx} We also discuss another related notion of holonomy, which we call \emph{strict Poisson holonomy}, and which allows one to discuss global splitting of an entire neighborhood of a symplectic leaf. The corresponding stability theorem states that if $S$ is transversely stable and has finite strict Poisson holonomy, there is a neighborhood of $S$ which is Poisson covered by a product $\tilde{S}\times N$, with $\tilde{S}$ a finite cover of $S$. Linear Poisson holonomy in turn can be discussed from the point of view of linear contravariant connections and, for each symplectic leaf, there is a notion of Bott contravariant connection. For a non-regular Poisson manifold, we do not have a normal bundle (over the whole of $M$) to the symplectic foliation. However, there is an appropriate notion of a basic connection on $M$: these are linear contravariant connections which preserve the Poisson tensor and restrict in each leaf to the Bott contravariant connection. Comparing a basic connection to a riemannian connection one is lead to ``exotic" or secondary Poisson characteristic classes. These are Poisson cohomology classes which give information on both the Poisson geometry and the topology of the symplectic foliation of $M$. In degree 1, this class actually coincides with the \emph{modular class} of $M$. This invariant was discussed recently by Weinstein in \cite{Weinstein:article:2}, where he shows that the modular class is an obstruction to the existence of measures in $M$ invariant under the hamiltonian flows. As a final note we remark that the most general setup for contravariant connections is in the context of Lie algebroids. Although we have omitted any references to Lie Algebroids, the results discussed here should go through without any major changes, and this will be discussed elsewhere. In a follow up to this paper (\cite{Fernandes:article:2}) we will discuss invariant connections. \section*{Acknowledgements} This paper was certainly influenced by some remarks made by Alan Weinstein after his talk at the Omega 99 Conference held in Lisbon, which showed he had the complete picture on contravariant connections on his mind. I also would like to thank my colleagues Ana Cannas da Silva and Miguel Abreu for additional comments and discussions. \section{Contravariant connections on principal bundles} \subsection{Contravariant cartan calculus} On a Poisson manifold there is a calculus on contravariant objects, analogous to the usual Cartan calculus on differential forms. We recall here some of the formulas and fix notation and conventions for later use. Proofs of the results stated in this introductory paragraph can be found in Vaisman's monograph \cite{Vaisman:book:1}. Let $M$ be a Poisson manifold and denote by $\Pi\in\X^2(M)$(\footnote{We denote by $\Omega^r(M)$ and $\X^r(M)$, respectively, the spaces of differential $r$-forms and $r$-multivector fields on a manifold $M$.}) the Poisson bivector field, so the Poisson bracket on $M$ is given by \begin{equation} \set{f_1,f_2}=\Pi(df_1,df_2),\qquad f_1,f_2\in C^\infty(M). \end{equation} We also have a bundle map $\#:T^*M\To TM$ defined by \begin{equation} \beta(\#\al)=\Pi(\al,\beta),\qquad \al,\beta\in T^*M. \end{equation} On the space of differential 1-forms $\Omega^1(M)$ the Poisson tensor induces a Lie bracket \begin{equation} \label{Lie:bracket:forms} [\al,\beta]=\Lie_{\#\al}\beta-\Lie_{\#\beta}\al-d(\Pi(\al,\beta)), \qquad \al,\beta\in \Omega^1(M), \end{equation} and for this Lie bracket and the usual Lie bracket on vector fields, the map $\#:\Omega^1(M)\To\X^1(M)$ is a Lie algebra homomorphism: \begin{equation} \label{eq:musical:homomorphism} \#[\al,\beta]=[\#\al,\#\beta]. \end{equation} We denote as usual by $X_f=\#(df)$ the hamiltonian vector field associated with the function $f\in C^\infty(M)$, and we have \begin{equation} \label{eq:bracket:function} [\al,f\beta]=f[\al,\beta]+\#\al(f)\beta=f[\al,\beta]-\left(i_{X_f}\al\right) \beta. \end{equation} The existence of a Lie bracket on the space of 1-forms allows one to mimic the algebraic definitions of $d$, $i_X$ and $\Lie_X$, to obtain contravariant versions of these operators. First, one defines the contravariant exterior differential $$\delta:\X^r(M)\To\X^{r+1}(M)$$ by: \begin{equation}\begin{split} &\delta Q(\al_0,\dots,\al_r)\\ &\quad =\frac{1}{r+1}\sum_{k=0}^r (-1)^{k+1}\#\al_k(Q(\al_0,\dots,\widehat{\al}_k,\dots,\al_r))\\ &\quad\quad+\frac{1}{r+1}\sum_{k0$ and a horizontal curve $\tilde{\gamma}(t)$ in $\nu(S)$, defined for $t\in[0,\eps)$, which satisfies: \[ \left\{ \begin{array}{l} \frac{d}{dt}\tilde{\gamma}(t)=h(\tilde{\gamma}(t),\al(t)), \qquad t\in[0,\eps),\\ \\ \tilde{\gamma}(0)=u. \end{array} \right.\] Moreover, we can choose a neighborhood $U_\gamma$ of $0\in\nu(S)|_x$, such that for each $u\in U_\gamma$ the lift $\tilde{\gamma}(t)$ with initial point $u$ is defined for all $t\in[0,1]$. If $(\gamma(t),\al(t))$ is a cotangent loop based at $x\in S$ then this lift gives, by passing from initial to end point, a diffeomorphism $H_S(\gamma,\al)$ of $U_\gamma$ into another neighborhood $V_\gamma$ of $0\in\nu(S)|_x$, with the property that $0$ is mapped to $0$. One extends the definition of $H_S$ for piecewise smooth cotangent loops in the obvious way. Denote by $\Agerm(F_x)$ the group of germs at $0$ of Poisson automorphisms of $F_x$ which map $0$ to $0$. \begin{prop}\hspace*{-1.8mm}{\bf .} \label{prop:Poisson:holonomy} Let $(\gamma,\al),(\gamma',\al')$ be cotangent loops based at $x\in S$, then: \begin{enumerate} \item[i)] $H_S(\gamma,\al)$ is an element of $\Agerm(F_x)$; \item[ii)] $H_S((\gamma,\al)\cdot(\gamma',\al'))=H_S(\gamma,\al)\circ H_S(\gamma',\al')$, where the dot denotes concatenation of cotangent loops. \end{enumerate} \end{prop} \medskip \proof Let $(\gamma(t),\al(t))$ be a cotangent curve in $S$. For each $t$, we have a trivialization of $p:\nu(S)\to S$ in a neighborhood of $\gamma(t)$ such that $p(x,y)=x$. If $\al(t)=\sum a(t) dx|_{\gamma(t)}+b(t)dy|_{\gamma(t)}$ we consider the 1-form with constant coefficients $\al_t=\sum a(t)dx+b(t)dy$. The lift of its restriction to $S$ defines the time-dependent vector field: \[ X_t=\#\tilde{\al}^{\parallel}_t+\#^{\perp}\tilde{\al}^{\perp}_t, \] where \[\tilde{\al}^{\parallel}_t\in (TF_{\gamma(t)})^0,\ \tilde{\al}^{\perp}_t\in T^*F_{\gamma(t)}\simeq(\#(TF_{\gamma(t)})^0)^0.\] For each $t$, the transverse component $\tilde{\al}^{\perp}_t$ is a closed 1-form in $F_{\gamma(t)}$. The lifts $\tilde{\gamma}$ of $\gamma$ are the integral curves of the vector field $X_t$. We claim that the flow $\phi^t$ of this vector field preserves the transverse Poisson structure $\Pi^{\perp}$ \begin{equation} \label{eq:holonomy:flow} (\phi^{-t})_*\Pi^{\perp}_{\phi^t(u)}=\Pi^{\perp}_u, \end{equation} so (i) follows. Part (ii) also follows since we have just shown that we can take $H_S(\gamma,\al)$ as the time-1 map of some flow. To prove (\ref{eq:holonomy:flow}) we observe that \[\frac{d}{dt} (\phi^{-t})_*\Pi^{\perp}_{\phi^t(u)}=(\phi^{-t})_* {\left[\frac{d}{dh} (\phi^{-h})_*\Pi^{\perp}_{\phi^{h}(\phi^t(u))}\right]}_{h=0},\] and we use the following lemma: \begin{lem}\hspace*{-1.8mm}{\bf .} \label{lemma:Lie:derivative} If $\al_1,\al_2\in T^*_uF_x\simeq(\#(T_uF_x)^0)^0$ then \[{\left[\frac{d}{dh} (\phi^{-h})_*\Pi^{\perp}_{\phi^{h}(u)}\right]}_{h=0}(\al_1,\al_2)= (\Lie_{X_t}\Pi)_u(\al_1,\al_2)\] \end{lem} Now we have \begin{align*} \Lie_{X_t}\Pi(\al_1,\al_2)&=\Lie_{\#\tilde{\al}^{\parallel}_t}\Pi(\al_1,\al_2)+ \Lie_{\#^{\perp}\tilde{\al}^{\perp}_t}\Pi(\al_1,\al_2)\\ &=\Lie_{\#\tilde{\al}^{\parallel}_t}\Pi(\al_1,\al_2)+ \Lie_{\#^{\perp}\tilde{\al}^{\perp}_t}\Pi^{\perp}(\al_1,\al_2) \end{align*} The transverse component vanishes since $\tilde{\al}^{\perp}_t$ is a closed form in the fiber, for each $t$. For the parallel component we write $\tilde{\al}^{\parallel}_t=\sum_i a_i dx^i$, and we compute \[ \Lie_{\#\tilde{\al}^{\parallel}_t}\Pi=\sum_i \left(a_i\Lie_{\#dx^i}\Pi+\#da_i\wedge\#dx^i\right).\] But $dx^i\in (TF_x)^0$ and since $\al_1,\al_2\in (\#(T_uF_x)^0)^0$ we conclude that \[ \Lie_{\#\tilde{\al}^{\parallel}_t}\Pi(\al_1,\al_2)= \sum_i a_i\Lie_{\#dx^i}\Pi(\al_1,\al_2)=0,\] so the parallel component also vanishes. It remains to prove Lemma \ref{lemma:Lie:derivative}. We note that for any $\al\in T^*_uF_x$ we have $q^*_{\phi^h(u)}(\phi^{-h})^*\al-(\phi^{-h})^*q_u^*\al\in(T F_{p(\phi^h(u))})^0$. Using this remark we find: \begin{align*} &{\left[\frac{d}{dh} (\phi^{-h})_*\Pi^{\perp}_{\phi^{h}(u)}\right]}_{h=0}(\al_1,\al_2)\\ &=\lim_{h\to 0}\frac{1}{h}\left[\Pi_{\phi^h(u)}(q^*_{\phi^h(u)}(\phi^{-h})^*\al_1, q^*_{\phi^h(u)}(\phi^{-h})^*\al_2)-\Pi_u(q_u^*\al_1,q_u^*\al_2)\right]\\ &= \Pi_{\phi^h(u)}\left[\Pi_{\phi^h(u)}((\phi^{-h})^*q_u^*\al_1,(\phi^{-h})^*q_u^*\al_2)- \Pi_u(q_u^*\al_1,q_u^*\al_2)\right]\\ &=(\Lie_{X_t}\Pi)_u(q^*_u\al_1,q^*_u\al_2), \end{align*} so the lemma follows. \qed \medskip Denoting by $\Omega_*(S,x)$ the group of piecewise smooth cotangent loops, we see that we have a map $H_S:\Omega_*(S,x)\to\Agerm(F_x)$, which will be called the \emph{Poisson holonomy} of the leaf $S$. Note that the Poisson holonomy map depends on the immersion $\tilde{i}:\nu(S)\to M$, but two different immersions lead to conjugate homomorphisms. \begin{ex}\hspace*{-1.8mm}{\bf .} {\em Let $S$ be a regular leaf of a Poisson manifold $M$. In decomposition \ref{eq:decompose:normal:bundle} we can identify $(T_u F_x)^0\simeq T^*_u S_u$ and $(T_u S_u)^0\simeq T^*_u F_u$, where $S_u$ is the symplectic leaf through $u$. It follows that the horizontal lift $h(u,\al)$ is the unique tangent vector in $T_u S_u$ which projects to $\#\al$. We conclude that for a regular leaf the Poisson holonomy coincides with the usual holonomy. } \end{ex} \begin{ex}\hspace*{-1.8mm}{\bf .} {\em \label{ex:holonomy:not:invariant} Let $\gg$ be some finite dimensional Lie algebra and consider on $M=\gg^*$ the canonical linear Poisson bracket. For the singular leaf $S=\set{0}$ we have $\nu(S)\simeq\gg^*$ with $p(u)\equiv 0$ and the decomposition \ref{eq:decompose:normal:bundle} collapses. Given a covector $\al\in T^*_0\gg^*=\gg$ we find $h(u,\al)=\#_u\al=\ad^*\al\cdot u$. It follows that for a constant cotangent loop $(0,\al)$ in $S$ we have $H_S(0,\al)=\Ad^*(\exp(\al))$, which of course is a Poisson automorphism of $F_0\simeq\gg^*$. } \end{ex} \subsection{Reduced Poisson holonomy} As example \ref{ex:holonomy:not:invariant} shows, Poisson holonomy is not a homotopy invariant. Following the construction given in \cite{Ginzburg:article:1} for the linear case, we can give a notion of \emph{reduced Poisson holonomy} which is homotopy invariant. For a Poisson manifold $M$ let us denote by $\Aut(M)$ the group of Poisson diffeomorphisms of $M$, and by $\Aut^0(M)$ its connected component of the identity: given $\phi\in\Aut^0(M)$ there exists a smooth family $\phi_t\in\Aut(M)$, $t\in[0,1]$, such that $\phi_0=\text{id}$, $\phi_1=\phi$, and $\phi_t$ is generated by a time-dependent vector field: \[ \frac{d\phi_t}{dt}=X_t\circ\phi_t.\] The vector field $X_t$ is an infinitesimal Poisson automorphism: \[ \Lie_{X_t}\Pi=0.\] We shall say that $\phi$ is a \emph{inner Poisson automorphism} or a \emph{hamiltonian automorphism} if there exists a smooth family of hamiltonian functions $h_t:M\to\Rr$ such that $X_t=X_{h_t}=\#dh_t$. The set $\Inn(M)\subset \Aut(M)$ of inner Poisson automorphisms is a normal subgroup, and we define the group of \emph{outer Poisson automorphisms} of $M$ to be the quotient $\Out(M)=\Aut(M)/\Inn(M)$ Recall that for a symplectic leaf $S$ we denote by $\Agerm(F_x)$ the group of germs at $0$ of Poisson automorphisms of $F_x$ which map $0$ to $0$. We shall also denote by $\Ogerm(F_x)$ the corresponding group of germs of outer Poisson automorphisms. \begin{prop}\hspace*{-1.8mm}{\bf .} \label{prop:reduced:holonomy} Let $S$ be a symplectic leaf of $M$, with Poisson holonomy $H_S:\Omega_*(S,x)\to\Agerm(F_x)$. If $(\gamma_1,\al_1)$ and $(\gamma_2,\al_2)$ are cotangent loops with $\gamma_1\sim\gamma_2$ homotopic then $H_S(\gamma_1,\al_1)$ and $H_S(\gamma_2,\al_2)$ represent the same equivalence class in $\Ogerm(F_x)$. \end{prop} \medskip \proof Since any piecewise smooth path $\gamma\subset S$ can be made into a cotangent path, by property (ii) in Proposition \ref{prop:Poisson:holonomy} it is enough to show that for every $x\in S$ there exists a neighborhood $U$ of $x$ in $S$ such that if $\gamma(t)\subset U$ is a piecewise smooth loop based at $x$ and $\al(t)\in T^*M$ is a piecewise smooth family with $\#\al=\dot{\gamma}$ then $H_S(\gamma,\al)\in\Inn(F_x)$. To see this we use the same notation as in the proof of Proposition \ref{prop:Poisson:holonomy}. In a trivializing neighborhood $U$ of $p:\nu(S)\to S$ containing $x$, we can decompose the vector field $X_t$ as: \[ X_t=\#\tilde{\al}^{\parallel}_t+\#^{\perp}\tilde{\al}^{\perp}_t,\] where \[\tilde{\al}^{\parallel}_t\in (TF_{\gamma(t)})^0,\ \tilde{\al}^{\perp}_t\in T^*F_{\gamma(t)}\simeq(\#(TF_{\gamma(t)})^0)^0.\] For each $t$, the transverse component $\tilde{\al}^{\perp}_t$ can be taken to be a closed 1-form in $F_{\gamma(t)}$. It is clear that the parallel component $\#\tilde{\al}^{\parallel}_t$ has no effect on the holonomy. Hence we can assume that $S=\set{x}$, $F_x=M$, $\gamma$ is a constant path and $\tilde{\al}^{\perp}_t=\tilde{\al}_t$, so \[ X_t=\#\tilde{\al}_t=\#dh_t,\] for some function $h_t$ defined in a neighborhood of $x$. Since $H_S(\gamma,\al)$ is the time-1 flow of this hamiltonian vector field we conclude that $H_S(\gamma,\al)\in \Inn(F_x)$. \qed \medskip Given a loop $\gamma$ in $S$ we shall denote by $\bar{H}_S(\gamma)\in\Ogerm(F_x)$ the equivalence class of $H_S(\gamma,\al)$ for some piece-wise smooth family $\al(t)$ with $\#\al(t)=\gamma(t)$. The map $\bar{H}_S:\Omega(S,x)\to\Ogerm(F_x)$ will be called the \emph{reduced Poisson holonomy homomorphism} of $S$. This maps extends to continuous loops and, by a standard argument, it induces a homomorphism $\bar{H}_S:\pi_1(S,x)\to\Ogerm(F_x)$ where $\pi_1(S,x)$ is the fundamental group (the use of the same letter to denote both these maps should not be the cause of any confusion). \subsection{Stability} The reduced Poisson holonomy of a leaf carries information on the behaviour of the Poisson structure in a neighborhood of the leaf. The simplest result in this direction can be obtained as follows: let us call $S$ \emph{transversely stable} if the transverse Poisson manifold $N$ is stable near $S\cap N$, i.~e., if $N$ has arbitrarily small neighborhoods of $N\cap S$ which are invariant under all hamiltonian automorphisms. \begin{thm}\hspace*{-1.8mm}{\bf .} \label{thm:local:stability:I} (Local Stability I) Let $S$ be a compact, transversely stable leaf, with finite reduced holonomy. Then $S$ is stable, i.~e., $S$ has arbitrarily small neighborhoods which are invariant under all hamiltonian automorphisms. Moreover, each symplectic leaf of $M$ near $S$ is a bundle over $S$ whose fiber is a finite union of symplectic leaves of the transverse Poisson structure. \end{thm} \medskip \proof Assume first that $S$ has trivial reduced holonomy. We fix an embedding $\tilde{i}:\nu(S)\to M$ as above and a base point $x_0\in S$. Also, we choose a Riemannian metric on $S$. By compactness of $S$, there exists a number $c>0$ such that every point $x\in S$ can be connected to $x_0$ by a smooth cotangent path of length $0$, there exists a neighborhood $U\subset D_\eps$ such that: \begin{enumerate} \item[i)] for any piecewise-smooth cotangent path in $S$, starting at $x_0$, with length $\le 2c$ and for any $u\in U$, there exists a lifting with initial point $u$; \item[ii)] the lifting of any cotangent loop based at $x_0$ with initial point $u\in U$ has end point in $U$; \item[iii)] $U$ is invariant under all hamiltonian automorphisms; \end{enumerate} In fact, let $(\gamma_1,\al_1),\dots,(\gamma_k,\al_k)$ be cotangent loops such that $\gamma_1,\dots,\gamma_k$ are generators of $\pi_1(S,x_0)$, and let $\phi_i$ be Poisson diffeomorphisms which represent the germs $H_S(\gamma_i,\al_i)$. Since the reduced holonomy is trivial, there is a neighborhood $U'$ of $0$ in $F_{x_0}=\nu(S)|_{x_0}$ such that $U'\subset\text{domain}(\phi_1)\cap\cdots\cap\text{domain}(\phi_k)$, and $\phi_i|U'\in\Inn(F_{x_0})$, for all i. Since $S$ is transversely stable, we can choose a smaller neighborhood $U\subset U'$ invariant under all hamiltonian automorphisms. Given $x\in S$ and a cotangent path $(\gamma,\al)$ connecting $x_0$ to $x$, let us denote by $\sigma_{(\gamma,\al)}:U\to F_{x}$ the diffeomorphism defined by lifting. It follows from i) and ii) above that if $(\gamma',\al')$ is a cotangent path homotopic to $(\gamma,\al)$ then $\sigma_{(\gamma,\al)}(U)=\sigma_{(\gamma',\al')}(U)$. It follows from iii) that $\sigma_{(\gamma,\al)}(U)$ is also invariant under all hamiltonian automorphisms. Let $V$ be a neighborhood of $S$ in $M$. There exists $\eps(x)>0$ such that for the corresponding $U_x\subset D_{\eps(x)}$ we have $\sigma_{(\gamma,\al)}(U_x)\subset V\cap F_x$. By compactness of $S$, we can choose $\eps>0$ (independent of $x\in S$) such that for the corresponding $U\subset D_{\eps}$ we have \[ \sigma_{(\gamma,\al)}(U)\subset V\cap F_x\] Set \[ V_0=\bigcup_{(\gamma,\al)}\sigma_{(\gamma,\al)}(U).\] Then $V_0\subset V$ is a open neighborhood of $S$ which is invariant under all hamiltonian automorphisms of $M$. If $u,u'\in V_0$ are two points in the same symplectic leaf such that $p(u)=p(u')=x$, then there is a path $\tilde{\gamma}$ in this symplectic leaf connecting these two points. It follows from the decomposition (\ref{eq:decompose:normal:bundle}) that there exists a cotangent loop $(\gamma,\al)$ in $S$ such that $\tilde{\gamma}$ is a horizontal lift of this loop. Thus $u'$ is the image of $u$ by $H_S(\gamma,\al)$ which is a hamiltonian automorphism of $V_0\cap F_x$. Therefore, $u$ and $u'$ lie in the same symplectic leaf of $V_0\cap F_x$. We conclude that each symplectic leaf of $M$ near $S$ is a bundle over $S$ whose fiber is a symplectic leaf of the transverse Poisson structure. Assume now that $S$ has finite reduced Poisson holonomy. We let $q:\tilde{S}\to S$ be a finite covering space such that $q_*\pi_1(\tilde{S})=\Ker\bar{H}_S\subset\pi_1(S)$. If we embed $\nu(S)$ into $M$ as above, and let $\nu(\tilde{S})$ be the pull back bundle of $\nu(S)$ over $\tilde{S}$, we have a unique Poisson structure in $\nu(\tilde{S})$ such that the natural map $\nu(\tilde{S})\to\nu(S)$ is a Poisson map. Moreover, the reduced Poisson holonomy of $\nu(\tilde{S})$ along $\tilde{S}$ is trivial, so we can apply the above argument to $\nu(\tilde{S})$ and the theorem follows. \qed \medskip \begin{rem}\hspace*{-1.8mm}{\bf .} {\em If a leaf $S$ is transversely stable and $x\in S$, let $N$ denote a stable neighborhood of $F_x$. For each cotangent path $(\gamma,\al)$, the Poisson holonomy $H_S(\gamma,\al)$ induces a homeomorphism of the orbit space of $N$, for the transverse Poisson structure, mapping zero to zero. If $(\gamma_1,\al_1)$ and $(\gamma_2,\al_2)$ are cotangent loops such that $H_S(\gamma_1,\al_1)$ and $H_S(\gamma_2,\al_2)$ represent the same class in $\Ogerm(F_x)$, then they induce the same germ of homeomorphism of the orbit space mapping zero to zero. In \cite{Dazord:article:1} holonomy of a general, transversely stable, foliation is defined using germs of homeomorphisms of the orbit space, which in the case of a Poisson manifold coincide with these ones. } \end{rem} \subsection{Strict Poisson holonomy} Another problem raised by the local splitting theorem and related to stability is whether one has a global splitting of an entire neighborhood of a leaf $S$. Note that if a neighborhood $V$ of $S$ has a Poisson splitting $S\times N$ then projection to the first factor is a Poisson map. This motivates the \begin{defn}\hspace*{-1.8mm}{\bf .} {\em \label{defn:Poisson:neighborhood} Let $M$ be a Poisson manifold and $i:S\hookrightarrow M$ a symplectic leaf of $M$. A \textsc{Poisson tubular neighborhood} of $S$ is a smooth immersion $\tilde{i}:\nu(S)\to M$ satisfying: \begin{enumerate} \item[i)] $\tilde{i}|_Z=i$, where $Z$ is the zero section of $\nu(S)$; \item[ii)] $\tilde{i}$ maps the fibers of $\nu(S)$ transversely to the symplectic foliation of $M$; \item[iii)] For the Poisson structure on $\nu(S)$ induced from $\tilde{i}$, the canonical projection $p:\nu(S)\to S$ is a Poisson map; \end{enumerate} } \end{defn} Suppose $S$ admits a Poisson tubular neighborhood. Then the regular distribution $\#(\Ker p_*)^0$ is integrable and $S$, identified with the zero section, is an integral leaf of this distribution. Hence, we can consider the holonomy of $S$ (in the usual sense) as a leaf of the corresponding foliation. We call this the \emph{strict Poisson holonomy} of $S$, and we denote by $\check{H}_S:\Omega(S,x)\to \Dgerm(F_x)$ the associated holonomy map, where $\Dgerm(F_x)$ denotes the group of germs of diffeomorphisms of $F_x$ which map $0$ to $0$. Strict Poisson holonomy is related to reduced Poisson holonomy as follows. \begin{prop}\hspace*{-1.8mm}{\bf .} Assume $S$ admits a Poisson tubular neighborhood. The map $\check{H}_S:\Omega(S,x)\to \Dgerm(F_x)$ has image inside $\Agerm(F_x)$ and the following diagram commutes: \[ \xymatrix{ \Omega(S,x)\ar[r]^{\check{H}_S} \ar[dr]_{\bar{H}_S} & \Agerm(F_x)\ar[d]\\ &\Ogerm(F_x)} \] \end{prop} \medskip \proof Fix a Poisson tubular neighborhood $p:\nu(S)\to S$ and consider the gene\-ra\-li\-zed connection in $\nu(S)$ defined by the distribution $\#(\Ker p_*)^0$. Given a loop $\gamma(t)$ in $S$ there exists a family of closed forms $\al^S_t\in \Omega^1(S)$ such that $\#\al^S_t(\gamma(t))=\dot{\gamma}(t)$. The horizontal lifts of this loop are integral curves of the time-dependent vector field \[ \check{X}_t=\#p^*\al^S_t.\] Since $dp^*\al^S_t=p^*d\al^S_t=0$, this vector field is an infinitesimal Poisson automorphism. We conclude that the holonomy maps $\check{H}_S(\gamma)$ are Poisson automorphisms. Moreover, in the notation of the proof of Proposition \ref{prop:reduced:holonomy}, we have $\check{X_t}=\#\al_t^\parallel$. It follows that if $(\gamma,\al)$ is a cotangent loop in $M$ then $H_S(\gamma,\al)$ and $\check{H}_S(\gamma)$ represent the same class in $\Out(F_x)$. \qed \medskip We can now state and prove the following splitting result: \begin{thm}\hspace*{-1.8mm}{\bf .} \label{thm:local:stability:II} (Local Stability II) Suppose $i:S\hookrightarrow M$ is a compact symplectic leaf of a Poisson manifold $M$ which admits a Poisson tubular neighborhood. Assume further that $S$ has finite strict Poisson holonomy and let $q:\tilde{S}\to S$ be the finite covering corresponding to $\Ker \check{H}_S\subset\pi_1(S,x)$. Then there is a neighborhood $V$ of $S$ and a finite covering Poisson map $\phi:\tilde{S}\times N\to V$, where $N$ is a transverse Poisson manifold to $S$. If $S$ is transversely stable, then we can choose $N$ and $V$ to be stable neighborhoods. \end{thm} \medskip \proof By a standard homotopy lifting argument, as in the end of the proof of Theorem \ref{thm:local:stability:I}, it is enough to consider the special case where the holonomy is trivial. We must then show that there is a neighborhood $V$ of $S$ and a Poisson diffeomorphism $\phi:S\times N\to V$, where $N$ is a transverse Poisson manifold to $S$. Again, we fix an embedding $\tilde{i}:\nu(S)\to M$ as above and a base point $x_0\in S$. Also, we choose a Riemannian metric on $S$. By compactness of $S$, there exists a number $c>0$ such that every point $x\in S$ can be connected to $x_0$ by a smooth cotangent path of length $0$ such that: for any piecewise-smooth cotangent path in $S$, starting at $x_0$, with length $\le 2c$ and for any $u\in D_\eps$, there exists a lifting with initial point $u$. Moreover, by shrinking $\eps$ if necessary, we can assume that the lifting of any cotangent loop based at $x_0$ with initial point $u$ also ends at $u$. In fact, let $(\gamma_1,\al_1),\dots,(\gamma_k,\al_k)$ be cotangent loops such that $\gamma_1,\dots,\gamma_k$ are generators of $\pi_1(S,x_0)$, and let $\phi_i$ be Poisson diffeomorphisms which represent the germs $\check{H}_S(\gamma_i,\al_i)$. Then, since the holonomy is trivial by assumption, there is a neighborhood $U$ of $0$ in $\nu(S)|_{x_0}$ such that $U\subset\text{domain}(\phi_1)\cap\cdots\cap\text{domain}(\phi_k)$, and $\phi_i|U=$identity, for all i. We need only to choose $\eps$ such that $D_\eps\subset U$. For each $u\in D_\eps$ we define a map $\sigma_u:S\to M$ as follows: let $x\in S$ and connect $x$ to $x_0$ by a cotangent path $(\gamma,\al)$ of length $r/2$. To compute these invariants one uses the contravariant derivative operator $D$ on $E$, associated with the contravariant connection $\Gamma$, and proceeds as follows. For covectors $\al,\beta\in T_x M$, the curvature tensor $R$ defines a linear map $R_{\al,\beta}=R(\al,\beta):F_x\to F_x$ which satisfies $R_{\al,\beta}=-R_{\beta,\al}$, and so $(\al,\beta)\to R_{\al,\beta}$ can be considered as a $\gl(E)$-valued bivector field. By fixing a basis of local sections, we have $F_x\simeq\Rr^q$ so we have $R_{\al,\beta}\in\gl_q(\Rr)$. (this matrix representation of $R_{\al,\beta}$ is defined only up to a change of basis in $\Rr^q$). Hence, if \[ P:\gl_q(\Rr)\times\cdots\times\gl_q(\Rr)\to\Rr\] is a symmetric, $k$-multilinear function, $\Ad(GL_q(\Rr))$-invariant, we a have a $2k$-vector field $\lambda(R)(P)$ on $M$ defined by \begin{equation} \label{eq:Chern-Weil:homomorphism:connection} \begin{split} &\lambda(R)(P)(\al_1,\dots,\al_{2k})\\ &\qquad \qquad=\sum_{\sigma\in S_{2k}} (-1)^{\sigma} P(R_{\al_{\sigma(1),\sigma(2)}},\dots,R_{\al_{\sigma(2k-1),\sigma(2k)}}). \end{split}\end{equation} It is easy to see that $\lambda(\Gamma)(P)=\lambda(R)(P)$, so this gives a procedure to compute the Poisson-Chern-Weil homomorphism and the Poisson-Pontrjagin classes. Similar considerations apply to other characteristic classes. One can define, e.~g., the Poisson-Chern classes $c_k(E,\Pi)$ of a complex vector bundle $E$ over a Poisson manifold, and they are just the images by $\#$ of the usual Chern classes of $E$. The fact that all these classes arise as images by $\#$ of some known classes is perhaps a bit disappointing. However, we shall see below that one can define Poisson secondary characteristic classes which are intrinsic of Poisson geometry, and which do not arise as images by $\#$ of some de Rham cohomology classes. \subsection{Secondary characteristic classes} We shall now introduce secondary characteristic classes of a Poisson manifold. We will see that these classes give information on the topology, as well as, the geometry of the symplectic foliation. As in the theory of (regular) foliations, these classes appear when we compare two connections, each from a distinguished class. On the Poisson manifold $M$, with $\dim M=m$, we consider the following data: \begin{enumerate} \item[i)] A basic connection $\Gamma^1$, with a contravariant derivative $D^1$; \item[ii)] A linear contravariant connection $\Gamma^0$ induced by a riemannian connection, so $D^0_\al=\nabla^0_{\#\al}$ with $\nabla^0 g=0$ for some riemannian metric $g$; \end{enumerate} Given an invariant, symmetric, $k$-multilinear function $P\in I^k(GL(m,\Rr))$ we consider the $(2k-1)$-vector field $\lambda(\Gamma^1,\Gamma^0)(P)$ given by (\ref{eq:invariants:2}). \begin{prop}\hspace*{-1.8mm}{\bf .} \label{prop:invariants:2:closed} If $k$ is odd, $\lambda(\Gamma^1,\Gamma^0)(P)$ is a closed $(2k-1)$-vector field. \end{prop} \medskip \proof According to (\ref{eq:delta:lambda:2}) we have \[ \delta\lambda(\Gamma^1,\Gamma^0)(P)=\lambda(\Gamma^1)(P)-\lambda(\Gamma^0)(P). \] and we claim that $\lambda(\Gamma^1)(P)=\lambda(\Gamma^0)(P)=0$ if $k$ is odd. The proof that $\lambda(\Gamma^0)(P)=0$ is standard: since there exists a metric such that $D^0 g=0$ we can reduce the structure group of $\Gamma^0$ to $O(m,\Rr)$, so the curvature bivector fields take their values in $\mathfrak{so}(m,\Rr)$. But if $A\in \mathfrak{so}(m,\Rr)$, we have $P_k(A)=0$ for any elementary symmetric function, since $k$ is odd. Hence we obtain $\lambda(\Gamma^0)(P)=0$. Consider now the connection $\Gamma^1$. Given $x\in M$ we choose local coordinates $(x^j,y^k)$ around $x$ as in the Weinstein splitting theorem: \[ \Pi=\sum_{i=1}^{n}\frac{\partial}{\partial x^i}\wedge \frac{\partial}{\partial x^{i+n}}+ \sum_{k,l}\phi_{kl}\frac{\partial}{\partial y^k}\wedge \frac{\partial}{\partial y^l},\] where $\phi_{kl}(x)=0$. Since $\Gamma^1$ is a basic connection, we have: \[ \Pi(D^1_\al dx^i,dx^j)=-\Pi(dx^i,D^1_\al dx^j),\qquad R^1(\al,\beta)dy^k|_x=0.\] It follows that $R^1(\al,\beta)_x$ is represented in the basis $(dx^j,dy^k)$ by a matrix of the form: \begin{equation} \label{eq:matrix:form} \left(\begin{array}{cc} B & 0 \\ C & 0 \end{array}\right), \end{equation} with $B$ a symplectic matrix. Now, if $A$ is any matrix of this form, it is clear that $\det(\mu I-A)=\det(\mu I-\tilde{A})$, where $\tilde{A}$ is the same as $A$ with $C=0$, i.~e., $\tilde{A}$ is symplectic. But if $\tilde{A}$ is symplectic, we have $P_k(A)=0$ for any elementary symmetric function, since $k$ is odd. Hence we obtain also $\lambda(\Gamma^1)(P)_x=0$. \qed \medskip Next we want to check that the Poisson cohomology class of \linebreak $\lambda(\Gamma^1,\Gamma^0)(P)$ is independent of the connections used to define it. Given connections $\Gamma^0,\Gamma^1,\Gamma^2$ we consider the family of connections $\Gamma^{s,t}$ whose connection vector fields are $\Lambda^{s,t}_j=(1-s-t)\Lambda^0_j+s\Lambda^1_j+t\Lambda^2_j$, where $(s,t)$ vary in the standard 2-simplex $\Delta_2$. We introduce a $(2k-2)$-vector field $\lambda(\Gamma^2,\Gamma^1,\Gamma^0)(P)$ given by a formula analogous to (\ref{eq:Chern-Weil:homomorphism:connection}) and (\ref{eq:invariants:2}): \begin{equation} \label{eq:invariants:3}\begin{split} &\lambda(\Gamma^2,\Gamma^1,\Gamma^0)(P)\\ &\qquad\qquad = k \sum_{\sigma\in S_{2k-2}} (-1)^{\sigma}\int_{\Delta_2} P(\Lambda^{1,0}_j,\Lambda^{2,0}_j,\Xi^{s,t}_j,\dots,\Xi^{s,t}_j)dtds. \end{split}\end{equation} Just like in the proof of Proposition \ref{prop:connection:invariance:1}, one shows that \begin{equation} \label{eq:delta:lambda:3}\begin{split} \delta\lambda(\Gamma^2,\Gamma^1,\Gamma^0)(P) =&\lambda(\Gamma^1,\Gamma^0)(P)- \lambda(\Gamma^2,\Gamma^0)(P)\\ &+\lambda(\Gamma^1,\Gamma^0)(P). \end{split}\end{equation} Now, we can prove \begin{prop}\hspace*{-1.8mm}{\bf .} \label{prop:independency:connection} The Poisson cohomology class $[\lambda(\Gamma^1,\Gamma^0)(P)]$ is independent of the connections used to define it. \end{prop} \medskip \proof Let $\Gamma^1$ and $\tilde{\Gamma}^1$ (resp.~$\Gamma^0$ and $\tilde{\Gamma}^0$) be basic connections (resp.~riemannian connections). It follows from (\ref{eq:delta:lambda:3}) that \begin{align*} \lambda(\Gamma^1,\Gamma^0)(P)- \lambda(\tilde{\Gamma}^1,\tilde{\Gamma}^0)(P)=& \delta\lambda(\tilde{\Gamma}^1,\Gamma^0,\tilde{\Gamma}^0)(P)- \delta\lambda(\Gamma^1,\tilde{\Gamma}^1,\Gamma^0)(P)\\ &+\lambda(\tilde{\Gamma}^1,\Gamma^1)(P) -\lambda(\Gamma^0,\tilde{\Gamma}^0)(P). \end{align*} Hence, it is enough to show that the Poisson cohomology classes of $\lambda(\tilde{\Gamma}^1,\Gamma^1)(P)$ and $\lambda(\Gamma^0,\tilde{\Gamma}^0)(P)$ are trivial. Consider first the basic connections $\tilde{\Gamma}^1$ and $\Gamma^1$. The linear combination $\Gamma^{1,t}=(1-t)\Gamma^1+t\tilde{\Gamma}^1$ is also a basic connection. If $x\in M$, we fix splitting coordinates $(x^j,y^k)$ around $x$ as in the proof of Proposition \ref{prop:invariants:2:closed}. Then we see that, with respect to the basis $\set{dx^j,dy^k}$, the matrix representations of $D^1_\al-\tilde{D}^1_\al$ and $R^t(\al,\beta)$ are of the form (\ref{eq:matrix:form}). Hence, we conclude that if $P\in I^k(GL(m,\Rr))$, with $k$ odd, \[P(\tilde{D}^1_{\al_1}-D^1_{\al_1},R^t(\al_2,\al_3),\dots, R^t(\al_{2k-2},\al_{2k-1}))=0.\] Therefore, $\lambda(\tilde{\Gamma}^1,\Gamma^1)(P)=0$, whenever $\tilde{\Gamma}^1$ and $\Gamma^1$ are basic connections. Now consider the riemannian connections $\Gamma^0$ and $\tilde{\Gamma}^0$. The linear combination $\Gamma^{0,t}=(1-t)\tilde{\Gamma}^0+t\Gamma^0$ is also a riemannian connection. All these connections are induced from covariant riemannian connections $\nabla^0$, $\tilde{\nabla}^0$ and $\nabla^{0,t}$, and we can define a $(2k-1)$-form $\lambda(\nabla^0,\tilde{\nabla}^0)(P)$ by a formula analogous to (\ref{eq:invariants:2}). Moreover, this form is closed (because $k$ is odd), and $ \#\lambda(\nabla^0,\tilde{\nabla}^0)(P)=\lambda(\Gamma^0,\tilde{\Gamma}^0)(P)$. It follows from the homotopy invariance of $H^*(M)$, as in the usual theory of characteristic classes of foliations (see \cite{Bott:lectures:1}, page 29), that \[ [\lambda(\nabla^0,\tilde{\nabla}^0)(P)]=[\lambda(\nabla^0,\nabla^0)(P)]=0.\] Hence, the Poisson cohomology class $[\lambda(\nabla^0,\tilde{\nabla}^0)(P)]$ vanishes. \qed \medskip \begin{rem}\hspace*{-1.8mm}{\bf .} {\em The assumption that the riemannian connections are of the special form $\nabla_{\#\al}$ was used in the proof to invoke the homotopy invariance of $H^*(M)$. Poisson cohomology $H^*_\Pi(M)$ is not homotopy invariant, so in defining the invariant $\lambda(\Gamma^1,\Gamma^0)(P)$ we cannot consider an arbitrary riemannian contravariant connection $\Gamma^0$. On the other hand, as we pointed out above, in general a Poisson manifold does not admit a Poisson connection of the form $\nabla_{\#\al}$. Hence, the basic connections are ``genuine'' contravariant connections, i.~e., not induced by any covariant connection. } \end{rem} We define the \emph{secondary characteristic classes} $\set{m_k(M)}$ of a Poisson manifold to be the Poisson cohomology classes \begin{equation} m_k(M)=[\lambda(\Gamma^1,\Gamma^0)(P_{k})]\in H^{2k-1}_\Pi(M),\qquad (k=1,3,\dots). \end{equation} If $M$ is a symplectic manifold then these classes obviously vanish. They also vanish if $M=S\times N$ where $S$ is symplectic and $N$ has the zero Poisson bracket. However, they do not vanish for a general, regular, Poisson manifold (see the examples below). Hence these characteristic classes give information on both the Poisson geometry and the topology of the symplectic foliation of $M$. In the next section we give some explicit computations of these classes, and in the following section we will show that the first class coincides with the modular class of $M$ (up to a scalar factor). \begin{rem}\hspace*{-1.8mm}{\bf .} {\em In general, one can only define the characteristic classes $m_k$ for $k$ odd. Assume, however, that $M$ admits flat riemannian connections and flat basic connections (we will see some non-trivial examples below). Then the proofs of Propositions \ref{prop:invariants:2:closed} and \ref{prop:independency:connection} can be carried through, in the class of flat connections, for \emph{any} $k$. Hence, in this case, one can define characteristic classes $m_k$ for \emph{any} $k$. } \end{rem} \subsection{Examples} We give a few types of Poisson manifolds where one can compute some of the secondary characteristic classes. \medskip \textbf{Euclidean spaces.} Consider a Poisson manifold $M\simeq\Rr^m$, so we have global coordinates $(x^1,\dots,x^m)$. To compute $\lambda(\Gamma^1,\Gamma^0)(P)$ we take as $\Gamma^0$ the flat connection determined by these global coordinates, and as $\Gamma^1$ we take the basic connection defined by \[ D_{dx^i} dx^j=[dx^i,dx^j]=\sum_{k}\frac{\partial\pi^{ij}}{\partial x^k}dx^k.\] Since $P_1(A)=\frac{1}{2\pi}\tr(A)$, we find immediately that the first characteristic class is \begin{equation} \label{eq:first:class} m_1(M)=\frac{1}{2\pi}\sum_{i,j} \frac{\partial\pi^{ij}}{\partial x^j}\frac{\partial}{\partial x^i}. \end{equation} To compute the second characteristic class, we note that $D^t_{dx^i} dx^j=(1-t)D_{dx^i} dx^j$, and we compute its curvature: \[ R^t(dx^i,dx^j)dx^k=-t(t-1) D_{[dx^i,dx^j]}dx^k.\] Now, \begin{align*} P_3(A,B,C)=\frac{1}{24\pi^3} &\left[\tr(ABC) -\frac{1}{2}(\tr A~\tr(BC)+\tr B~\tr(CA)\right.\\ &\left.\qquad +\tr C~\tr(AB))-\frac{1}{2}\tr A~\tr B~\tr C \right] \end{align*} and the expression for the characteristic class $m_3(M)$ is a certain homogeneous polynomial of degree $5$ involving the derivatives of order $\le 3$ of the components $\pi^{ij}$ of the Poisson tensor. \medskip \textbf{Linear Poisson structures.} Let $M=\gg^*$ with the Lie-Poisson structure determined by the Lie algebra $\gg$. Then, from the previous example, we see that the first class is represented by the constant vector field \[ m_1(\gg^*)(v)=\frac{1}{2\pi}\tr(\ad v).\] In this case both the basic connection and the riemannian connection are flat and so we can consider the classes $m_k$ for \emph{any} $k$. The computations simplify considerably, and we see that all classes can be represented by constant multivector fields. For example, a straightforward computation shows that \begin{align*} m_2(\gg^*)(v_1,v_2,v_3)&=\frac{3!}{4\pi^2}K_2(v_1,[v_2,v_3]),\\ m_3(\gg^*)(v_1,\dots,v_5)&=\frac{1}{8\pi^3} \sum_{\sigma\in S_5} K_3(v_{\sigma(1)},[v_{\sigma(2)},v_{\sigma(3)}],[v_{\sigma(4)},v_{\sigma(5)}]) \end{align*} where we have set \[ K_j(v_1,\dots,v_j)=\tr(\ad v_1\cdots\ad v_j).\] Note that $K_2$ is just the killing form. In this case it is possible to give a general formula for all characteristic classes: \begin{equation*}\begin{split} &m_k(\gg^*)(v_1,\dots,v_{2k-1})\\ &\qquad=\frac{1}{(2\pi)^k}\sum_{\sigma\in S_{2k-1}} K_k(v_{\sigma(1)},[v_{\sigma(2)},v_{\sigma(3)}],\dots, [v_{\sigma(2k-2)},v_{\sigma(2k-1)}]). \end{split}\end{equation*} The proof of these formulas involves a certain amount of computation using Newton's identities for the elementary symmetric polynomials. Incidentally, we note that the classes $m_k$ are $\ad$-invariant since each $K_j$ is an $\ad$-invariant multilinear form. Therefore, the classes $m_k(\gg^*)$ represent certain cohomology classes in the Lie algebra cohomology of $\gg$. \medskip \textbf{Poisson-Lie Groups.} Let $G$ be a connected Poisson-Lie group (see, e.~g., \cite{Lu:article:1}). Then the set of left invariant 1-forms $\Omega^1_{\text{Inv}}(G)$ is closed for the Lie bracket defined by the Poisson bracket. Hence we can define a basic connection $D^1$ in $G$ by requiring that \begin{equation} \label{eq:connection:group} D_\al\beta=[\al,\beta], \qquad \forall \al,\beta\in\Omega^1_{\text{Inv}}(G). \end{equation} This connection is flat. Let $D^0=\nabla_{\#\al}$ be the unique left invariant connection in $G$ which for left invariant vector fields is given by \[ \nabla_X Y=[X,Y],\qquad \forall X, Y\in\gg.\] This connection is also flat. We compute $\lambda(D^1,D^0)(P)$ and, generalizing the previous example, the classes $m_k(G)$ are all represented by the left invariant multivector fields: \begin{equation*}\begin{split} &m_k(G)(\al_1,\dots,\al_{2k-1})\\ &\qquad=\frac{1}{(2\pi)^k}\sum_{\sigma\in S_{2k-1}} K_k(\al_{\sigma(1)},[\al_{\sigma(2)},\al_{\sigma(3)}],\dots, [\al_{\sigma(2k-2)},\al_{\sigma(2k-1)}]) \end{split}\end{equation*} where $\al_1,\dots,\al_n\in\Omega^1_{\text{Inv}}(G)$. In these formulas, $[~,~]$, $\ad$ and $K_k$ are relative to the Lie algebra $\gg^*=\Omega^1_{\text{Inv}}(G)$. \vskip 10 pt \begin{rem}\hspace*{-1.8mm}{\bf .} {\em Note that if the Poisson bracket in $G$ is not trivial, the contravariant connection defined by (\ref{eq:connection:group}) is \emph{not} left invariant, because left translation in the group is not a Poisson map. These type of connections are studied in a complement to the present paper, where we deal with invariant connections (\cite{Fernandes:article:2}). } \end{rem} \medskip \textbf{Regular Poisson manifolds.} Let $M$ be a regular Poisson manifold of dimension $m$ and corank $q$. First choose some riemannian connection determining a splitting \[ T^*(M)=T^*(\mathcal{S})\oplus \nu^*(\mathcal{S}),\] where $T^*(\mathcal{S})$ (resp.~$\nu^*(\mathcal{S})$) is the cotangent (resp.~conormal) bundle to the symplectic foliation. We have a riemannian connection $D^0$ such that: \[ D^0_\al(\beta+\gamma)=\nabla_{\#\al}^{0,\parallel}\beta+ \nabla_{\#\al}^{0,\perp}\gamma, \] where $\beta$ and $\gamma$, are sections of $T^*(\mathcal{S})$ and $\nu^*(\mathcal{S})$, and $\nabla^{0,\parallel}$ and $\nabla^{0,\perp}$, are covariant riemannian connections in these bundles. Because $M$ is regular, we can also choose a covariant connection $\nabla^{1,\parallel}$ on $TM$ such that $\nabla^{1,\parallel}\Pi=0$. We define the basic connection $D^1$ on $M$ by setting \[ D^1_\al(\beta+\gamma)=\nabla_{\#\al}^{1,\parallel}\beta+ \nabla_{\#\al}^{1,\perp}\gamma, \] where $\nabla^{1,\perp}$ is a basic connection in $\nu(\mathcal{S})$ in the usual sense of foliation theory (see \cite{Bott:lectures:1}, p. 33). A computation shows that \[ \lambda(D^1,D^0)(P)=\#\lambda(\nabla^{1,\perp},\nabla^{0,\perp})(\tilde{P}), \] where $\tilde{P}$ is the obvious restriction of $P\in I^*(GL_m(\Rr))$ to $I^*(GL_q(\Rr))$. It is well known in foliation theory (see \cite{Bott:lectures:1}, p.~66) that the forms \begin{align*} c_k&=\lambda(\nabla^{1,\perp})(\tilde{P}_{k}),\qquad (1\le k\le q)\\ h_{2k-1}&=\lambda(\nabla^{1,\perp},\nabla^{0,\perp})(\tilde{P}_{2k-1}), \qquad (1\le 2k-1\le q), \end{align*} satisfy \begin{align} \label{eq:Gelfand:Fuks} dc_k&=0,\qquad (1\le k\le q)\\ dh_{2k-1}&=c_{2k-1},\qquad (1\le 2k-1\le q). \end{align} and so they can be used to define a homomorphism of graded algebras \[ H^*(WO_q)\to H^*(M),\] where $H^*(WO_q)$ is the relative Gelfand-Fuks cohomology of formal vector fields in $\Rr^q$. This homomorphism is independent of the connections and its image are the exotic or secondary characteristic classes of foliation theory. In this respect, the Poisson secondary characteristic classes are simpler than the corresponding ones in foliation theory: the $(2k-1)$-forms $\lambda(\nabla^{1,\perp},\nabla^{0,\perp})(\tilde{P}_{k})$ are not closed in general, but are closed along the symplectic leaves, so its image under $\#$ is a closed $(2k-1)$-vector field and, hence, define Poisson cohomology classes. Therefore, one has \begin{equation} m_{2k-1}(M)=[\#h_{2k-1}] \end{equation} but, in general, $m_{2k-1}$ is not in the image of $\#:H^*(M)\to H^*_{\Pi}(M)$. Still, one can sometimes relate the two types of secondary characteristic classes. Take, for example, the Godbillon-Vey class which by definition is the cohomology class $w=[h_1c_1^q]\in H^{2q+1}(M)$ (it follows from relations (\ref{eq:Gelfand:Fuks}) that $d(h_1c_1^q)=c_1^{q+1}=0$, so $h_1c_1^{q}$ does define a cohomology class). \begin{prop}\hspace*{-1.8mm}{\bf .} If a regular Poisson manifold has a non-trivial Godbillon-Vey class then it has a non-trivial first Poisson secondary characteristic class. \end{prop} \medskip \proof If $m_1(M)=[\#h_1]$ is trivial, we have $\#h_1=\#df$ for some smooth function $f$, i.~ e., $h_1(\#\al)=df(\#\al)$. But $h_1$ is defined up to a 1-form in the differential ideal that gives the symplectic foliation, so $h_1\wedge (dh_1)^ q=0$ and the the Godbillon-Vey class must vanish. \qed \medskip On the other hand, it is perfectly possible for the Godbillon-Vey class to vanish while $m_1(M)\not=0$. One such example is provided by the Reeb foliation in $S^3$ with the leafwise area form (see \cite{Ginzburg:article:1, Weinstein:article:2} for details on this example). Another consequence of this relationship is that, for a regular Poisson manifold $M$, the characteristic classes $m_k(M)=0$, for $2k-1>q=\text{corank}(M)$. As a special case, let us consider a Poisson manifold of corank 1. The only non-vanishing class is $m_1(M)$. If the symplectic foliation is transversely orientable, let $Z$ be a trivializing section of the normal bundle. Let $\theta$ be the corresponding 1-form that trivializes the conormal bundle. There exists a 1-form $\eta$ such that \[ d\theta=\eta\wedge\theta.\] For $\nabla^{1,\perp}$ we choose a basic connection in $\nu(\mathcal{S})$ such that \[ \nabla^{1,\perp}_{X}Z=\eta(X)Z.\] For $\nabla^{0,\perp}$ we choose a riemannian connection such that \[ \nabla^{0,\perp}_{X}Z=0.\] These choices give \[\lambda(\nabla^{1,\perp},\nabla^{0,\perp})(\tr)=\eta,\] so we conclude that \[ m_1(M)=\frac{1}{2\pi}[\#\eta].\] In fact, in this case we have $h_1=\frac{1}{2\pi}\eta$ so $w=\frac{1}{4\pi^2}\eta\wedge d\eta$ represents the Godbillon-Vey class. If the symplectic foliation is not transversely orientable one can pass to a double cover and apply the same reasoning. \subsection{The Modular Class} \label{section:moudlar:class} The modular class of a Poisson manifold is an obstruction lying in the first Poisson cohomology group $H^1_\Pi(M)$ to the existence of a transverse invariant measure (see \cite{Weinstein:article:2} for details on the modular class). It can be defined as follows: Let $\mu$ be any measure in $M$ with associated divergence operator $\text{div}_{\mu}X\equiv \Lie_X\mu/\mu$. Then one checks that the map $f\mapsto \text{div}_\mu \#df$ is a derivation of $C^\infty(M)$ so defines a vector field $v_{\mu}$, called the \emph{modular vector field} associated with the measure $\mu$. This vector field is an infinitesimal automorphism of $\Pi$. If $\mu'=a\mu$ is another measure, we have $v_{\mu'}=v_{\mu}+\#d\log a= v_{\mu}+\delta\log a$, so in fact the modular class \[ \mod(M)\equiv[v_\mu]\in H^1_{\Pi}(M)\] is well defined and independent of $\mu$. The examples in the previous section when compared to the computations of the modular class done in \cite{Weinstein:article:2} suggest the following \begin{thm}\hspace*{-1.8mm}{\bf .} For any Poisson manifold $M$ \begin{equation} \label{eq:invariant:modular:class} m_1(M)=\frac{1}{2\pi}\text{mod~}(M). \end{equation} \end{thm} \medskip \proof Choose a basic connection $D^1$ and a riemannian connection $D^0$ relative to some metric on $M$. Let $\mu$ be the measure defined by this metric. We claim that \begin{equation} \label{eq:invariant:modular:field} \lambda(D^1,D^0)(\tr)=v_\mu, \end{equation} so (\ref{eq:invariant:modular:class}) follows. Observe that it is enough to show that (\ref{eq:invariant:modular:field}) holds on the regular points of $M$, since the set of regular points is an open dense set and both sides are smooth vector fields on $M$. So assume that $x\in M$ is a regular point and pick Darboux coordinates $(x^1,\dots,x^m)$. If $g=\left(\seq{dx^i,dx^j}\right)$ is the $m\times m$-matrix of inner products of the $dx^i$'s, we have \[ \mu=(\det g)^{\frac{1}{2}}dx^1\wedge\cdots\wedge dx^m.\] As in the proofs of the previous section, relative to the basis \linebreak $\set{dx^1,\dots,dx^m}$, the operator $D^1_\al$ has a matrix representation by a traceless matrix, so we only need to understand what is the matrix representation, relative to this basis, of the riemannian connection $D^0_\al=\nabla^0_{\#\al}$. Since $\nabla^0$ is a metric connection, parallel transport preserves the volume, and we have for any smooth function $f\in C^\infty(M)$: \begin{align*} 0=&\nabla^0_{\#df}\mu\\ =&\#df((\det g)^{\frac{1}{2}})dx^1\wedge\cdots\wedge dx^m+\\ & +(\det g)^{\frac{1}{2}}(\nabla^0_{\#df}dx^1\wedge\cdots\wedge dx^m+ \cdots+dx^1\wedge\cdots\wedge \nabla^0_{\#df}dx^m)\\ =&\left(\#df((\det g)^{\frac{1}{2}})+(\det g)^{\frac{1}{2}}\tr \nabla^0_{\#df}\right) dx^1\wedge\cdots\wedge dx^m. \end{align*} So we conclude that: \begin{equation} \label{eq:modular:aux:1} \tr (D^1_{df}-D^0_{df})\mu=\#df((\det g)^{\frac{1}{2}})dx^1\wedge\cdots\wedge dx^m. \end{equation} Now recall that $(x^1,\dots,x^m)$ were Darboux coordinates around a regular point, so the form $dx^1\wedge\cdots\wedge dx^m$ is preserved by the hamiltonian flows, and we have \[ \Lie_{\#df}(dx^1\wedge\cdots\wedge dx^m)=0.\] Hence, we conclude that: \begin{equation} \label{eq:modular:aux:2} \Lie_{\#df}\mu=\#df((\det g)^{\frac{1}{2}})dx^1\wedge\cdots\wedge dx^m. \end{equation} Comparing (\ref{eq:modular:aux:1}) and (\ref{eq:modular:aux:2}) gives \[ \tr (D^1_{df}-D^0_{df})=\text{div}_\mu \#df,\] so relation (\ref{eq:invariant:modular:field}) holds. \qed \medskip If $(\gamma(t),\al(t))$, $t\in[0,1]$, is a cotangent path and $X$ is a vector field, one defines the integral \[\int_{(\gamma,\al)}X=-\int_0^1 i_{X(\gamma(t))}\al(t) dt.\] (For basic properties of this integral see \cite{Ginzburg:article:1}). As a corollary of the theorem and the Ginzburg and Golubev formula (\ref{eq:Ginzburg:Golubev}), we obtain: \begin{cor}\hspace*{-1.8mm}{\bf .} Let $(\gamma,\al)$ be a cotangent loop in the symplectic leaf $S$. Then \begin{equation} \det H_S^L(\gamma,\al)=\exp(\int_{(\gamma,\al)}tr(D^1-D^0)), \end{equation} where the determinant is relative to the transverse measure induced by the volume element of the metric associated with $D^0$. \end{cor} % ------------------------------------------------------------------------ \def\lllll{} \begin{thebibliography}{\footnotesize \bibitem{Bott:lectures:1} R.~Bott, \emph{Lectures on characteristic classes and foliations}, Lectures on Algebraic and Differential Topology, Lec.~ Notes in Math. Vol. 279, Springer, Berlin, 1972. \bibitem{Dazord:article:1} P.~Dazord, \emph{Feuilletages \`{a} singularit\'{e}s}, Indag. 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