Geometry of bracket-generating distributions of step 2 on graded manifolds

Abstract

A $Z_2-$graded analogue of bracket-generating distribution is given. Let $\cd$ be a distribution of rank $(p,q)$ on an $(m,n)$-dimensional graded manifold $\cm,$ we attach to $\cd$ a linear map $F$ on $\cd$ defined by the Lie bracket of graded vector fields of the sections of $\cd.$ Then $\mathcal{D}$ is a bracket-generating distribution of step $2$, if and only if $F$ is of constant rank $(m-p, n-q)$ on $\cm$.