Characteristic vector fields of generic distributions of corank 2

Abstract

We study generic distributions $D\subset TM$ of corank 2 on manifolds $M$ of dimension $n⩾5$. We describe singular curves of such distributions, also called abnormal curves. For $n$ even the singular directions (tangent to singular curves) are discrete lines in $D(x)$, while for $n$ odd they form a Veronese curve in a projectivized subspace of $D(x)$, at generic $x\in M$. We show that singular curves of a generic distribution determine the distribution on the subset of $M$ where they generate at least two different directions. In particular, this happens on the whole of $M$ if $n$ is odd. The distribution is determined by characteristic vector fields and their Lie brackets of appropriate order. We characterize pairs of vector fields which can appear as characteristic vector fields of a generic corank 2 distribution, when $n$ is even.