= Poisson structure
{c}
{title2=$\{f,g\}=\omega^{ab}f_ag_b$}
A <Poisson structure> is an antisymmetric smooth bivector field whose induced function bracket satisfies the <Jacobi identity>. In coordinates its condition is the vanishing cyclic sum of $\omega^{dc}\partial_d\omega^{ab}$. Applying Jacobi to coordinate functions proves necessity; expanding arbitrary brackets cancels the second derivatives and proves sufficiency. A function bracket-commuting with every function is a <Casimir function of a Poisson manifold>.
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