For any non-zero vector field $v$ and a volume form $\omega$ there exists (locally) a positive function $f$ such that $L_{fv}\omega=0$ (Indeed, one can take coordinates such that $v=\frac{\partial}{\partial x_1}$, $\omega=A(x_1,...,x_n)dx_1\wedge ...\wedge dx_n$. In these coordinates the condition $L_{fv}\omega=0$ is equivalent to an ordinary differential equation $\frac{\partial f}{\partial x_1}+\frac{1}{A}\frac{\partial A}{\partial x_1}f=0$). x_1}f=0$, I am using the homotopy formula $L_u=i_ud+di_u$ here with $u=fv$). Hence, for any distribution (of constant rank) there exists a local basis of divergent-free vector fields - take any basis of vector fields and multiply each vector field from it on a suitable $f$.
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For any non-zero vector field $v$ and a volume form $\omega$ there exists (locally) a positive function $f$ such that $L_{fv}\omega=0$ (Indeed, one can take coordinates such that $v=\frac{\partial}{\partial x_1}$, $\omega=A(x_1,...,x_n)dx_1\wedge ...\wedge dx_n$. In these coordinates the condition $L_{fv}\omega=0$ is equivalent to an ordinary differential equation $\frac{\partial f}{\partial x_1}+\frac{1}{A}\frac{\partial A}{\partial x_1}f=0$). Hence, for any distribution (of constant rank) there exists a basis of divergent-free vector fields - take any basis of vector fields and multiply each vector field from it on a suitable $f$. |
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