The sharp map

Let $M$ be a manifold, and $\Pi$ a Poisson bivector.

Remark: $(i _{\al_x}\Pi )(\bt x)=\scalar{\Pi}{\bt_x\wedge \al_x}=\Pi_x(\al_x, \bt_x)$ for all $\al_x, \bt_x\in T_x^*M$.

Properties:

• $\char93 _{\Pi}$ is a bundle map on $M$. It is also called the anchor of $(M, \Pi)$.
• Being a bundle map it induces a map on sections
  
  $$\char93 _{\Pi}\:\Om^1(M)\to \gX(M),\quad \al\mapsto i _{\al}\Pi$$
  
  

• in particular on exact 1-forms one easily has $\char93 _{\Pi}(df)=X_f$. In fact
  
  $$\scalar{\char93 _{\Pi}(df)}{dg}=\Pi(df, dg)=\{f,g\}=\scalar{X_f}{dg}.$$
  
  
  Remark then, that a vector field is uniquely determined by its contractions with exact 1-forms (locality of vector fields).
• Local expression
  
  $$\char93 _{\Pi}\left(\sum_{i=1}^n a_i dx_i\right)=\sum_{i,j=1}^n\Pi_{ij}a_i\del_{x_j}$$
  
  
  
  
  $$\char93 _{\Pi}(dx_j) = \Pi\left(\sum_{i=1}^n a_idx_i, dx_j\right) =\sum_{i=1}^n\Pi(a_i dx_i, dx_j)=\sum_{i=1}^n a_i\Pi_{ij}.$$
  
  
  If $\Pi_{ij}$ are smooth, then so is $\char93 _{\Pi}$.
• $\im \char93 _{\Pi,x}=\Ham_x(M)$ - vector subspace of $T_xM$. This is an easy consequence of $\char93 _{\Pi}(df)=X_f$.

The word generalized refers to the fact that we do not require $\dim S_x M$ to be constant in $x$. The fact that $\im\char93 _{\Pi}$ is locally generated by Hamiltonian vector fields proves that $\im\char93 _{\Pi}$ is a differentiable distribution. It will be called the characteristic distribution of the Poisson manifold $(M, \Pi)$.



Remarks:

• The reason for the name is that in local coordinates
  
  $$\rho(x)=\rank(\Pi_{ij}(x))=\rank(\{X_i, X_j\}(x)).$$
  
  

• $\rho\: M\to\bZ$; from the differentiability of the distribution it follows that $\rho(x)$ is lower semicontinuos function of $x$, i.e. it cannot decrease in a neghbourhood of $x$. Indeed, take $v_1,\hdots, v_r$ - basis of $(\im\char93 _{\Pi})_{x_0}$, $X_1, \hdots, X_r$ corresponding local vector fields, then $X_1(x), \hdots, X_r(x)$ are linearly independent in a neighbourhood of $x_0$.