Coordinate Jacobi and Schouten self-bracket criterion for Poisson bivectors

On an overlap, write $J_i^a=\mathrm{\partial }{y}^a\mathrm{/}\mathrm{\partial }{x}^i$. By [F2], ${\mathrm{\partial }}_{x^i}f=J_i^a{\mathrm{\partial }}_{y^a}f$ and likewise for $g$. Then [F1] gives ${\pi }_x^{ij}({\mathrm{\partial }}_{x^i}f)({\mathrm{\partial }}_{x^j}g)={\pi }_x^{ij}{J}_i^a{J}_j^b({\mathrm{\partial }}_{y^a}f)({\mathrm{\partial }}_{y^b}g)={\pi }_y^{ab}({\mathrm{\partial }}_{y^a}f)({\mathrm{\partial }}_{y^b}g),$ so the local bracket is independent of the chart. Skew-symmetry and real-bilinearity follow from ${\pi }^{ij}=-{\pi }^{ji}$ and the coordinate formula.

The coordinate product rule gives $\{f,gh{\}}_{\pi }={\pi }^{ij}{f}_i(g_{j}h+g{h}_j)=\{f,g{\}}_{\pi }h+g\{f,h{\}}_{\pi }$. Skew-symmetry gives the derivation rule in the first entry as well. Hence the bracket is a biderivation.

In a chart, expand the cyclic Jacobiator $\{f,\{g,h{\}}_{\pi }{\}}_{\pi }+\{g,\{h,f{\}}_{\pi }{\}}_{\pi }+\{h,\{f,g{\}}_{\pi }{\}}_{\pi }$. Its first-derivative terms are $J^{ijk}{f}_i{g}_j{h}_k$. The second-derivative terms cancel in pairs: the terms with $g_{\mathrm{\ell }j}$ have coefficients ${\pi }^{i\mathrm{\ell }}{\pi }^{jk}$ and ${\pi }^{k\mathrm{\ell }}{\pi }^{ij}$, which cancel after interchanging $j,\mathrm{\ell }$ in the second term and using skew-symmetry; the terms with $h_{\mathrm{\ell }k}$ cancel after interchanging $\mathrm{\ell },k$; the terms with $f_{\mathrm{\ell }i}$ cancel after interchanging $\mathrm{\ell },i$. Thus the displayed Jacobiator formula holds for all smooth $f,g,h$.

If every $J^{ijk}$ vanishes, step 2.2 makes every local Jacobiator zero, so the glued bracket on $C^{\mathrm{\infty }}(M,\mathbb{R})$ satisfies Jacobi. Conversely, assume that global bracket satisfies Jacobi. Fix a point $p$ in a chart $x:U\to x(U)\subseteq {\mathbb{R}}^n$ and indices $i,j,k$. Choose $0<r<R$ with the closed coordinate ball of radius $R$ about $x(p)$ contained in $x(U)$. Define $\theta (t)=0$ for $t\le 0$ and $\theta (t)=e^{-1\mathrm{/}t}$ for $t>0$, and put $\chi (q):=\frac{\theta (R^2-\parallel q-x(p){\parallel }^2)}{\theta (R^2-\parallel q-x(p){\parallel }^2)+\theta (\parallel q-x(p){\parallel }^2-r^2)}\mathrm{.}$ The function $\theta$ is smooth at $0$ because all its right derivatives there tend to zero, and the denominator defining $\chi$ is everywhere positive. Thus $\chi$ is smooth, equals $1$ near $x(p)$, and vanishes outside the closed ball, which is compactly contained in $x(U)$. The functions $f,g,h$ defined on $U$ by $(\chi \circ x)x^i,(\chi \circ x)x^j,(\chi \circ x)x^k$ and extended by zero outside $U$ are therefore smooth global functions and agree with the corresponding coordinates near $p$. By step 2.2 their global Jacobiator at $p$ is $J^{ijk}(p)$, so global Jacobi forces $J^{ijk}(p)=0$. Since $p$ and the indices were arbitrary, every coefficient vanishes in every chart. The Jacobiator is formed from the chart-independent bracket of step 1.1, so the criterion is coordinate-independent. By [F3], the Schouten self-bracket vanishes exactly when these coefficients vanish, since $2$ is invertible over $\mathbb{R}$. This proves the claimed Poisson and Schouten criteria and justifies the bracket in General Poisson manifolds from bivector fields.