About poisson manifolds?

Question

A Poisson manifold is a pair $(M, \{\cdot, \cdot\})$ where $M$ is a smooth manifold and $\{\cdot, \cdot\}$ is a Lie bracket on the $\mathbb R$-algebra $C^\infty(M)$ satisfying $$\{f, gh\}=\{f, g\}h+g\{f, h\}.$$ It is said that one might show there exists only one $\omega\in \Omega^2(M)$ ($2$-form on $M$) such that $$\{f, g\}=\omega(df, dg),$$ for all $f, g\in C^\infty(M)$. Can anyone explain-me what $\omega(df, dg)$ mean?

It must be a function so it should be something like $$\omega(df, dg)(p)=\omega_p(df_p(?), dg_p(??)),$$ where $?, ??$ live in the tangent space $T_pM$. However, I can't see some reasonable elements for using in $?$ and $??$.

Thanks

Answer 1

As you already remarked in the comments, a Poisson structure on a manifold induces a bivector field $w\in\Gamma(\Lambda^2TM)$. It is possible to associate a $2$-form to $w$ in a (somewhat) canonical way only if we have a prescribed way to identify $TM$ and $T^*M$, for example:

1. if we have a Riemannian metric on $M$, or
2. if we have a symplectic structure on $M$, or
3. if the Poisson structure is regular of maximal rank, then the map $\#_w:T^*M\to TM$ induced by $w$ is bijective; in this case the obtained $2$-form is a symplectic form (non-degeneracy comes from bijectivity of $\#_w$, closeness essentially from the Jacobi identity).